Given a smooth quartic surface of $\mathbb{P}^3$, we may recover its Picard rank thanks to the numerical evaluation of some of its periods. Such a variety is a $K3$ surface. Its middle cohomology group $H^2(X)$ has rank 22, and its canonical bundle is trivial: there is a unique holomorphic form $\omega$, up to scaling. The kernel of the map $\gamma \mapsto \int_{\gamma} \omega$ is a sublattice of $H_2(X)$ called the Néron-Severi group of $X$. The rank of this lattice is called the Picard rank (or Picard number, or Néron-Severi rank) of $X$.

The LLL algorithm can be used to heuristically recover this kernel from high-precision numerical approximations of the periods. This computation is not certified and can fail in two ways, in principle. First, the algorithm may miss integer relations if the number of digits in the coefficients is not smaller than the number of significant digits computed in the periods. Second, it may recover fake integer relations reflecting a numerical coincidence that a higher precision computation would detect. In practice, we never observed these phenomena. See (Lairez & Sertöz, 2019) for a discussion of these issues.

Throughout the following examples, we used the following method. The holomorphic 2-form of a given quartic $X = V(P)$ can be identified as the residue of the only irreducible rational form of $H^3(\mathbb{P}^3 \setminus X)$ with pole order 1 (Griffiths, 1969, Eq. 8.6). Explicitly the periods are given by $\int_{\gamma} \Omega_3 P$, with $\Omega_3$ the volume form of $\mathbb{P}^3$ and $\gamma \in H_3(\mathbb{P}^3 \setminus X)$. We may then use the steps up to 5 of Section 3.8 to compute a basis of $\mathcal{T}(Y)$. We then compute the periods of the holomorphic form on this basis with the methods of Section 3.7.1. This yields 24 numerical approximations of complex numbers $\alpha_i \in \mathbb{C}$. Cycles on which the periods vanish induce integer linear relations between these numbers. Of these relations, three come from the exceptional divisors of the blowup $\pi : Y \to X$ (Section 5.1). As the Néron-Severi lattice is characterised by the vanishing of the periods, we may use these relations to compute it. All in all, the Picard rank $\rho(X)$ of $X$ is given by

$$ (80) \qquad \rho(X) = 22 - \dim_{\mathbb{Q}} \langle \alpha_1, \dots, \alpha_{24} \rangle. $$

We can apply the LLL algorithm to compute $\dim_{\mathbb{Q}} \langle \alpha_1, \dots, \alpha_{24} \rangle$ and heuristically recover the Picard rank.

Altogether the holomorphic periods of 530 smooth quartic surfaces were computed for the results presented in this section. The computations were run on a cluster, using 32 cores. The average time needed to compute the holomorphic periods of one quartic surface was around 40 minutes, ranging from 16 minutes up to 13 hours. The median time was 28 minutes. The computation took longer than 90 minutes for only 24 surfaces. It should be noted that the cause of lengthy computations seems to stem from the choice of the fibration $X \dasharrow \mathbb{P}^1$ rather than be intrinsic to the surface itself. Indeed, the limiting factor is the integration step, and the cases where the computation takes a lot of time seem to always be due to the integration on one single pathological edge. In all the cases we looked at closely, picking another generic fibration reduced the computation time, to around the expected 40 minutes. However, we have for now no way to choose a fibration that is well adapted to the computation of a given surface *a priori*.

### 6.2.1. Families studied by Bouyer.
In this section we numerically verify the Picard ranks of the generic elements of the families of quartic surfaces of $\mathbb{P}^3$ given by Bouyer (2018, Theorem 4.9). These families are generated by polynomials of the form

$$ (81) \qquad [A, B, C, D, E] \stackrel{\text{def}}{=} A(x^4 + y^4 + z^4 + w^4) + Bxyzw + C(x^2y^2 + z^2w^2) \\ \qquad \qquad + D(x^2z^2 + y^2w^2) + E(x^2w^2 + y^2z^2), $$

with 5 parameters $A, B, C, D$ and $E$. More precisely, the families are given respectively by polynomials $[A, B, C, D, E]$, $[A, (DE - 2AC)/A, C, D, E]$, $[A, 0, C, D, 2AC/D]$, $[A, B(2A - B)/A, B, B, B]$ and $[A, 0, C, 0, 0]$. The theorem of Bouyer states that the generic Picard rank of these families are respectively 16, 17, 18, 19 and 19.
Given a smooth quartic surface of $\mathbb{P}^3$, we may recover its Picard rank thanks to the numerical evaluation of some of its periods. Such a variety is a $K3$ surface. Its middle cohomology group $H^2(X)$ has rank 22, and its canonical bundle is trivial: there is a unique holomorphic form $\omega$, up to scaling. The kernel of the map $\gamma \mapsto \int_{\gamma} \omega$ is a sublattice of $H_2(X)$ called the Néron-Severi group of $X$. The rank of this lattice is called the Picard rank (or Picard number, or Néron-Severi rank) of $X$.

The LLL algorithm can be used to heuristically recover this kernel from high-precision numerical approximations of the periods. This computation is not certified and can fail in two ways, in principle. First, the algorithm may miss integer relations if the number of digits in the coefficients is not smaller than the number of significant digits computed in the periods. Second, it may recover fake integer relations reflecting a numerical coincidence that a higher precision computation would detect. In practice, we never observed these phenomena. See (Lairez & Sertöz, 2019) for a discussion of these issues.

Throughout the following examples, we used the following method. The holomorphic 2-form of a given quartic $X = V(P)$ can be identified as the residue of the only irreducible rational form of $H^3(\mathbb{P}^3 \setminus X)$ with pole order 1 (Griffiths, 1969, Eq. 8.6). Explicitly the periods are given by $\int_{\gamma} \Omega_3 P$, with $\Omega_3$ the volume form of $\mathbb{P}^3$ and $\gamma \in H_3(\mathbb{P}^3 \setminus X)$. We may then use the steps up to 5 of Section 3.8 to compute a basis of $\mathcal{T}(Y)$. We then compute the periods of the holomorphic form on this basis with the methods of Section 3.7.1. This yields 24 numerical approximations of complex numbers $\alpha_i \in \mathbb{C}$. Cycles on which the periods vanish induce integer linear relations between these numbers. Of these relations, three come from the exceptional divisors of the blowup $\pi : Y \to X$ (Section 5.1). As the Néron-Severi lattice is characterised by the vanishing of the periods, we may use these relations to compute it. All in all, the Picard rank $\rho(X)$ of $X$ is given by

$$ (80) \qquad \rho(X) = 22 - \dim_{\mathbb{Q}} \langle \alpha_1, \dots, \alpha_{24} \rangle . $$

We can apply the LLL algorithm to compute $\dim_{\mathbb{Q}} \langle \alpha_1, \dots, \alpha_{24} \rangle$ and heuristically recover the Picard rank.

Altogether the holomorphic periods of 530 smooth quartic surfaces were computed for the results presented in this section. The computations were run on a cluster, using 32 cores. The average time needed to compute the holomorphic periods of one quartic surface was around 40 minutes, ranging from 16 minutes up to 13 hours. The median time was 28 minutes. The computation took longer than 90 minutes for only 24 surfaces. It should be noted that the cause of lengthy computations seems to stem from the choice of the fibration $X \dasharrow \mathbb{P}^1$ rather than be intrinsic to the surface itself. Indeed, the limiting factor is the integration step, and the cases where the computation takes a lot of time seem to always be due to the integration on one single pathological edge. In all the cases we looked at closely, picking another generic fibration reduced the computation time, to around the expected 40 minutes. However, we have for now no way to choose a fibration that is well adapted to the computation of a given surface *a priori*.

