For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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103 views

E: $y^2+y=x^3$ an elliptic curve over $F_{2}$. How to prove the number of $E(F_{2^n})$ = $2^n+1$ if n is odd, …

Let E be the elliptic curve $y^2 + y = x^3$ over $F_2$. Prove $ $#E($F_{2^n})$$ = \left\{ \begin{array}{ll} 2^n+1 & \quad n=odd \\ 2^n+1-2(-2)^{n/2} & \quad ...
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2answers
309 views

Moduli space of isogeny classes of elliptic curves

The modular curve $Y(1)$ classifies isomorphism classes of elliptic curves, namely its $K$-points for any field $\mathbb Q\subseteq K\subseteq \mathbb C$ correspond via the $j$-invariant to $\mathbb C$...
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1answer
78 views

Is it correct to say $ x^3+2x+1=y^2 $ is an elliptic curve?

I'm a bit confused about the definition on elliptic curve. For example, can we say that $x^3+2x+1=y^2$ is an elliptic curve? My opinion is that it is not an elliptic curve as the definition given in ...
5
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1answer
164 views

Elliptic Curves Without Geometry

Unfortunately geometry terrifies me, so I was hoping to understand the basic theory of elliptic curves algebraically (via their function fields). Let F be a transcendence degree 1 extension of $\...
5
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1answer
315 views

How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
2
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1answer
81 views

Minimal Discriminant of An Elliptic Curve

I want to determine the minimal discriminant of $$ y^2 + xy = x^3-x^2-50x+111 $$ as an elliptic curve over the rationals. I managed to reduce it to the form $y^2=x^3+Ax+B$ where $A,B$ are rational, ...
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1answer
137 views

$L$-functions of elliptic curves over $\mathbb{Q}$

How to find out the $P_{v}(E/\mathbb{Q},X)$ theoretically given below in the definition of $L$-functions for elliptic curves over $\mathbb{Q}$ $?$ Please cite some references for the same. For an ...
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1answer
31 views

Generate point with desired order on a curve

I'm looking to construct a set of elliptic curve parameters for the Tate pairing. I'm working with an elliptic curve over the field $F_{q^{12}}$ where $q$ is a prime number of magnitude around $~2^{...
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0answers
63 views

Elliptic curve which attains potential good reduction over an Artin-Schreier extension.

I am looking for an elliptic curve $E$ over the field $\overline{\mathbb{F}_{p}}((t))$, which attains good reduction over an Artin-Schreier extension of $\overline{\mathbb{F}_{p}}((t))$, i.e.: an ...
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2answers
304 views

Question about kernel of reduction mapping of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has good (nonsingular) reduction $\tilde{E}$ modulo some prime $p$. Denote their groups $E(\mathbb{Q})$ and $\tilde{E}(\mathbb{F}_p)$ respectively....
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2answers
74 views

From $y^2=x^3+Ax^2+Bx$ to $y^2+(1-c)xy-by=x^3-bx^2$

I have two question How can I transfer with a change of coordinates from $$y^2=x^3+Ax^2+Bx$$ to $$y^2+(1-c)xy-by=x^3-bx^2?$$ In a note of Prof. Lozano "Elliptic Curves, Modular Forms and their L-...
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1answer
72 views

Correspondence between prime ideals and Galois orbits of affine points on an elliptic curve

In notes of prof. W.Stein - http://wstein.org/edu/2010/581b/stein-algebraic_number_theory.pdf - the first paragraph of page 112 has the following told: "When $K$ is a perfect field, the prime ideals ...
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1answer
76 views

Finding the inverse of P on the generalized weierstrass equation

If P = (x, y) = ∞ is on a monic cubic polynomial, then −P is the other finite point of intersection of the curve and the vertical line through P. Show that −P = (x, −$a_{1}x$ − $a_3$ − y). (Hint: This ...
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1answer
94 views

Semistable Elliptic Curves

For a general representation $\rho: G_{\mathbb{Q}} \rightarrow \operatorname{GL}(V)$, where $V$ is a two dimensional $\overline{\mathbb{F}}_p$ vector space, the level $N(\rho)$ in Serre's conjecture ...
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1answer
115 views

Elliptic curves: Why over the complex numbers?

