The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q).
From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form E: y2 = x3 + ax2 + bx. In this thesis, we generalize this formula, using a purely group theoretic approach, and utilize this generalization to find the rank of curves of the form E: y2 = x3 + c. To do this, we review a few well-known homomorphisms on the curve E: y2 = x3 + ax2 + bx as in Tate and Silverman's Elliptic Curves, and study analogous homomorphisms on E: y2 = x3 + c and relevant facts.
","abstract_html":"<p>The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points <em>E</em>(<strong><em>Q</em></strong>) of an elliptic curve <em>E</em> form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of <strong><em>Z </em></strong><em></em>and finite cyclic groups. The number of finite copies of <strong><em>Z</em></strong> is called the rank of <em>E</em>(<strong><em>Q</em></strong>).</p> <p>From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form <em>E</em>: <em>y</em><sup>2</sup> = <em>x</em><sup>3</sup> + a<em>x<sup>2</sup></em> + b<em>x</em>. In this thesis, we generalize this formula, using a purely group theoretic approach, and utilize this generalization to find the rank of curves of the form <em>E</em>: <em>y</em><sup>2</sup> = <em>x</em><sup>3</sup> + <em>c</em>. To do this, we review a few well-known homomorphisms on the curve <em>E</em>: <em>y</em><sup>2</sup> = <em>x</em><sup>3</sup> + a<em>x<sup>2</sup></em> + b<em>x</em> as in Tate and Silverman's <em>Elliptic Curves</em>, and study analogous homomorphisms on <em>E</em>: <em>y</em><sup>2</sup> = <em>x</em><sup>3</sup> + <em>c </em>and relevant facts.</p>","abstract_has_math":false,"creators":["Mecklenburg, Trinity"],"institution":null,"degree_name":"Master of Arts in Mathematics","degree_level":"Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Han, Ilseop"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-06-01T07:00:00Z","date_published":"2015-06-01T07:00:00Z","updated_at":"2026-07-24T01:52:53Z","subjects":["rank","elliptic curves","y^2=x^3+c","Algebra"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.lib.csusb.edu/etd/186","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Han, Ilseop"]},{"key":"dc:creator","label":"Author","values":["Mecklenburg, Trinity"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2015-05-22T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Arts in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["rank","elliptic curves","y^2=x^3+c","Algebra"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.lib.csusb.edu/etd/186"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q).
From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form E: y2 = x3 + ax2 + bx. In this thesis, we generalize this formula, using a purely group theoretic approach, and utilize this generalization to find the rank of curves of the form E: y2 = x3 + c. To do this, we review a few well-known homomorphisms on the curve E: y2 = x3 + ax2 + bx as in Tate and Silverman's Elliptic Curves, and study analogous homomorphisms on E: y2 = x3 + c and relevant facts.
"]},{"key":"dc:title","label":"Title","values":["Elliptic Curves"]}]}],"canonical_facts":{"dc:contributor":["Han, Ilseop"],"dc:creator":["Mecklenburg, Trinity"],"dc:date.available":["2015-05-22T07:00:00Z"],"dc:description.abstract":["The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q).
From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form E: y2 = x3 + ax2 + bx. In this thesis, we generalize this formula, using a purely group theoretic approach, and utilize this generalization to find the rank of curves of the form E: y2 = x3 + c. To do this, we review a few well-known homomorphisms on the curve E: y2 = x3 + ax2 + bx as in Tate and Silverman's Elliptic Curves, and study analogous homomorphisms on E: y2 = x3 + c and relevant facts.
"],"dc:identifier":["https://scholarworks.lib.csusb.edu/etd/186"],"dc:subject":["rank","elliptic curves","y^2=x^3+c","Algebra"],"dc:title":["Elliptic Curves"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Arts in Mathematics"]},"updated_at":"2026-07-24T01:52:53Z"}