### 6.2.1. Families studied by Bouyer.
In this section we numerically verify the Picard ranks of the generic elements of the families of quartic surfaces of $\mathbb{P}^3$ given by Bouyer (2018, Theorem 4.9). These families are generated by polynomials of the form

$$ (81) \qquad [A, B, C, D, E] \stackrel{\text{def}}{=} A(x^4 + y^4 + z^4 + w^4) + Bxyzw + C(x^2y^2 + z^2w^2) \\ \qquad \qquad + D(x^2z^2 + y^2w^2) + E(x^2w^2 + y^2z^2), $$

with 5 parameters $A, B, C, D$ and $E$. More precisely, the families are given respectively by polynomials $[A, B, C, D, E]$, $[A, (DE - 2AC)/A, C, D, E]$, $[A, 0, C, D, 2AC/D]$, $[A, B(2A - B)/A, B, B, B]$ and $[A, 0, C, 0, 0]$. The theorem of Bouyer states that the generic Picard rank of these families are respectively 16, 17, 18, 19 and 19.
To generate elements of these 5 families, we simply pick integers $A, B, C, D, E$ randomly in the interval $[-100, 100]$ and consider the quartics defined by the above polynomials. If these quartics are smooth we may compute the Picard rank as above. Otherwise we pick other values for $A, B, C, D, E$.

We checked that the values our method yields for the Picard rank coincide with the values given by the theorem for the varieties corresponding to 56 sets of values for $A, B, C, D, E$. This gives numerical evidence of the results of Bouyer (2018).

We found singular examples were the Picard rank was not generic, for $[A, B, C, D, E]$ and $[A, (DE - 2AC)/A, C, D, E]$ with $(A, B, C, D, E) = (49, 92, -51, 19, -51)$. The corresponding polynomials are

$$ (82) \qquad [A, B, C, D, E] = 49x^4 - 51x^2y^2 + 49y^4 + 19x^2z^2 - 51y^2z^2 + 49z^4 + 92xyzw \\ \qquad \qquad - 51x^2w^2 + 19y^2w^2 - 51z^2w^2 + 49w^4, $$

and

$$ (83) \qquad [A, (DE - 2AC)/A, C, D, E] = 49x^4 - 51x^2y^2 + 49y^4 + 19x^2z^2 - 51y^2z^2 + 49z^4 \\ \qquad \qquad + \frac{4029}{49}xyzw - 51x^2w^2 + 19y^2w^2 - 51z^2w^2 + 49w^4. $$

The Picard ranks were 1 higher than the generic value for their respective families, i.e. 17 and 18.

### 6.2.2. Picard rank of symmetric polynomials.
In this section we compute the Picard rank of families of quartic surfaces defined by a symmetric polynomial. The defining equation of a quartic in $\mathbb{P}^3$ is a homogeneous polynomial in 4 variables, say $x, y, z$ and $w$. We consider the families of polynomials that are symmetric in some of these variables. Up to a permutation of the variables, there are 4 such families:

(1) polynomials symmetric in all the variables,

$$ \forall \sigma \in \mathfrak{S}_{\{x,y,z,w\}} \quad P(x,y,z,w) = P(\sigma(x), \sigma(y), \sigma(z), \sigma(w)), $$

(2) polynomials symmetric in three variables, say $x, y$ and $z$,

$$ P(x, y, z, w) = P(x, z, y, w) = P(y, x, z, w) = P(y, z, x, w) = P(z, x, y, w) = P(z, y, x, w) $$

(3) polynomials where $x$ and $y$ are symmetric, as well as $z$ and $w$

$$ P(x, y, z, w) = P(y, x, z, w) = P(x, y, w, z) = P(y, x, w, z), $$

(4) and polynomials where $x$ and $y$ are symmetric

$$ P(x, y, z, w) = P(y, x, z, w). $$

A basis of the vector space of such polynomials is given by products of elementary symmetric polynomials. We may thus generate a priori generic (i.e. with minimal Picard rank) elements of the family by picking random coefficients as in the previous section. In practice we pick random integer coefficients in the interval $[-5, 5]$.

Doing so, we observe respectively

(1) a Picard rank of 17 for 113 elements, 18 for one element, and 19 for one element,

(2) a Picard rank of 14 for 100 elements, and 15 for one element,

(3) a Picard rank of 12 for 107 elements,

(4) a Picard rank of 8 for 114 elements.

This leads us to conjecture that the generic Picard ranks of these families are respectively 17, 14, 12 and 8.
To generate elements of these 5 families, we simply pick integers $A, B, C, D, E$ randomly in the interval $[-100, 100]$ and consider the quartics defined by the above polynomials. If these quartics are smooth we may compute the Picard rank as above. Otherwise we pick other values for $A, B, C, D, E$.

We checked that the values our method yields for the Picard rank coincide with the values given by the theorem for the varieties corresponding to 56 sets of values for $A, B, C, D, E$. This gives numerical evidence of the results of Bouyer (2018).

We found singular examples where the Picard rank was not generic, for $[A, B, C, D, E]$ and $[A, (DE - 2AC)/A, C, D, E]$ with $(A, B, C, D, E) = (49, 92, -51, 19, -51)$. The corresponding polynomials are

$$ (82) \qquad [A, B, C, D, E] = 49x^4 - 51x^2y^2 + 49y^4 + 19x^2z^2 - 51y^2z^2 + 49z^4 + 92xyzw \\ \qquad \qquad - 51x^2w^2 + 19y^2w^2 - 51z^2w^2 + 49w^4, $$

and

$$ (83) \qquad [A, (DE - 2AC)/A, C, D, E] = 49x^4 - 51x^2y^2 + 49y^4 + 19x^2z^2 - 51y^2z^2 + 49z^4 \\ \qquad \qquad + \frac{4029}{49}xyzw - 51x^2w^2 + 19y^2w^2 - 51z^2w^2 + 49w^4. $$

The Picard ranks were 1 higher than the generic value for their respective families, i.e. 17 and 18.

### 6.2.2. Picard rank of symmetric polynomials.
In this section we compute the Picard rank of families of quartic surfaces defined by a symmetric polynomial. The defining equation of a quartic in $\mathbb{P}^3$ is a homogeneous polynomial in 4 variables, say $x, y, z$ and $w$. We consider the families of polynomials that are symmetric in some of these variables. Up to a permutation of the variables, there are 4 such families:

(1) polynomials symmetric in all the variables,

$$ \forall \sigma \in \mathfrak{S}_{\{x,y,z,w\}} \quad P(x,y,z,w) = P(\sigma(x), \sigma(y), \sigma(z), \sigma(w)), $$

(2) polynomials symmetric in three variables, say $x, y$ and $z$,

$$ P(x, y, z, w) = P(x, z, y, w) = P(y, x, z, w) = P(y, z, x, w) = P(z, x, y, w) = P(z, y, x, w) $$

(3) polynomials where $x$ and $y$ are symmetric, as well as $z$ and $w$

$$ P(x, y, z, w) = P(y, x, z, w) = P(x, y, w, z) = P(y, x, w, z), $$

(4) and polynomials where $x$ and $y$ are symmetric

$$ P(x, y, z, w) = P(y, x, z, w). $$

A basis of the vector space of such polynomials is given by products of elementary symmetric polynomials. We may thus generate a priori generic (i.e. with minimal Picard rank) elements of the family by picking random coefficients as in the previous section. In practice we pick random integer coefficients in the interval $[-5, 5]$.