I am a math undergrad, so much of the literature on elliptic curves escapes me. I'm trying to understand why one considers elliptic curves over the complex numbers. Specifically, this part of the ...
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1answer
101 views

Elliptic curves in projective form question

Let $K$ be any field with Char $K \neq 2, 3$, and let $\varepsilon : F ( X_0 ;X_1 ;X_2 ) = X_1^2 X_2- ( X_0^3 +AX_0 X_2^2 + BX_2^3 )$ ; with $A, B \in K$, be an elliptic curve. Let $P$ be a point on $...
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1answer
39 views

Birational Transformations question

so I'm wondering, is there a birational transformation one can make to the equation $Y^2 = X^m + f_{m-1}X^{m-1} + ... + f_0$, where all $f_i \in \mathbb{Q}$ so it is of the form $Y^2 = X^m + g_{m-1}X^{...
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1answer
42 views

Why is the answer set limited here?

This question is based on pp $67$ - $68$ of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation $f(x,y) = x^2+y^2$ We can ...
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1answer
32 views

On the rank of $y^2=x^3+a^2x^2-a^4x$

How can I prove that the rank of $y^2=x^3+a^2x^2-a^4x$ is zero where $a$ is rational and positive?
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1answer
61 views

Rational Number Form

I was reading Rational Points on Elliptic Curves by Silverman and Tate and they state: Every non-zero rational number may be uniquely written in the form $\frac{m}{n}p^v$, where $m,n$ are integers ...
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1answer
546 views

Rational points of order 2 on elliptic curves

I'm working on a question that's asking me to list all the $\mathbb Q$-rational points of order 2 and all the $\mathbb C$-rational points of order 2 for some elliptic curves. I've made the following ...
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0answers
30 views

Reduction of isogenies at bad primes

Let $L$ be a number field and $E,E'$ two elliptic curves defined over $L$. Suppose $\varphi\colon E\to E'$ is an isogeny defined over $L$. Let $\mathfrak p$ be a prime of bad reduction for $E,E'$. ...
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1answer
78 views

Weierstrass normal form

How can I show that the Weierstrass normal form $u^3 + v^3 = \alpha$,with $x=12\alpha/(u+v)$ and $y=36\alpha (u-v)/(u+v)$, satisfy $y^2=x^3-432α^2$ ?
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1answer
136 views

Are $10^{10}$-digit-numbers too big for Lenstra's elliptic curve method (ECM)?

I would like to search prime factors of the numbers $$10^{10^{10}}-113$$ and $$10^{10^{10}}+13$$ Both numbers have no prime factor below $10^9$. Are these numbers still too big for ECM ? I also ...
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1answer
59 views

How to determine the group structure of $E(\mathbb{R})$ for an elliptic curve $E/\mathbb{R}$

Using Weierstrass' $\wp$ function it can be proved that the group of complex points on an elliptic curve $E /\mathbb{C}: y^2 = x^3 + ax + b$ satisfies $E(\mathbb{C}) \cong \mathbb{R}/\mathbb{Z} \oplus ...
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1answer
60 views

Complex Multiplication of $y^2=x^3+B$

I would like to find out what the complex endomorphism for the class of elliptic curves given by $$y^2=x^3+B$$ looks like. I know that for the class of elliptic curves $$y^2=x^3+Ax,$$ the complex ...
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1answer
63 views

Solutions of elliptic curve in finite field

If I take the following elliptic formula over a finite field of size $17$: $$y^2 = x^3 + 2x + 3$$ The solutions for $x = 2$ would be $7$ and $10$. Because $7^2=49$ and $49 \equiv 15 \bmod 17$ $...
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214 views

Using an elliptic curve to create pseudo random number

I recently started learning about encryption. I read about how elliptic curves can be used to create pseudo random numbers (and how the nsa might have abused this fact to create a backdoor in ...
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1answer
80 views

Formal group of an elliptic curve, from Silverman's the arithmetic of an elliptic curves

In the beginning of page 120, to establish the formal group law for an elliptic curve, the book adds 2 points $(z_1,w_1)$ and $(z_2,w_2)$, where $w_1 = w(z_1), w_2 = w(z_2)$ using the group law. It ...
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0answers
59 views

Inflection points on elliptic curves over a field of characteristic 2

I'm looking at the elliptic curve $C:={\cal Z}(XY^2+ZX^2+YZ^2)$ in the field $k:=\overline{\mathbb{F}_2}$. I want to prove that this curve has 9 inflection points. Since the characteristic of $k$ is 2,...
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1answer
75 views

A special cubic curve

How can I transfer following cubic curve to a Weierstrass normal form? $$2x^2y+4xy^2+2y^3-2axy-ay^2+a=0,$$ where $a$ is a fixed rational number.
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1answer
47 views

