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

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2
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1answer
91 views

Infinity of Right Triangle Elliptic Curve

Translate the congruent number problem into elliptic curve, we conclude that an integer $n\in\mathbb Z^+$ is area of a right triangle with $a,b,c\in\mathbb{Q}$ if and only if the corresponding ...
4
votes
1answer
101 views

Computing cohomology groups of elliptic curves

I'm skimming through Silverman's text to recall some theory of elliptic curves that I've learned in undergrad. In practice however, I'm having trouble actually computing the cohomology groups. For ...
2
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1answer
107 views

Fields where all smooth projective curves of genus $1$ are elliptic curves

A common definition of an elliptic curve over a field $k$, is that it is a smooth projective curve of genus $1$ (defined over $k$) with a distinguished $k$-rational point. The distinguished point is ...
5
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1answer
50 views

deg functions and maps

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic ...
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0answers
31 views

deg of composition on supersingular curve

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Let we have algebraic function $f \in \bar{\mathbb{F}_q}(E)$ with div($f) = \sum_{i=0}^{i=n}n_iP_i$. Then div$(f) \circ [q] = ...
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0answers
34 views

number of solutions eqauation on supersingular elliptic curve

To Frobenius endomorphism on supersingular elliptic curve I want to prove that equation $\pi_q(X) = A$ has 1 solution for any point $A \in E(\bar{\mathbb{F}_q}))$ where $E$ is supersingular. Is it ...
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0answers
63 views

Frobenius endomorphism on supersingular elliptic curve

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Is it true that for every point $P$ $q$-Frobenius endomorphism $\pi_q$ can be write as $A + [q]B$ where $P = A + B$? It is true if ...
2
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0answers
72 views

Pole of differential

Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$. We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
3
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1answer
302 views

Find all integer solutions to $x^2+4=y^3$. [duplicate]

Find all integer solutions to $x^2+4=y^3$. Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
3
votes
1answer
123 views

Question about quadratic twists of elliptic curves

Let $E$ be an elliptic curve and $d$ be a squarefree integer. If $E'$ and $E$ are isomorphic over $\mathbb{Q}(\sqrt{d})$, must $E'$ be a quadratic twist of $E$?
3
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2answers
233 views

Isomorphism of Elliptic Curves:

In Stinson's Cryptography Theory and Practice, a theorem is given without proof: Theorem 6.1 Let $E$ be an elliptic curve defined over $Z_p$, where $p$ is prime and $p > 3$. Then there exist ...
3
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1answer
59 views

Explicitly computing finite subgroups on elliptic curves

I have a simple cubic curve, say $y^2 = x^3 - x.$ Is there a simple way to find a small finite subgroup of points lying on this curve? (with respect to the elliptic curve group law.) Otherwise, does ...
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0answers
44 views

Multiples of a point in a non-elliptic curve

Let $E:y^2-xy+y-x^3=0$ over a field $K$, $P=(0,0)$. If $\text{char}(K)\neq2$, $E$ is an elliptic curve for which I can easily get $n*P$. Now, something goes wrong with $\text{char}(K)=2$: Basically, I ...
2
votes
2answers
66 views

Relation between Galois representation and rational $p$-torsion

Let $E$ be an elliptic curve over $\mathbb{Q}$. Does the image of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under the mod $p$ Galois representation tell us whether or not $E$ has rational ...
7
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3answers
160 views

The genus of a curve with a group structure

I'm reading Milne's Elliptic Curves and came across this statement: If a nonsingular projective curve has a group structure defined by polynomial maps, then it has genus 1. In this question a similar ...
5
votes
2answers
196 views

Other ways to compute the torsion subgroup of elliptic curves

Suppose I have a family of elliptic curves $E_{n}/\mathbb{Q}$. I would like to determine the torsion subgroup of $E_{n}(\mathbb{Q})$ denoted by $E_{n}(\mathbb{Q})_{\textrm{tors}}$. Two ways to do this ...
2
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2answers
229 views

How do you determine if an elliptic curve over a finite field is cyclic?