Doing so, we observe respectively

(1) a Picard rank of 17 for 113 elements, 18 for one element, and 19 for one element,

(2) a Picard rank of 14 for 100 elements, and 15 for one element,

(3) a Picard rank of 12 for 107 elements,

(4) a Picard rank of 8 for 114 elements.

This leads us to conjecture that the generic Picard ranks of these families are respectively 17, 14, 12 and 8.
**Remark 15.** Alice Garbagnati pointed out that lower bounds on the Picard rank follow from properties of K3 surfaces with automorphisms. These bounds match the heuristic computation of the rank we obtained numerically. More precisely, the automorphisms of $\mathbb{P}^3$ that permute the coordinates induce automorphisms of the K3 surfaces of the aforementioned families. Denote $\sigma$ such an automorphism and $\sigma^*$ the induced isometry on $H_2(S)$. Depending on the nature of the automorphism, either the transcendental lattice is included in $H_2(X)^{\sigma^*}$ (the sublattice fixed by $\sigma^*$) or $H_2(X)^{\sigma^*}$ is included in the Néron-Severi lattice. The ranks of $H_2(X)^G$ for all finite group actions G are known (Nikulin, 1979; Artebani et al., 2011; Hashimoto, 2012; Xiao, 1996), and this yields lower bounds for the Picard ranks. However, the question of rigorously proving that the lower bounds match the Picard ranks seems to be still open.

6.2.3. Two isomorphic rank 2 smooth quartic surfaces in $\mathbb{P}^3$. Oguiso (2017, Thm. 1.4) gives an example of two isomorphic smooth quartic K3 surfaces $S_1$ and $S_2$, that are isomorphic as abstract varieties but not Cremona equivalent (i.e. there is no birational automorphism of $\mathbb{P}^3$ inducing an isomorphism $S_1 \simeq S_2$). Defining equations $f_1$ and $f_2$ for $S_1$ and $S_2$ are given by

$$ (84) \qquad \begin{aligned} f_1 = x^3y + x^2y^2 - xy^3 + x^3z + 2x^2yz - xy^2z - y^3z + x^2z^2 - xyz^2 - 2y^2z^2 - yz^3 - z^4 \\ + x^3w - 2xy^2w - 2xyzw - xz^2w + yz^2w + xyw^2 - y^2w^2 - z^2w^2 + xw^3 + yw^3 \end{aligned} $$

and

$$ (85) \qquad \begin{aligned} f_2 = x^4 &+ 3x^3z - x^2yz + 3x^2z^2 - 4xyz^2 - y^2z^2 + xz^3 - 3yz^3 - x^2yw + 2xy^2w \\ &+ y^3w - 2x^2zw - 2xyzw + 3y^2zw - 3xz^2w - 3yz^2w - 2z^3w + 4xyw^2 \\ &+ 2y^2w^2 + 3yzw^2 - z^2w^2 + 2xw^3 + 2yw^3 + zw^3 + w^4 \end{aligned} $$

The goal of this section is to present a method allowing one to numerically verify that these two surfaces are indeed isomorphic.

A *Hodge isometry* between two varieties X and Y is a morphism $X \to Y$ that induces an isometry of the homology groups $H_2(X) \simeq H_2(Y)$ that respects the Hodge decomposition on the complexifications $H_2(X, \mathbb{C})$ and $H_2(Y, \mathbb{C})$. When X is a K3 surface, its Hodge decomposition is given by the holomorphic periods (indeed $H^{2,0}(X)$ has rank 1 and is the complex conjugate of $H^{0,2}(X)$).

By the global Torelli theorem for K3 surfaces (Huybrechts, 2016, Thm. 5.3), it is sufficient to find a Hodge isometry between the second homology groups of two K3 surfaces to prove that they are isomorphic. In order to recover such an isometry, we proceed in the following way.

Using the methods presented in this paper, we compute the holomorphic period vectors of $S_i$ in some basis of homology $\gamma_1^i, \dots, \gamma_{22}^i$ of $H_2(S_i)$ for $i = 1, 2$. This allows to (heuristically) recover the Néron-Severi sublattice NS($S_i$), which we find has rank 2. The transcendental lattice Tr($S_i$) is then simply the orthogonal complement of NS($S_i$) in $H_2(S_i)$. The lattice NS($S_i$) $\oplus$ Tr($S_i$) is a full rank sublattice of $H_2(S_i)$, which may have positive index. In order to find a Hodge isometry, we look for an isometry between these sublattices that extends to an isometry between the full homology lattices.

More explicitly, let $\omega_1$ and $\omega_2$ be the holomorphic forms of $S_1$ and $S_2$ respectively, $\gamma_1^1, \dots, \gamma_{22}^1 \in H_2(S_1)$ and $\gamma_1^2, \dots, \gamma_{22}^2 \in H_2(S_2)$ bases of cohomology. Let $I_1, I_2$ be the intersection matrices in these bases, and $\pi_1 = (\int_{\gamma_1^1} \omega_1)_{1 \le j \le 22}$ and $\pi_2 = (\int_{\gamma_1^2} \omega_2)_{1 \le j \le 22}$ the row vectors of the periods of the holomorphic form. Then a Hodge isometry is the data of a matrix $A \in GL_{22}(\mathbb{Z})$ and a scalar $\lambda \in \mathbb{C}$ such that

$$ (86) \qquad \pi_2 A = \lambda \pi_1 \text{ and } {}^t A I_2 A = I_1. $$
**Remark 15.** Alice Garbagnati pointed out that lower bounds on the Picard rank follow from properties of K3 surfaces with automorphisms. These bounds match the heuristic computation of the rank we obtained numerically. More precisely, the automorphisms of $\mathbb{P}^3$ that permute the coordinates induce automorphisms of the K3 surfaces of the aforementioned families. Denote $\sigma$ such an automorphism and $\sigma^*$ the induced isometry on $H_2(S)$. Depending on the nature of the automorphism, either the transcendental lattice is included in $H_2(X)^{\sigma^*}$ (the sublattice fixed by $\sigma^*$) or $H_2(X)^{\sigma^*}$ is included in the Néron-Severi lattice. The ranks of $H_2(X)^G$ for all finite group actions $G$ are known (Nikulin, 1979; Artebani et al., 2011; Hashimoto, 2012; Xiao, 1996), and this yields lower bounds for the Picard ranks. However, the question of rigorously proving that the lower bounds match the Picard ranks seems to be still open.

