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

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Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$.

$$\Large\textbf{Problem}$$ Let $E:y^2 = x^3 + 1$ be an elliptic curve. For each prime $5 \leq p \leq 13$, describe the group $E(\mathbb{F}_p)$, the Mordell-Weil group. $$\Large\textbf{Attempts and ...
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1answer
79 views

Is $y^2 =$ quartic in $x$ smooth at infinity?

Let $q(x)\in K(x)$ be a quartic polynomial in x with distinct roots over the algebraically closed field $K$. Consider the curve $C\subset \Bbb P^2$ given by $y^2-q(x)$. Is $C$ smooth? Well, at least ...
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1answer
82 views

Trivial torsion subgroup

I am just wondering, suppose we have a curve $y^2 = x^3+ax + b$ defined over $\mathbb{Q}$ and suppose for simplicity $a,b \in \mathbb{Z}$. Can we say something about the torsion subgroup with the only ...
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2answers
224 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
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0answers
70 views

Elliptic curves in $\Bbb P^3$

How can I check that a curve inside of $\Bbb P^3$ is an elliptic curve? Specifically, let $C$ be the plane cubic $$C:aX^3+bY^3+cZ^3=0$$ and $\phi:\Bbb P^2\to \Bbb P^3$ given by ...
4
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1answer
127 views

Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $ E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650 $ $ [\text{Cremona}:35a2] $ $ E_{2}: y^{2}+y=x^{3}+x^{2}-x $ $ [\text{Cremona}:35a3] $ We know that ...
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1answer
536 views

Finding points on an elliptic curve

I have an elliptic curve $$x^3+17x+5 \mod 59$$ $P = (4,14)$ is given and I need to find point $8P$. to calculate $8P$, I first calculated $2P$ by using the equation sigma = 3x^2+a/2y = ...
2
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1answer
189 views

Order of a point on an Elliptic Curve

I am currently struggling with the determination the order of a point on an elliptic curve. We had to do the following exercise: $C = V(y^2+x^3-1)$ and $P = (0,1)$. Now Wikipedia told me that I can ...
3
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1answer
237 views

Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
3
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1answer
94 views

Isogeny of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime. Then what does it mean by "$E$ has a $\mathbb{Q}$-isogeny of degree $p$"?
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Rationality of divisors at infinity

In an attempt to clarify to myself some terminology (ant the scope of the Riemann-Roch theorem), I would like to ask for examples of genus $1$ curves of the form $$C : y^2 = ax^4 + b$$ where $a, b \in ...
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2answers
151 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
4
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1answer
34 views

Part of verifying that the Weil pairing $e_m$ is well-defined.

As part of a homework problem, I need to show that the value of $e_m(P,Q)$ is independent of the choice of a point $S \in E[m] \setminus \{\mathcal{O},P,-Q,P-Q\}$, where $E[m]$ is the collection of ...
0
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1answer
74 views

Solving an equation in charcateristic 2 in sage OR finding 3-torsion points of an elliptic curve over field with char 2

Problem: show that an elliptic curve over a field of char 2 has nontrivial 3-torsion points Method: I used SAGE to unwind the duplication formula for an elliptic curve given in short Weierstrass form ...
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2answers
58 views

Geometric picture of 3-torsion points on an elliptic curve

I'm faced with what seems a paradox. If we have an elliptic curve $E/\Bbb C$ in Weierstrass form so that $\mathcal O_E$ is at infinity, then the addition law is quite easy to picture geometrically. In ...
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1answer
100 views

Silverman exercise 3.1 proving that two polynomials are relatively prime iff the discriminant is non-zero

Silverman, p. 104: Show that the polynomials $$f=x^4−b_4x^2−2b_6x−b8 \qquad \text{and}\qquad g=4x^3+b_2x^2+2b_4x+b_6$$ appearing in the duplication formula (III.2.3d) are relatively prime if ...
3
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1answer
96 views

rational points on particular elliptic curve

I do have a few books that discuss elliptic curves, however... What are the rational points on $$ y^2 = 4 x^3 - 4 x = 4 x(x-1)(x+1)? $$ I think it ought to be $(-1,0), (0,0), (1,0).$ Maybe it's ...
4
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2answers
135 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
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1answer
35 views

Why does an isogeny not ramify?