I know the group order and the points of the elliptic curve $y^2 = x^3 + Ax + B$, but I am confused on how to determine if they from a cyclic group The curve $y^2 = x^3 + 2x +2$ in $\Bbb F_{11}$ ...
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2answers
129 views

Rank $2$ Elliptic Curves

I'm on a quest for some rank $2$ elliptic curves. My question is actually twofold: Is there a way to easily construct a curve with this property? Is there a database of elliptic curves with given ...
5
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2answers
61 views

Property of the $p^n$-Selmer group

Consider the $p^n$-Selmer group of an elliptic curve $E/\mathbb{Q}$, $\operatorname{Sel}_{p^n}(E)$. Must we always have $\operatorname{Sel}_{p^n}(E) \cong (\mathbb{Z}/p^n\mathbb{Z})^{s}$ for some $s$? ...
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0answers
90 views

On the elliptic curve $x^4+y^4 =193z^2$

Given the simultaneous Diophantine equations, $$u^2+v^2=w^2\tag{1}$$ $$x^4+y^4 = (u^6+v^6)t^2\tag{2}$$ the only solutions seem to be for the first Pythagorean triple $u,v,w = 3,4,5$ which yield the ...
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0answers
158 views

Every smooth cubic curve has a flex point

I want to show that every smooth irreducible plane cubic $C$ has a flex point, i.e. a point $P$ with $i_P(C, T_C(P)) = 3)$ (where $T_C(P)$ is the tangent to $C$ at $P$). I know how to do this in ...
4
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3answers
114 views

Example of an elliptic curve with trivial torsion subgroup and rank 0

What is an example of an elliptic curve over $\mathbb{Q}$ with trivial torsion subgroup and rank 0?
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0answers
87 views

Some basic questions about Jacobians of curves

Let $C$ be a curve defined over $\mathbb{Q}$, of positive genus. Let $J$ denote its Jacobian. I would like to ask a couple of basic (I presume) questions: 0) Why is $J$ an algebraic variety? 1) For ...
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1answer
44 views

Elliptic curve and restriction

Let $E$ be an elliptic curve. Let $\xi$ be a class of $H^{1}(\mathbb{Q}, E[m])$ unramified at the prime $\ell$. Then $\xi$ restricted to $H^{1}(I_{\ell}, E[m])$ where $I_{\ell}\subset ...
4
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1answer
135 views

Mathematics for Pleasure of a Beginner

I've just read "The Music of the Primes" by Marcus du Sautoy, it is worth a read. I'm not from a maths background, but I'd like to develop a deeper understanding of the concepts. The poetry of math is ...
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0answers
60 views

Do points on elliptic curves exist where the denominators of point multiples grows more slowly than normal?

Looking at prime multiples of $P=[1,1]$ on the curve $y^2=x^3+x-1$ the size of the denominator grows quite rapidly. So ...
2
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1answer
76 views

Given a real number, how do I produce an elliptic curve with j-invariant equal to that number?

I have formula for computing the j-invariant but I was wondering if given number $j$, is there a formula for getting a curve $y^2=x^3+a_2x^2+a_4x+a_6$ with j-invariant j?
2
votes
1answer
81 views

$H^{0}$ cohomology group and elliptic curve

Let $E$ be an elliptic curve with good reduction at $\ell$. Is it possible that one can have $H^{0}(\mathbb{Q}_{\ell}, E[\ell]) = E[\ell]$?
3
votes
1answer
90 views

projective cubic curve to complex projectie space

Suppose we are given the equation $$ y^2z = x(x - z)(x - 2z) $$ I would like to define a degree two map $g$ on this curve into complex projective space. I hate to say I am already lost here - how do I ...
10
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2answers
125 views

Solve : $ab(a+b)(a-b)=c^2-1$

As we know that $ab(a+b)(a-b)=c^2$ has no integer solution in $Z^+$.However, it seems that $$ab(a+b)(a-b)=c^2-1$$ has infinite positive integer solutions,could you prove it? Here are some of them: ...
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0answers
84 views

Good source of problems for Knapp's Elliptic Curves?

I'm studying elliptic curves (and eventually modular forms) out of Knapp's book because of the softer algebraic geometry prereqs. It's incredibly accessible but the problem is that I don't know I can ...
9
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2answers
201 views

Intuitively, what is the height of a point on an abelian variety?

I have been reading through Silverman's classic text on elliptic curves and I just can't seem to wrap my head around the height functions. It just kind of shows up. What exactly does the height ...
2
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0answers
172 views

A strong form of the Nagell-Lutz theorem

The motivation of this question can be found in Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order? Given the elliptic curve: $$C:y²=x³+ax+b$$ ...
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0answers
52 views

Is it possible to say that there is a curve $C$ such that its rank exactly $r$?

