Question: Agnes of Austria was a member of the house that ruled the Holy Roman Empire beginning in what year?

Context paragraphs:
Battle of Leitzersdorf: The Battle of Leitzersdorf was a battle between the Holy Roman Empire and the Kingdom of Hungary in 1484.  Fuelled by the earlier conflicts of Matthias Corvinus and Frederick III, Holy Roman Emperor it marked the end of anti-Ottoman preparations and initiations of a holy war.  It was the only open field battle of the Austro-Hungarian War, and the defeat meant – in long terms – the loss of the Archduchy of Austria for the Holy Roman Empire.

House of Habsburg: The House of Habsburg ( ; ] ), also called House of Austria, was one of the most influential and outstanding royal houses of Europe.  The throne of the Holy Roman Empire was continuously occupied by the Habsburgs between 1438 and 1740.  The house also produced emperors and kings of the Kingdom of Bohemia, Kingdom of England ("Jure uxoris" King), Kingdom of Germany, Kingdom of Hungary, Kingdom of Croatia, Second Mexican Empire, Kingdom of Ireland ("Jure uxoris" King), Kingdom of Portugal, and Spain, as well as rulers of several Dutch and Italian principalities.  From the 16th century, following the reign of Charles V, the dynasty was split between its Austrian and Spanish branches.  Although they ruled distinct territories, they nevertheless maintained close relations and frequently intermarried.

Agnes of Austria (1281–1364): Agnes of Austria (18 May 1281 – 10 June 1364) was a daughter of Albert I of Germany and his wife Elisabeth of Tirol.  She was Queen of Hungary by marriage.  She was a member of the House of Habsburg.

Imperial Treasury, Vienna: The Imperial Treasury (German: "Kaiserliche Schatzkammer" ) at the Hofburg Palace in Vienna, Austria contains a valuable collection of secular and ecclesiastical treasures covering over a thousand years of European history.  The entrance to the treasury is at the Schweizerhof (Swiss Courtyard), the oldest part of the palace, which was rebuilt in the sixteenth century in the Renaissance style under Holy Roman Emperor Ferdinand I.  The Imperial Treasury is affiliated with the Kunsthistorisches Museum, and houses in 21 rooms a collection of rare treasures that were compiled by the Imperial House of Habsburg over the course of centuries, including the Imperial Crown, Orb, and Sceptre of Austria, and the Imperial Regalia of the Emperors and Kings of the Holy Roman Empire, including the Imperial Crown of the Holy Roman Empire.

List of Holy Roman Empresses: Holy Roman Empress or Empress of the Holy Roman Empire is the title given to the consort (wife) or regent of the Holy Roman Emperor.  The elective dignity of Holy Roman Emperor was restricted to males only, therefore there was never a Holy Roman Empress regnant, though women such as Theophanu or Maria Theresa of Austria, who controlled the power of rule, served as de facto Empresses regnant.

Karl Anselm, 4th Prince of Thurn and Taxis: Karl Anselm, 4th Prince of Thurn and Taxis, full German name: "Karl Anselm Fürst von Thurn und Taxis" (2 June 1733, Frankfurt am Main, Free Imperial City of Frankfurt, Holy Roman Empire – 13 November 1805, Winzer bei Regensburg, Electorate of Bavaria, Holy Roman Empire) was the fourth Prince of Thurn and Taxis, Postmaster General of the Imperial Reichspost, and Head of the Princely House of Thurn and Taxis from 17 March 1773 until his death on 13 November 1805.  Karl Anselm served as "Prinzipalkommissar" at the Perpetual Imperial Diet in Regensburg for Joseph II, Holy Roman Emperor and Francis II, Holy Roman Emperor from 1773 to 1797.

Agnes of Poitou: Agnes of Poitou, also called Agnes of Aquitaine or Empress Agnes (  1025 – 14 December 1077), a member of the House of Poitiers, was German queen from 1043 and Holy Roman Empress from 1046 until 1056.  From 1056 to 1061 she acted as regent of the Holy Roman Empire during the minority of her son Henry IV.