6.2.3. Two isomorphic rank 2 smooth quartic surfaces in $\mathbb{P}^3$. Oguiso (2017, Thm. 1.4) gives an example of two isomorphic smooth quartic K3 surfaces $S_1$ and $S_2$, that are isomorphic as abstract varieties but not Cremona equivalent (i.e. there is no birational automorphism of $\mathbb{P}^3$ inducing an isomorphism $S_1 \simeq S_2$). Defining equations $f_1$ and $f_2$ for $S_1$ and $S_2$ are given by

$$ (84) \qquad \begin{aligned} f_1 = x^3 y + x^2 y^2 - xy^3 + x^3 z + 2x^2 yz - xy^2 z - y^3 z + x^2 z^2 - xyz^2 - 2y^2 z^2 - yz^3 - z^4 \\ + x^3 w - 2xy^2 w - 2xyzw - xz^2 w + yz^2 w + xyw^2 - y^2 w^2 - z^2 w^2 + xw^3 + yw^3 \end{aligned} $$

and

$$ (85) \qquad \begin{aligned} f_2 = x^4 &+ 3x^3 z - x^2 yz + 3x^2 z^2 - 4xyz^2 - y^2 z^2 + xz^3 - 3yz^3 - x^2 yw + 2xy^2 w \\ &+ y^3 w - 2x^2 zw - 2xyzw + 3y^2 zw - 3xz^2 w - 3yz^2 w - 2z^3 w + 4xyw^2 \\ &+ 2y^2 w^2 + 3yzw^2 - z^2 w^2 + 2xw^3 + 2yw^3 + zw^3 + w^4 \end{aligned} $$

The goal of this section is to present a method allowing one to numerically verify that these two surfaces are indeed isomorphic.

A Hodge isometry between two varieties $X$ and $Y$ is a morphism $X \to Y$ that induces an isometry of the homology groups $H_2(X) \simeq H_2(Y)$ that respects the Hodge decomposition on the complexifications $H_2(X, \mathbb{C})$ and $H_2(Y, \mathbb{C})$. When $X$ is a K3 surface, its Hodge decomposition is given by the holomorphic periods (indeed $H^{2,0}(X)$ has rank 1 and is the complex conjugate of $H^{0,2}(X)$).

By the global Torelli theorem for K3 surfaces (Huybrechts, 2016, Thm. 5.3), it is sufficient to find a Hodge isometry between the second homology groups of two K3 surfaces to prove that they are isomorphic. In order to recover such an isometry, we proceed in the following way.

Using the methods presented in this paper, we compute the holomorphic period vectors of $S_i$ in some basis of homology $\gamma_1^i, \dots, \gamma_{22}^i$ of $H_2(S_i)$ for $i = 1, 2$. This allows to (heuristically) recover the Néron-Severi sublattice NS($S_i$), which we find has rank 2. The transcendental lattice Tr($S_i$) is then simply the orthogonal complement of NS($S_i$) in $H_2(S_i)$. The lattice NS($S_i$) $\oplus$ Tr($S_i$) is a full rank sublattice of $H_2(S_i)$, which may have positive index. In order to find a Hodge isometry, we look for an isometry between these sublattices that extends to an isometry between the full homology lattices.

More explicitly, let $\omega_1$ and $\omega_2$ be the holomorphic forms of $S_1$ and $S_2$ respectively, $\gamma_1^1, \dots, \gamma_{22}^1 \in H_2(S_1)$ and $\gamma_1^2, \dots, \gamma_{22}^2 \in H_2(S_2)$ bases of cohomology. Let $I_1, I_2$ be the intersection matrices in these bases, and $\pi_1 = (\int_{\gamma_1^1} \omega_1)_{1 \le j \le 22}$ and $\pi_2 = (\int_{\gamma_1^2} \omega_2)_{1 \le j \le 22}$ the row vectors of the periods of the holomorphic form. Then a Hodge isometry is the data of a matrix $A \in GL_{22}(\mathbb{Z})$ and a scalar $\lambda \in \mathbb{C}$ such that

$$ (86) \qquad \pi_2 A = \lambda \pi_1 \text{ and } {}^t A I_2 A = I_1. $$
Let $N_i \in \mathbb{Z}^{22 \times 2}$ be the coordinate matrix of a basis of the Néron-Severi group NS($S_i$) and
$T_i \in \mathbb{Z}^{22 \times 20}$ be the coordinate matrix of a basis of the transcendental lattice Tr($S_i$). As these
sublattices of $H_2(S_i)$ are algebraic invariants, we have the identities

$$
(87) \quad AT_1 = T_2 B \quad \text{and} \quad AN_1 = N_2 C
$$

for some invertible matrices $B \in GL_{20}(\mathbb{Z})$ and $C \in GL_2(\mathbb{Z})$.

Then $\lambda\pi_1 T_1 = \pi_2 A T_1 = \pi_2 T_2 B$. In particular, coefficient wise we have

$$
(88) \quad \lambda(\pi_1 T_1)_i = \sum_j (\pi_2 T_2)_j B_{ji},
$$

which allows us to recover the integers $B_{ji}$ using the LLL algorithm. We find that $\lambda = 1$ with the
choice $\omega_i = \text{Res}(\Omega/f_i)$.

Furthermore

$$
(89) \quad {}^t C N_2 I_2 N_2 C = {}^t N_1 I_2 A N_1 = {}^t N_1 I_1 N_1 .
$$

This yields 4 quadratic equations in the coefficients of $C$, to which we may find integer solutions.
There are infinitely many solutions to this system, and not all yield a Hodge isometry – they only
do if the corresponding $A$ is an invertible integer matrix.

Indeed we have

$$
(90) \quad A \left( \begin{array}{c|c} T_1 & N_1 \end{array} \right) = \left( \begin{array}{c|c} T_2 & N_2 \end{array} \right) \left( \begin{array}{c|c} \frac{B}{0} & 0 \\ \hline 0 & C \end{array} \right),
$$

and thus

$$
(91) \quad A = \left( \begin{array}{c|c} T_2 & N_2 \end{array} \right) \left( \begin{array}{c|c} B & 0 \\ \hline 0 & C \end{array} \right) \left( \begin{array}{c|c} T_1 & N_1 \end{array} \right)^{-1} \in \mathrm{GL}_{22}(\mathbb{Z}).
$$

We pick solutions for $C$ and check whether they satisfy this condition. We then verify that the conditions of (86) are also satisfied. Using this method, we found that there is indeed a Hodge isometry between $S_1$ and $S_2$ up to high precision, which confirms that they are isomorphic.

6.2.4. *An explicit equation of a smooth quartic K3 surface with given Picard rank.* In this section we provide complements to Table 6.1 of Lairez and Sertöz (2019), for which examples of defining equations of quartic surfaces of Picard rank 2, 3 and 5 were missing. Additionally, we have verified the known entries of this table using the method presented in this paper.

In addition to the example(s) of Section 6.2.3, Picard rank 2 smooth quartic K3 surfaces were
found by testing generic polynomials with coefficients in $\{-1, 0, 1\}$. Following the construction of
the proof of Oguiso (2012, Thm. 1.7), we may construct an equation of a smooth quartic surface
with Picard rank 3. We also stumbled upon a rank 5 example, completing the missing entries. The
completed table can be found in Table 3.

Such polynomials are too large (in terms of the number of monomials) for the Picard rank to be
recovered using the previous method of Sertöz (2019) with current numerical integration software.
Using the SageMath implementation of the algorithm presented in this paper, we were able to
numerically recover the Picard rank of these smooth quartic K3 surfaces in less than an hour each
on a laptop.