The following argument is, I believe, based on the premise that an isogeny (or a morphism of curves that is a group homomorphism) doesn't ramify: Considering the multiplication by $n$ map $[n]$ on a ...
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1answer
89 views

Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In ...
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1answer
92 views

Odd torsion of elliptic curves are isomorphic

$C: Y^2=X(X^2+aX+b)$ $D: Y^2=X(X^2+a_1X+b_1)$ where $a,b,\in\mathbb Z a_1=-2a,b_1=a^2-4b,b(a^2-4b)\neq0$ Let $C_{oddtors}(\mathbb Q)$ denote the set of torsion elements of $C(\mathbb Q)$ which ...
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1answer
101 views

Finding J-invariant of Legendre form of Elliptic Curve

PROBLEM: Put the Legendre equation $y^2 = x(x − 1)(x − λ)$ into Weierstrass form and use this to show that the j-invariant is j = $2^8\frac{(λ2 − λ + 1)^3}{λ^2(λ − 1)^2}$ . Recall: Weierstrass ...
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1answer
39 views

How to prove $-P = (x, -a_1x - a_3 - y)$ for an Elliptic Curve of the General Weirstrass equation for P not the identity?

Let $P = (x,y) \ne \{\infty\}$. Then $-P$ is the other finite point of intersection of the curve and the vertical line through $P$. General Weirstrass equation: E: $a_1y^2+a_3xy+a_5y = ...
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1answer
160 views

solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
3
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3answers
61 views

Elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup

Show that an elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup. We've been told that for this problem, we are not allowed to use Mazur's Theorem. ...
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2answers
48 views

Is torsion subgroup of elliptic curve birationally invariant?

It's probably a very basic question: Having two birationally equivalent elliptic curves over $\mathbb{Q}$ - is the torsion subgroup unchanged under the birational equivalence?
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Show that the curve $2Y^2 = X^4 - 17$ has no points in $\mathbb{Q}$

There is a hint to show that if there were points in $\mathbb{Q}$, then there would exist $r,s, t \in \mathbb{Z}$ with $\gcd(r, t) = 1$ such that $2s^2 = t^4 - 17r^4$ , and then show that any prime ...
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1answer
142 views

Discriminants and Weierstrass form of elliptic curves

I'm confused by what appears to be contradictory information. In this post, the claim is made that "Every elliptic curve over $\mathbb{Q}$ can be written in the form $y^{2}= x^{3}+ax+b$ where ...
3
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0answers
65 views

Number of points over elliptic curve is p+1 given…

Suppose that -1 is not a square in $\mathbb{Z_p}$. Show that the number of points on the elliptic curve $y^2=x^3+ax$ over $\mathbb{Z_p}$ is $p+1$. Hint: Use the fact that $x^3+ax$ is an odd function. ...
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1answer
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Show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$

I've been asked to show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$ - I've managed to show that it is birationally equivalent to the curve $Y^2 = 2X^4 - 34$ (as suggested in the ...
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1answer
50 views

Why are we not allowed to rescale the variables of the equation of an elliptic curve independently one from the other?

It seems that in whatever proof of the theorem that an elliptic curve can be put in Weierstrass form that you look at, the next step after getting an equation: $$\alpha Y^2Z + a_1XY Z + a_3Y Z^2= ...
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2answers
136 views

Elliptic curves on a K3 surface

Let $X$ be an elliptic K3 surface. Let $\alpha$ be a smooth curve of genus $\geq3$. Define $$d(\alpha)=\min\lbrace \epsilon\cdot \alpha \ | \ \epsilon \mbox{ is an elliptic curve on } X \rbrace, $$ ...
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4answers
488 views

Two circles intersect in two points

Take for example two circles $$\begin{cases}x^2+y^2=1\\x^2+y^2-x-y=0\end{cases}$$ These two circles intersect in two points namely $(0,1)$ and $(1,0)$. But by Bezout's theorem they must intersect four ...
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Is it possible to do elliptic curve cryptography over $\mathbb{Q}$ instead of a finite field?

Whenever I read about elliptic curve cryptography (ECC), the writer always works over a finite field. But as I understand it there is no group-theoretic reason not to use $\mathbb{Q}$ as the ...
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1answer
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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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215 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 ...
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1answer
75 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 ...
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152 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 ...
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1answer
209 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 ...
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1answer
58 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
112 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
26 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 ...
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0answers
53 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
216 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 $\tilde{E}(\mathbb{F}_p)$ ...
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69 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 ...
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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
52 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
75 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
92 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
67 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 ...