Imperial Army (Holy Roman Empire): The Imperial Army (German: "Kaiserliche Armee" ), Imperial Troops ("Kaiserliche Truppen"), Exercitus Imperatoris Romani, or Imperialists ("Kaiserliche") for short, was a name used for several centuries, especially to describe soldiers recruited for the Holy Roman Emperor during the Early Modern Period.  The Imperial Army of the Emperor should not be confused with the Army of the Holy Roman Empire ("Reichsarmee"), which could only be deployed with the consent of the Imperial Diet.  The Imperialists effectively became a standing army of troops under the Habsburg emperor from the House of Austria, which is why they were also increasingly described in the 18th century as "Austrians", although its troops were recruited not just from the Archduchy of Austria but from all over the Holy Roman Empire of the German Nation.

Duchy of Württemberg: The Duchy of Württemberg (German: "Herzogtum Württemberg" ) was a duchy located in the south-western part of the Holy Roman Empire.  It was a member of the Holy Roman Empire from 1495 to 1806.  The dukedom's long survival for nearly four centuries was mainly due to its size, being larger than its immediate neighbors.  During the Protestant Reformation, Württemberg faced great pressure from the Holy Roman Empire to remain a member.  Württemberg resisted repeated French invasions in the 17th and 18th centuries.  Württemberg was directly in the path of French and Austrian armies who were engaged in the long rivalry between the House of Bourbon and the House of Habsburg.  In 1803, Napoleon raised the duchy to be the Electorate of Württemberg of the Holy Roman Empire, and when he abolished the Empire in 1806, the Electorate was elevated as the Kingdom of Württemberg.

List of monarchs of Prussia: The monarchs of Prussia were members of the House of Hohenzollern who were the hereditary rulers of the former German state of Prussia from its founding in 1525 as the Duchy of Prussia.  The Duchy had evolved out of the Teutonic Order, a Roman Catholic crusader state and theocracy located along the eastern coast of the Baltic Sea.  The Teutonic Knights were under the leadership of a Grand Master, the last of whom, Albert, converted to Protestantism and secularized the lands, which then became the Duchy of Prussia.  The Duchy was initially a vassal of the Kingdom of Poland, as a result of the terms of the Prussian Homage whereby Albert was granted the Duchy as part of the terms of peace following the Prussian War.  When the main line of Prussian Hohenzollerns died out in 1618, the Duchy passed to a different branch of the family, who also reigned as Electors of Brandenburg in the Holy Roman Empire.  While still nominally two different territories, Prussia under the suzerainty of Poland and Brandenburg under the suzerainty of the Holy Roman Empire, the two states are known together historiographically as Brandenburg-Prussia.  Following the Second Northern War, a series of treaties freed the Duchy of Prussia from any vassalage to any other state, making it a fully sovereign Duchy in its own right.  This complex situation (where the Hohenzollern ruler of the independent Duchy of Prussia was also a subject of the Holy Roman Emperor as Elector of Brandenburg) laid the eventual groundwork for the establishment of the Kingdom of Prussia in 1701.  For diplomatic reasons, the rulers of the state were known as the King in Prussia from 1701 to 1772; largely because they still owed fealty to the Emperor as Electors of Brandenburg, the "King in Prussia" title (as opposed to "King of Prussia") avoided offending the Emperor.  As the Prussian state grew through several wars and diplomatic moves throughout the 18th century, it became apparent that Prussia had become a Great Power that did not need to submit meekly to the Holy Roman Empire.  By 1772, the pretense was dropped, and the style "King of Prussia" was adopted.  Thus it remained until 1871, when in the aftermath of the Franco-Prussian War, the King of Prussia Wilhelm I was crowned German Emperor.  From that point forward, though the Kingdom of Prussia retained its status as a constituent state of the German Empire, all remaining Kings of Prussia also served as German Emperor, and that title took precedence.

Supporting paragraphs: Agnes of Austria (1281–1364) House of Habsburg
Answer: 1438