Size of automorphism group of element of $Y_0(5)$

The modular curve $Y_0(5)$ parametrizes elliptic curves $E$ with an isogeny of degree five. So an element of $Y_0(5)$ can be interpreted as $E \xrightarrow{\phi} E'$. Suppose we are working over a ...
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1answer
209 views

Weierstrass to general form

Can we go from short Weierstrass equation equation $y^2=x^3+Ax^2+Bx+C$ to general $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$?
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3answers
259 views

$ax^3+by^3+cz^3=0$ and Elliptic curves

What is relation between $ax^3+by^3+cz^3=0$ and Elliptic curves?
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100 views

Elliptic Curve and Conjugation

If I consider an elliptic curve $C$ as a Riemann surface cut out in $\mathbb{C}P^2$ by a homogenous cubic, and if that cubic is defined over $\mathbb{R}$, then I think we have a conjugation map $C\...
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Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
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1answer
181 views

Linear Equivalence of Divisors on Projective Plane Cubic

I'm self-studying Miranda's Algebraic Curves and Riemann Surfaces and am uncertain of how I'm supposed to solve problem V.2c on linearly equivalent divisors: Let $X$ be the projective plane cubic ...
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1answer
101 views

Embedding of elliptic curves into $\mathbb{P}^2$ by arbitrary line bundle of degree $3$

Let $E$ be a complex elliptic curve, with distinguished point $x_0 \in E$. Any divisor of degree three is equivalent to the divisor $D=x+2x_0$. If $x=x_0$, it is well known that $L(D)$ has an explicit ...
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0answers
76 views

Trivial divisor on elliptic curve

Suppose $E$ is an elliptic curve over $k$, and $(E,+)$ is an abelian group(suppose we fix some closed point as identity). Let $[p]$ denote the Weil divisor corresponding to the closed point $p \in E$....
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1answer
128 views

Are all elliptic curves from $w^3 = \text{cubic}(z)$ isomorphic?

I've been playing around with Riemann surfaces of cubics, and it seems to me that all surfaces obtained as coverings of the Riemann sphere from equations of the form $w^3 = q(z)$, where $q(z)$ is a ...
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1answer
67 views

Projective coordinates for elliptic curves

If we consider an elliptic curve projectively, it is a homogeneous form in $3$ variables say $x$, $y$ and $z$. How is this related to the Thue equations (homogeneous forms in $2$ variables)? I'm ...
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1answer
41 views

Uniqueness of points in Elliptic Curve addition

When working on a curve E, is the point yielded by P + Q (some P and Q on E) completely unique? What I mean is there are no other points on E sharing the same x or y value. Thanks!
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Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
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1answer
81 views

Calculating Non-Singular Map of Elliptic Curve

I have a function y^2 = x^3 + Ax + B mod p. I know the curve has a singularity as the discriminate is zero mod p. I'm trying to isolate the non-singular points of the curve by mapping the singularity ...
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1answer
79 views

Prove that the Frobenius map is a homomorphism

I want to prove that the Frobenius map $\phi$ is a homomorphism from the group of points on an elliptic curve $E(F_{2^k})$ to itself (endomorphism). It is trivial to check that if a point $P \in E$ ...
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0answers
158 views

Attacking Elliptic Curve Cryptography Problem with a Bad Reduction $\pmod p$

I'm working on a crypto problem as a puzzle and unfortunately my math isn't at the level I need it to be to answer the question. I have been given a prime $p$, a curve $E$ defined over $F(p)$, a ...
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1answer
168 views

Representing Points of Jacobian in Magma

Although I understand the Mumford representation of points on the Jacobian (of a genus 2 hyperelliptic curve), I don't understand how Magma represents such points. I would guess the confusion arises ...
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1answer
90 views

Structure of $C(F_5)$ from Rational Points on Elliptic Curves

In the book Rational Points on Elliptic Curves by Silverman/Tate one examines the elliptic curve $y^2 = x^3 + x + 1$ over $F_5$. One can then easily determine the group $$ C(F_5) = \lbrace \mathcal{O}...
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1answer
48 views

ellipse chord length along its axis.

how to determine the position in an ellipse, where the chord length is equal to its minor axis and perpendicular to the major axis? Is there any equation for it?
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1answer
119 views

Counting the number of elliptic curves with certain discriminant/conductor

I'm looking for some references regarding the above topic. To be more specific, references that address questions such as Given $D > 0$, how many elliptic curves over $\mathbb{Q}$ are there with ...