**6.3. An example from Feynman integrals: the Tardigrade Graph.** The methods presented here allowed the study of the geometry of a parametrised Feynman integral corresponding to the Tardigrade graph. The associated Feynman integral is a relative period integral in the sense of Bloch et al. (2006) and Brown (2017) for a family of singular quartic surfaces defined by the Feynman graph in Fig. 9, as argued in Bourjaily et al. (2020), Bourjaily et al. (2019) and characterised completely in Doran et al. (2023). This example is interesting because it goes beyond the scope of this paper, as the quartic surfaces associated to this graph are not smooth. What's more, it shows
Let $N_i \in \mathbb{Z}^{22 \times 2}$ be the coordinate matrix of a basis of the Néron-Severi group NS($S_i$) and
$T_i \in \mathbb{Z}^{22 \times 20}$ be the coordinate matrix of a basis of the transcendental lattice Tr($S_i$). As these
sublattices of $H_2(S_i)$ are algebraic invariants, we have the identities

$$
(87) \quad AT_1 = T_2 B \quad \text{and} \quad AN_1 = N_2 C
$$

for some invertible matrices $B \in GL_{20}(\mathbb{Z})$ and $C \in GL_2(\mathbb{Z})$.

Then $\lambda\pi_1 T_1 = \pi_2 A T_1 = \pi_2 T_2 B$. In particular, coefficient wise we have

$$
(88) \quad \lambda(\pi_1 T_1)_i = \sum_j (\pi_2 T_2)_j B_{ji},
$$

which allows us to recover the integers $B_{ji}$ using the LLL algorithm. We find that $\lambda = 1$ with the
choice $\omega_i = \text{Res}(\Omega/f_i)$.

Furthermore

$$
(89) \quad {}^t C {}^t N_2 I_2 N_2 C = {}^t N_1 {}^t A I_2 A N_1 = {}^t N_1 I_1 N_1 .
$$

This yields 4 quadratic equations in the coefficients of $C$, to which we may find integer solutions.
There are infinitely many solutions to this system, and not all yield a Hodge isometry – they only
do if the corresponding $A$ is an invertible integer matrix.

Indeed we have

$$
(90) \quad A \left( \begin{array}{c|c} T_1 & N_1 \end{array} \right) = \left( \begin{array}{c|c} T_2 & N_2 \end{array} \right) \left( \begin{array}{c|c} \frac{B}{0} & 0 \\ \hline 0 & C \end{array} \right),
$$

and thus

$$
(91) \quad A = \left( \begin{array}{c|c} T_2 & N_2 \end{array} \right) \left( \begin{array}{c|c} B & 0 \\ \hline 0 & C \end{array} \right) \left( \begin{array}{c|c} T_1 & N_1 \end{array} \right)^{-1} \in \mathrm{GL}_{22}(\mathbb{Z}).
$$

We pick solutions for $C$ and check whether they satisfy this condition. We then verify that the conditions of (86) are also satisfied. Using this method, we found that there is indeed a Hodge isometry between $S_1$ and $S_2$ up to high precision, which confirms that they are isomorphic.

6.2.4. *An explicit equation of a smooth quartic K3 surface with given Picard rank.* In this section we provide complements to Table 6.1 of Lairez and Sertöz (2019), for which examples of defining equations of quartic surfaces of Picard rank 2, 3 and 5 were missing. Additionally, we have verified the known entries of this table using the method presented in this paper.

In addition to the example(s) of Section 6.2.3, Picard rank 2 smooth quartic K3 surfaces were
found by testing generic polynomials with coefficients in $\{-1, 0, 1\}$. Following the construction of
the proof of Oguiso (2012, Thm. 1.7), we may construct an equation of a smooth quartic surface
with Picard rank 3. We also stumbled upon a rank 5 example, completing the missing entries. The
completed table can be found in Table 3.

Such polynomials are too large (in terms of the number of monomials) for the Picard rank to be
recovered using the previous method of Sertöz (2019) with current numerical integration software.
Using the SageMath implementation of the algorithm presented in this paper, we were able to
numerically recover the Picard rank of these smooth quartic K3 surfaces in less than an hour each
on a laptop.

**6.3. An example from Feynman integrals: the Tardigrade Graph.** The methods presented here allowed the study of the geometry of a parametrised Feynman integral corresponding to the Tardigrade graph. The associated Feynman integral is a relative period integral in the sense of Bloch et al. (2006) and Brown (2017) for a family of singular quartic surfaces defined by the Feynman graph in Fig. 9, as argued in Bourjaily et al. (2020, 2019) and characterised completely in Doran et al. (2023). This example is interesting because it goes beyond the scope of this paper, as the quartic surfaces associated to this graph are not smooth. What's more, it shows how an approach
<table><caption>**Table 3.** Example polynomial for each Picard number. The new rows are Picard numbers 2, 3 and 5.</caption><thead><tr><th>Defining polynomial</th><th>Picard number</th></tr></thead><tbody><tr><td>$wx^3 + w^3y + y^4 + xz^3 + z^4$</td><td>1</td></tr><tr><td>$-x^2y^2 + xy^3 - y^4 + x^3z + x^2yz - xy^2z - y^3z + xyz^2 - y^2z^2 + xz^3 - yz^3 + x^3w$</td><td>2</td></tr><tr><td>$-x^2yw - xy^2w - y^2zw - z^3w - x^2w^2 - xyw^2 + y^2w^2 + yzw^2 + yw^3 - zw^3$</td><td>3</td></tr><tr><td>$x^4 - y^4 + z^4 - w^4 + (x-y)(z+w)yw - (x+y)(z-w)y^2$</td><td>4</td></tr><tr><td>$x^3y + z^4 + y^3w + zw^3$</td><td>4</td></tr><tr><td>$5x^4 + x^3y - xy^3 - 5y^4 + x^3z + x^2yz - xy^2z - y^3z + x^2z^2 + 2xyz^2 + 3y^2z^2 + 2xz^3 + 2yz^3 + 2z^4 - x^3w + x^2yw + xy^2w - y^3w + x^2zw - y^2zw - 2xz^2w + 2yz^2w + 2z^3w$</td><td>5</td></tr><tr><td>$-2x^2w^2 - 2xyw^2 - 2y^2w^2 - 2xzw^2 - 2yzw^2 + 2z^2w^2 + 2xw^3 - 2yw^3 - 2zw^3 - 4w^4$</td><td>6</td></tr><tr><td>$x^3y + y^4 + z^3w + yw^3 + zw^3$</td><td>7</td></tr><tr><td>$w^3x + x^4 + wx^2z + x^3z + xy^2z - y^3z + wxz^2 + x^2z^2 - xz^3 + z^4$</td><td>8</td></tr><tr><td>$x^3y + z^4 + y^3w + xw^3 + w^4$</td><td>9</td></tr><tr><td>$w^4 + wx^2y + y^4 + x^3z - xy^2z + z^4$</td><td>10</td></tr><tr><td>$x^3y + z^4 + y^3w + w^4$</td><td>11</td></tr><tr><td>$w^4 + x^4 + x^2y^2 + y^4 - w^3z - xy^2z + x^2z^2 + z^4$</td><td>12</td></tr><tr><td>$x^3y + y^4 + z^3w + x^2w^2 + w^4$</td><td>13</td></tr><tr><td>$w^4 + w^3x + wy^3 + y^2z^2 + wz^3 + x^2z^3$</td><td>14</td></tr><tr><td>$x^3y + y^4 + z^3w + yw^3 + w^4$</td><td>15</td></tr><tr><td>$x^3y + y^3z + z^4 + xy^2w + zw^3$</td><td>16</td></tr><tr><td>$x^3y + y^4 + z^3w + xyw^2 + y^2w^2 + w^4$</td><td>17</td></tr><tr><td>$x^3y + x^3z + y^3z + yz^3 + w^4$</td><td>18</td></tr><tr><td>$x^3y + z^4 + y^3w + xyzw + xw^3$</td><td>19</td></tr><tr><td>$x^3y + z^4 + y^3w + xw^3$</td><td>20</td></tr></tbody></table>

how an approach relying on effective Picard-Lefschetz theory can be used to compute the periods of varieties given as elliptic fibrations. Finally, it shows that our algorithm manages examples from physics that were previously out of reach.

