# Elliptic curves with j-invariant equals 0 or 1728 over a finite prime field.

Extracta Mathematicae (1991)

• Volume: 6, Issue: 2-3, page 145-147
• ISSN: 0213-8743

top

## Abstract

top
Let p be a prime number, p ≠ 2,3 and Fp the finite field with p elements. An elliptic curve E over Fp is a projective nonsingular curve of genus 1 defined over Fp. Each one of these curves has an isomorphic model given by an (Weierstrass) equation E: y2 = x3 + Ax + B, A,B ∈ Fp with D = 4A3 + 27B2 ≠ 0. The j-invariant of E is defined by j(E) = 1728·4A3/D.The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants equals to 0 or 1728, and the connection between these cardinalities and some expressions of sum of squares.

## How to cite

top

Munuera Gómez, Carlos. "Elliptic curves with j-invariant equals 0 or 1728 over a finite prime field.." Extracta Mathematicae 6.2-3 (1991): 145-147. <http://eudml.org/doc/39938>.

@article{MunueraGómez1991,
abstract = {Let p be a prime number, p ≠ 2,3 and Fp the finite field with p elements. An elliptic curve E over Fp is a projective nonsingular curve of genus 1 defined over Fp. Each one of these curves has an isomorphic model given by an (Weierstrass) equation E: y2 = x3 + Ax + B, A,B ∈ Fp with D = 4A3 + 27B2 ≠ 0. The j-invariant of E is defined by j(E) = 1728·4A3/D.The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants equals to 0 or 1728, and the connection between these cardinalities and some expressions of sum of squares.},
author = {Munuera Gómez, Carlos},
journal = {Extracta Mathematicae},
keywords = {Teoría algebraica de números; Curvas elípticas; Campos finitos; Números primos; Invariantes; rational points of elliptic curve defined over a finite field; deterministic algorithm},
language = {eng},
number = {2-3},
pages = {145-147},
title = {Elliptic curves with j-invariant equals 0 or 1728 over a finite prime field.},
url = {http://eudml.org/doc/39938},
volume = {6},
year = {1991},
}

TY - JOUR
AU - Munuera Gómez, Carlos
TI - Elliptic curves with j-invariant equals 0 or 1728 over a finite prime field.
JO - Extracta Mathematicae
PY - 1991
VL - 6
IS - 2-3
SP - 145
EP - 147
AB - Let p be a prime number, p ≠ 2,3 and Fp the finite field with p elements. An elliptic curve E over Fp is a projective nonsingular curve of genus 1 defined over Fp. Each one of these curves has an isomorphic model given by an (Weierstrass) equation E: y2 = x3 + Ax + B, A,B ∈ Fp with D = 4A3 + 27B2 ≠ 0. The j-invariant of E is defined by j(E) = 1728·4A3/D.The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants equals to 0 or 1728, and the connection between these cardinalities and some expressions of sum of squares.
LA - eng
KW - Teoría algebraica de números; Curvas elípticas; Campos finitos; Números primos; Invariantes; rational points of elliptic curve defined over a finite field; deterministic algorithm
UR - http://eudml.org/doc/39938
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.