# Tagged Questions

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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
### Counting elements of $y^2 - y = x^3$ in finite fields
The problem I have to solve is the following: Let $p$ be a prime number with $p \equiv 2$ mod $3$. Let $E$ be the elliptic curve given by $y^2 - y = x^3$. Show that $\#E(\mathbb{F}_p) = p+1$ and ...