<figure><img src="image_13.png" /><figcaption>**Figure 9.** The tardigrade graph</figcaption></figure>

The Tardigrade graph corresponds to a family of K3 surfaces given as the minimal resolution of a family of generically singular quartic surfaces in $\mathbb{P}^3$. For example, one element of this family, on which the computation was performed, is the quartic defined by the equation derived in Doran
<table><caption>**Table 3.** Example polynomial for each Picard number. The new rows are Picard numbers 2, 3 and 5.</caption><thead><tr><th>Defining polynomial</th><th>Picard number</th></tr></thead><tbody><tr><td>wx<sup>3</sup> + w<sup>3</sup>y + y<sup>4</sup> + xz<sup>3</sup> + z<sup>4</sup></td><td>1</td></tr><tr><td>-x<sup>2</sup>y<sup>2</sup> + xy<sup>3</sup> - y<sup>4</sup> + x<sup>3</sup>z + x<sup>2</sup>yz - xy<sup>2</sup>z - y<sup>3</sup>z + xyz<sup>2</sup> - y<sup>2</sup>z<sup>2</sup> + xz<sup>3</sup> - yz<sup>3</sup> + x<sup>3</sup>w</td><td>2</td></tr><tr><td>-x<sup>2</sup>yw - xy<sup>2</sup>w - y<sup>2</sup>zw - z<sup>3</sup>w - x<sup>2</sup>w<sup>2</sup> - xyw<sup>2</sup> + y<sup>2</sup>w<sup>2</sup> + yzw<sup>2</sup> + yw<sup>3</sup> - zw<sup>3</sup></td><td>3</td></tr><tr><td>x<sup>4</sup> - y<sup>4</sup> + z<sup>4</sup> - w<sup>4</sup> + (x - y)(z + w)yw - (x + y)(z - w)y<sup>2</sup></td><td>4</td></tr><tr><td>x<sup>3</sup>y + z<sup>4</sup> + y<sup>3</sup>w + zw<sup>3</sup></td><td>4</td></tr><tr><td>5x<sup>4</sup> + x<sup>3</sup>y - xy<sup>3</sup> - 5y<sup>4</sup> + x<sup>3</sup>z + x<sup>2</sup>yz - xy<sup>2</sup>z - y<sup>3</sup>z + x<sup>2</sup>z<sup>2</sup> + 2xyz<sup>2</sup> + 3y<sup>2</sup>z<sup>2</sup> + 2xz<sup>3</sup> + 2yz<sup>3</sup></td><td>5</td></tr><tr><td>+2z<sup>4</sup> - x<sup>3</sup>w + x<sup>2</sup>yw + xy<sup>2</sup>w - y<sup>3</sup>w + x<sup>2</sup>zw - y<sup>2</sup>zw - 2xz<sup>2</sup>w + 2yz<sup>2</sup>w + 2z<sup>3</sup>w</td><td>5</td></tr><tr><td>-2x<sup>2</sup>w<sup>2</sup> - 2xyw<sup>2</sup> - 2y<sup>2</sup>w<sup>2</sup> - 2xzw<sup>2</sup> - 2yzw<sup>2</sup> + 2z<sup>2</sup>w<sup>2</sup> + 2xw<sup>3</sup> - 2yw<sup>3</sup> - 2zw<sup>3</sup> - 4w<sup>4</sup></td><td>6</td></tr><tr><td>x<sup>3</sup>y + y<sup>4</sup> + z<sup>3</sup>w + yw<sup>3</sup> + zw<sup>3</sup></td><td>7</td></tr><tr><td>w<sup>3</sup>x + x<sup>4</sup> + wx<sup>2</sup>z + x<sup>3</sup>z + xy<sup>2</sup>z - y<sup>3</sup>z + wxz<sup>2</sup> + x<sup>2</sup>z<sup>2</sup> - xz<sup>3</sup> + z<sup>4</sup></td><td>8</td></tr><tr><td>x<sup>3</sup>y + z<sup>4</sup> + y<sup>3</sup>w + xw<sup>3</sup> + w<sup>4</sup></td><td>9</td></tr><tr><td>w<sup>4</sup> + wx<sup>2</sup>y + y<sup>4</sup> + x<sup>3</sup>z - xy<sup>2</sup>z + z<sup>4</sup></td><td>10</td></tr><tr><td>w<sup>4</sup> + x<sup>4</sup> + x<sup>2</sup>y<sup>2</sup> + y<sup>4</sup> - w<sup>3</sup>z - 2xy<sup>2</sup>z + x<sup>2</sup>z<sup>2</sup> + z<sup>4</sup></td><td>11</td></tr><tr><td>x<sup>3</sup>y + y<sup>4</sup> + z<sup>3</sup>w + x<sup>2</sup>w<sup>2</sup> + w<sup>4</sup></td><td>12</td></tr><tr><td>w<sup>4</sup> + w<sup>3</sup>x<sup>4</sup> + wy<sup>3</sup> + y<sup>2</sup>z<sup>2</sup> + wz<sup>3</sup> + x<sup>2</sup>z<sup>3</sup></td><td>13</td></tr><tr><td>x<sup>3</sup>y + y<sup>4</sup> + z<sup>3</sup>w + yw<sup>3</sup> + w<sup>4</sup></td><td>14</td></tr><tr><td>x<sup>3</sup>y + y<sup>3</sup>z + z<sup>4</sup> + xy<sup>2</sup>w + zw<sup>3</sup></td><td>15</td></tr><tr><td>x<sup>3</sup>y + y<sup>4</sup> + z<sup>3</sup>w + xyw<sup>2</sup> + y<sup>2</sup>w<sup>2</sup> + w<sup>4</sup></td><td>16</td></tr><tr><td>x<sup>3</sup>y + y<sup>4</sup> + z<sup>4</sup> + x<sup>2</sup>w<sup>2</sup> + zw<sup>3</sup></td><td>17</td></tr><tr><td>x<sup>3</sup>y + x<sup>3</sup>z + y<sup>3</sup>z + yz<sup>3</sup> + w<sup>4</sup></td><td>18</td></tr><tr><td>x<sup>3</sup>y + z<sup>4</sup> + y<sup>3</sup>w + xyzw + xw<sup>3</sup></td><td>19</td></tr><tr><td>x<sup>3</sup>y + z<sup>4</sup> + y<sup>3</sup>w + xw<sup>3</sup></td><td>20</td></tr></tbody></table>

relying on effective Picard-Lefschetz theory can be used to compute the periods of varieties given as elliptic fibrations. Finally, it shows that our algorithm manages examples from physics that were previously out of reach.