The motivation of this question can be found in Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order? Given the elliptic curve: $$C:y²=x³+ax+b$$ ...
3
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1answer
110 views

Finding number of solutions to an equation in $\mathbb F_p$

$p=3 \pmod 4$ is a prime. Let $b\in \mathbb F_p^*$. Show that the equation $v^2=u^4-4b$ has $p-1$ solutions $(u,v)$ with $u, v \in \mathbb F_p$. If we write the given equation as $v+u^2=x$ and ...
2
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1answer
182 views

Elliptic curve as an intersection of quadrics

Let $E$ be an elliptic curve. If one starts with embedding associated with invertible sheaf $\mathcal{O}(3x)$ where $x$ is some point on $E$ then one gets cubic in $\mathbb{P}^2$ and this embedding is ...
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0answers
58 views

Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not?

The motivation to this question can be found in How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$ We know that there is an isomorphism ...
4
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1answer
68 views

How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$

Let $C$ be an elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the ...
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0answers
48 views

Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order?

Let $C$ an elliptic curve over $\mathbb Q$. Assume that the rank of $C(ℚ)$ is equal to $r$. Then the cardinality of a maximal independent set in $C(ℚ)$ is $r$, thus there exists $r$ independent points ...
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1answer
63 views

What is the point $\{∞\}$?

The set of all rational points in an elliptic curve $C$ over $ℚ$ is denoted by $C(ℚ)$ and called the Mordell-Weil group, i.e., $C(ℚ)=\{\text{points on } $C$ \text{ with coordinates in } ℚ\}∪\{∞\}$. ...
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125 views

What is the complex *algebraic* moduli of elliptic curves?

It's well-known that $SL_2(\mathbb{Z}) \backslash \mathfrak{h}$ is a coarse moduli space for complex elliptic curves. Thus, I would expect this to be related to the pullback of $\mathcal{M}_{ell} ...
3
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1answer
118 views

Stark's formula for the j-invariant

In his paper On the "gap" in Heegner's proof (which you can find : here) Stark gives the following formula for the $j$-invariant (for some $\tau \in \mathcal{H}$ and $q=e^{2i\pi\tau}$) $$ j(\tau) = ...
2
votes
1answer
120 views

Inverse Scalar Multiplication of a point over elliptic curve

I was implementing point arithmetic operation, and was exploring the properties of point arithmetic, and I am unable to conclude whether $$ k^{-1}(kP) = P $$ where P is a point over elliptic curve $ ...
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3answers
282 views

Diophantine equation $x^2 + 32x = y^3$

I am trying to find all solutions to the Diophantine equation $x^2 + 32x = y^3$. I think that the first step is to factorise: $x(x+32)=y^3$. If $x$ is odd, then $x+32$ is also odd. The common ...
10
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3answers
331 views

Integer solutions for $x^3+2=y^2$?

I've heard a famous result that $26$ is the only integer, such that $26-1=25$ is a square number and $26+1=27$ is a cubic number.In other words, $(x,y)=(5,3)$ is the only solution for $x^2+2=y^3$. ...
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0answers
98 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
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0answers
76 views

Two quartic polynomials to be made a square?

Given two generally non-square quartic polynomials that are to be simultaneously made squares for particular values of $x$, $$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5 = y_1^2$$ ...
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vote
1answer
51 views

What is the upper bound of order of $ E(F_{599})$

Can you please help me to slove this problem: Let E be the elliptic curve $y^{2}=x^{3}+1$ over the finite field $F_{599}$. Using Hasse's theorem find What is the upper bound of order of $ E(F_{599})$ ...
4
votes
2answers
149 views

Rank of the elliptic curve $y^2=x^3+px$

I need to prove that the rank of the curve $y^2=x^3+px$ is $0$, if $p\equiv 7 \pmod {16}$ is a prime. Using the standard technique, we need to show that none of the following two equations admits an ...
6
votes
1answer
91 views

Why is $H^1(\text{Gal}(\overline{K}/K),A)=\lim_{\rightarrow}H^1\left(L/K, A^{\text{Gal}(\overline{K}/L)}\right)$

I'm doing a course on elliptic curves, and I'm stuck on this. Here are the definitions: Let $G$ be a group and let $A$ be a $G$-module (that is, a $\mathbb{Z}[G]$-module). We have the "cochains" ...