<figure><img src="image_14.png" /><figcaption>**Figure 9.** The tardigrade graph</figcaption></figure>

The Tardigrade graph corresponds to a family of $K3$ surfaces given as the minimal resolution of a family of generically singular quartic surfaces in $\mathbb{P}^3$. For example, one element of this family, on which the computation was performed, is the quartic defined by the equation derived in Doran
et al. (2023, §8)

$$ (92) \qquad \begin{aligned} & 6124x^4 - 24782692x^3x + 24962401977x^2x^2 - 20842243972xx^3 + 4331388844x^4 + 6124w^4 \\ & \quad + 13827992x^3z - 27919677996x^2xz - 119291704836xx^2z + 50444249752x^3z \\ & \quad + 7840306116x^2z^2 - 1895725740xxz^2 + 168749562396x^2z^2 + 38842829528xz^3 \\ & \quad + 168487393048xz^3 + 48321305644z^4 - 101996680x^3w + 204653103868x^2xw \\ & \quad - 85803990572xx^2w + 10300568x^3w - 115176640844x^2zw - 460049503942xxzw \\ & \quad - 18769272012x^2zw - 311785995116xz^2w - 108990818964xz^2w - 62891049316z^3w \\ & \quad + 417760330428x^2w^2 - 126779372xxw^2 + 10306692x^2w^2 + 179497287052xzw^2 \\ & \quad + 37592148xzw^2 + 22845754953z^2w^2 - 101996680xw^3 + 12248xw^3 - 22389124zw^3. \end{aligned} $$

Using a variation on the methods presented in this paper, of which we give an overview below, we were able to recover algebraic invariants of this family, namely

(1) its generic Picard rank, 11,

(2) the Picard lattice, i.e. the kernel of the holomorphic period map $\gamma \mapsto \int_{\gamma} \omega$,

(3) and an embedding of the Picard lattice in the standard K3 lattice.

The quartic surfaces considered generically have 4 nodal singularities. We want to obtain the periods of their minimal desingularisation. Despite the presence of singularities, we may still consider a Lefschetz fibration of these varieties and compute the monodromy matrices around the critical values. From these monodromy matrices, we recover a fibration of a formal smoothing of the singular quartic surface which yields a description of its homology. The homology of the smoothing of the singular quartic is isometric to that of its desingularisation. Therefore this allows us to compute a description of the homology of the K3 surface. We can then integrate the holomorphic form following Section 3.7. This shows that we may use the effective Picard-Lefschetz theory presented in this paper to recover the homology of hypersurfaces with nodal singularities.

In practice, however, in an effort to reduce the runtime, we may instead consider an elliptic fibration of the K3 surface directly. This has two main benefits:

* The order of the Picard-Fuchs equations that need to be integrated diminishes from 7 to 3 (as the genus of the homology of the fibre goes from 3 for a quartic curve to 1 for an elliptic curve). This greatly decreases the runtime of the computation down to less than a minute.

* We do not need to consider a modification of the K3 surface, which means there are no superfluous exceptional divisors to remove.

In order to obtain an elliptic fibration of the K3 surface, we project the K3 surface away from one of its singular points. This yields a double cover of $\mathbb{P}^2$ ramified along a sextic curve. Up to a change of variable, the defining equation of this sextic curve can be made quartic in one of the variables. Taking another variable as a parameter, we obtain an affine equation of the form $y^2 = p(x, t)$ with $p$ quartic in $x$, which gives an elliptic fibration of the K3 surface parametrised by $t$. This fibration has 17 singular fibres, two of which are $I_4$ fibres, one is an $I_2$ and the rest are $I_1$'s as per the Kodaira classification of singular fibers of elliptic surfaces. We identify the type of the singular fibres by looking at their monodromy matrices. Passing again to a smoothing of the fibration, we are able to recover the homology and perform the integration. Further details about this computation are given by Doran et al. (2023, App. B).

## REFERENCES

Artebani, M., Sarti, A., & Taki, S. (2011). K3 surfaces with non-symplectic automorphisms of prime order. With an appendix by Shigeyuki Kondō. *Math. Z.*, **268**(1-2), 507–533.
https://doi.org/10.1007/s00209-010-0681-x
et al. (2023, §8)

$$ (92) \qquad \begin{aligned} & 6124x^4 - 24782692x^3x + 24962401977x^2x^2 - 20842243972xx^3 + 4331388844x^4 + 6124w^4 \\ & \qquad + 13827992x^3z - 27919677996x^2xz - 119291704836xx^2z + 50444249752x^3z \\ & \qquad + 7840306116x^2z^2 - 1895725740xxz^2 + 168749562396x^2z^2 + 38842829528xz^3 \\ & \qquad + 168487393048xz^3 + 48321305644z^4 - 101996680x^3w + 204653103868x^2xw \\ & \qquad - 85803990572xx^2w + 10300568x^3w - 115176640844x^2zw - 460049503942xxzw \\ & \qquad - 18769272012x^2zw - 311785995116xz^2w - 108990818964xz^2w - 62891049316z^3w \\ & \qquad + 417760330428x^2w^2 - 126779372xxw^2 + 10306692x^2w^2 + 179497287052xzw^2 \\ & \qquad + 37592148xzw^2 + 22845754953z^2w^2 - 101996680xw^3 + 12248xw^3 - 22389124zw^3. \end{aligned} $$

Using a variation on the methods presented in this paper, of which we give an overview below, we were able to recover algebraic invariants of this family, namely

(1) its generic Picard rank, 11,

(2) the Picard lattice, i.e. the kernel of the holomorphic period map $\gamma \mapsto \int_{\gamma} \omega$,

(3) and an embedding of the Picard lattice in the standard $K3$ lattice.

The quartic surfaces considered generically have 4 nodal singularities. We want to obtain the periods of their minimal desingularisation. Despite the presence of singularities, we may still consider a Lefschetz fibration of these varieties and compute the monodromy matrices around the critical values. From these monodromy matrices, we recover a fibration of a formal smoothing of the singular quartic surface which yields a description of its homology. The homology of the smoothing of the singular quartic is isometric to that of its desingularisation. Therefore this allows us to compute a description of the homology of the $K3$ surface. We can then integrate the holomorphic form following Section 3.7. This shows that we may use the effective Picard-Lefschetz theory presented in this paper to recover the homology of hypersurfaces with nodal singularities.

In practice, however, in an effort to reduce the runtime, we may instead consider an elliptic fibration of the $K3$ surface directly. This has two main benefits:

* The order of the Picard-Fuchs equations that need to be integrated diminishes from 7 to 3 (as the genus of the homology of the fibre goes from 3 for a quartic curve to 1 for an elliptic curve). This greatly decreases the runtime of the computation down to less than a minute.

* We do not need to consider a modification of the $K3$ surface, which means there are no superfluous exceptional divisors to remove.

In order to obtain an elliptic fibration of the $K3$ surface, we project the $K3$ surface away from one of its singular points. This yields a double cover of $\mathbb{P}^2$ ramified along a sextic curve. Up to a change of variable, the defining equation of this sextic curve can be made quartic in one of the variables. Taking another variable as a parameter, we obtain an affine equation of the form $y^2 = p(x, t)$ with $p$ quartic in $x$, which gives an elliptic fibration of the $K3$ surface parametrised by $t$. This fibration has 17 singular fibres, two of which are $I_4$ fibres, one is an $I_2$ and the rest are $I_1$'s as per the Kodaira classification of singular fibers of elliptic surfaces. We identify the type of the singular fibres by looking at their monodromy matrices. Passing again to a smoothing of the fibration, we are able to recover the homology and perform the integration. Further details about this computation are given by Doran et al. (2023, App. B).

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Griffiths, P. A. (1969). On the periods of certain rational integrals I. *Ann. Math.*, **90**(3), 460. https://doi.org/10/dq3zc6

Griffiths, P. A. (1968). Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties. *Amer. J. Math.*, **90**, 568–626.

Griffiths, P. A., & Harris, J. (1978). *Principles of algebraic geometry*. Wiley-Interscience.

Grothendieck, A. (1966). On the de Rham cohomology of algebraic varieties. *Publ. Mathématiques IHÉS*, (29), 95–103. https://doi.org/10/cxsrzs

Haraoka, Y. (2020). *Linear differential equations in the complex domain: From classical theory to forefront*. Springer. https://doi.org/10/gr8d3g

Hartshorne, R. (1977). *Algebraic geometry*. Springer.

Hashimoto, K. (2012). Finite symplectic actions on the $K_3$ lattice. *Nagoya Math. J.*, **206**, 99–153. https://doi.org/10.1215/00277630-1548511

Hatcher, A. (2002). *Algebraic topology*. Cambridge University Press.

Hirzebruch, F. (1956). Der Satz von Riemann-Roch in Faisceau-theoretischer Formulierung: einige Anwendungen und offene Fragen. In *Proceedings of ICM 1954*. Erven P. Noordhoff N. V.

Huybrechts, D. (2016). *Lectures on $K_3$ Surfaces*. Cambridge University Press. https://doi.org/10/fh6x

Kauers, M., Jaroschek, M., & Johansson, F. (2015). Ore polynomials in Sage. In J. Gutierrez, J. Schicho, & M. Weimann (Eds.), *Computer Algebra and Polynomials: Applications of Algebra and Number Theory* (pp. 105–125). Springer. https://doi.org/10/f3s5ht

Lairez, P., & Sertöz, E. C. (2019). A numerical transcendental method in algebraic geometry: Computation of Picard groups and related invariants. *SIAM J. Appl. Algebra Geom.*, **3**(4), 559–584. https://doi.org/10/ggck6n

Lamotke, K. (1981). The topology of complex projective varieties after S. Lefschetz. *Topology*, **20**(1), 15–51. https://doi.org/10/dw8m2q

Lefschetz, S. (1924). *L'analyse situs et la géométrie algébrique*. Gauthier-Villars.

Lenstra, A. K., Lenstra, H. W., & Lovàsz, L. (1982). Factoring polynomials with rational coefficients. *Math. Ann.*, **261**(4), 515–534. https://doi.org/10/c3j8hx

Mather, J. (2012). Notes on Topological Stability. *Bull. Am. Math. Soc.*, **49**(4), 475–506. https://doi.org/10/gg2nbz

Mezzarobba, M. (2010). NumGfun: A package for numerical and analytic computation with D-finite functions. *Proc. ISSAC 2010*, 139–146. https://doi.org/10/cg7w72

Mezzarobba, M. (2016). *Rigorous multiple-precision evaluation of D-finite functions in Sagemath*. *arXiv*: 1607.01967.

Molin, P., & Neurohr, C. (2019). Computing period matrices and the Abel-Jacobi map of superelliptic curves. *Math. Comp.*, **88*(316), 847–888. https://doi.org/10/ggck8t

Narumiya, N., & Shiga, H. (2001). The mirror map for a family of $K_3$ surfaces induced from the simplest 3-dimensional reflexive polytope. In *Proceedings on moonshine and related topics* (pp. 139–161). AMS.

Neurohr, C. (2018). *Efficient integration on Riemann surfaces and applications*. http://oops.uni-oldenburg.de/3607/1/neueff18.pdf

Nikulin, V. V. (1979). Finite automorphism groups of Kählerian surfaces of type $K_3$. *Tr. Mosk. Mat. O.-va*, **38**, 75–137.

Niyogi, P., Smale, S., & Weinberger, S. (2008). Finding the homology of submanifolds with high confidence from random samples. *Discrete Comput. Geom.*, **39**(1), 419–441. https://doi.org/10/b7qcdg
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Oguiso, K. (2017). Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations. *Taiwanese J. Math.*, **21**(3), 671–688. https://doi.org/10/gr7hm6

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UNIVERSITÉ PARIS-SACLAY, INRIA, 91120 PALAISEAU, FRANCE

Email address: pierre.lairez@inria.fr

INSTITUT DE PHYSIQUE THEORIQUE, UNIVERSITÉ PARIS-SACLAY, CEA, CNRS, F-91191 GIF-SUR-YVETTE CEDEX, FRANCE; UNIVERSITÉ PARIS-SACLAY, INRIA, 91120 PALAISEAU, FRANCE

Email address: eric.pichon@polytechnique.edu

INSTITUT DE PHYSIQUE THEORIQUE, UNIVERSITÉ PARIS-SACLAY, CEA, CNRS, F-91191 GIF-SUR-YVETTE CEDEX, FRANCE

Email address: pierre.vanhove@ipht.fr
Oaku, T., & Takayama, N. (1999). An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation. *J. Pure Appl. Algebra*, **139**(1-3), 201-233.
https://doi.org/10/dqgzc6

Oguiso, K. (2012). *Smooth quartic K3 surfaces and Cremona transformations II*. arXiv: 1206.5049.

Oguiso, K. (2017). Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations. *Taiwanese J. Math.*, **21**(3), 671–688. https://doi.org/10/gr7hm6

Pham, F. (1965). Formules de Picard-Lefschetz généralisées et ramification des intégrales. *B. Soc. Math. Fr.*, **79**, 333–367. https://doi.org/10/ggck9f

Preparata, F. P., & Shamos, M. I. (1985). *Computational geometry. An introduction*. New York etc.: Springer-Verlag.

Sertöz, E. C. (2019). Computing periods of hypersurfaces. *Math. Comput.*, **88**(320), 2987–3022.
https://doi.org/10/ggck7t

Shiga, H. (1979). One attempt to the K3 modular function. I. *Ann. Sc. Norm. Super. Pisa Cl. Sci.* **(4)**, 6, 609–635.

Swierczewski, C. (2017). *Abelfunctions: A library for computing with Abelian functions, Riemann surfaces, and algebraic curves*. https://github.com/abelfunctions/abelfunctions

The Sage Developers. (2023). *SageMath, the Sage Mathematics Software*. https://doi.org/10/gr8dhc

Tretkoff, C. L., & Tretkoff, M. D. (1984). Combinatorial group theory, Riemann surfaces and differential equations. In *Contributions to group theory* (pp. 467–519, Vol. 33). AMS. https://doi.org/10/fzq8bb

van der Hoeven, J. (1999). Fast evaluation of holonomic functions. *Theoret. Comput. Sci.*, **210**(1), 199–215. https://doi.org/10/b95scc

Xiao, G. (1996). Galois covers between K3 surfaces. *Ann. Inst. Fourier*, **46**(1), 73–88. https://doi.org/10.5802/aif.1507

UNIVERSITÉ PARIS-SACLAY, INRIA, 91120 PALAISEAU, FRANCE

Email address: pierre.lairez@inria.fr

INSTITUT DE PHYSIQUE THEORIQUE, UNIVERSITÉ PARIS-SACLAY, CEA, CNRS, F-91191 GIF-SUR-YVETTE CEDEX, FRANCE; UNIVERSITÉ PARIS-SACLAY, INRIA, 91120 PALAISEAU, FRANCE

Email address: eric.pichon@polytechnique.edu

INSTITUT DE PHYSIQUE THEORIQUE, UNIVERSITÉ PARIS-SACLAY, CEA, CNRS, F-91191 GIF-SUR-YVETTE CEDEX, FRANCE

Email address: pierre.vanhove@ipht.fr