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Given an order in a Quaternion algebra, determine if a type of maximal orders contains it Let $B$ be a definite quaterni

Computing ideals of a given norm in Quaternion algebra Let $B$ be the quaternion algebra over $\mathbb{Q}$ ramified at p

Computing ideals of a given norm in Quaternion algebra Let $B$ be the quaternion algebra over $\mathbb{Q}$ ramified at p

Computing ideals of a given norm in Quaternion algebra Let $B$ be the quaternion algebra over $\mathbb{Q}$ ramified at p

Computing ideals of a given norm in Quaternion algebra Let $B$ be the quaternion algebra over $\mathbb{Q}$ ramified at p

Computing ideals of a given norm in Quaternion algebra Let $B$ be the quaternion algebra over $\mathbb{Q}$ ramified at p

Computing ideals of a given norm in Quaternion algebra Let $B$ be the quaternion algebra over $\mathbb{Q}$ ramified at p

glitch on isogeny_ell_graph When I try to plot the supersingular $l$-iosgeny graph of elliptic curves over $\mathbb{F}_{

glitch on isogeny_ell_graph When I try to plot the supersingular $l$-iosgeny graph of elliptic curves over $\mathbb{F}_{

glitch on isogeny_ell_graph When I try to plot supersingular $l$-iosgeny graphs of elliptic curves over $\mathbb{F}_{p^2

I have found two elliptic curves over a number field whose reductions are isomorphic to supersingular elliptic curves (used Deuring Lifting Theorem). How can I check which reduction is isomorphic to which supersingular elliptic curve?

sage: H = hilbert_class_polynomial(-64)
sage: S.<l> = H.splitting_field()
sage: R = H.roots(ring = S)
sage: for r in R:
....:     j0 = r[0]
....:     E = EllipticCurve(j=j0)
....:     I = S.ideal(p)
....:     Ebar = E.reduction(I)
....:     Ebar
....:     Ebar.j_invariant()
Elliptic Curve defined by y^2 = x^3 + (67*lbar+26)*x + (54*lbar+71) over Residue field in lbar of Fractional ideal (179)
lbar
Elliptic Curve defined by y^2 = x^3 + (112*lbar+44)*x + (125*lbar+171) over Residue field in lbar of Fractional ideal (179)
178*lbar + 35


sage: p = 179
sage: F.<i> = GF(p^2, modulus=x^2+1)
sage: Rp = H.roots(ring=F)
sage: for r in Rp:
....:     j0 = r[0]
....:     EllipticCurve(j=j0)
....:     j0
Elliptic Curve defined by y^2 = x^3 + (10*i+35)*x + (155*i+121) over Finite Field in i of size 179^2 
99*i + 107
Elliptic Curve defined by y^2 = x^3 + (169*i+35)*x + (24*i+121) over Finite Field in i of size 179^2
80*i + 107

How to get the numeric value of a generator defining number field? Suppose $K$ is a number field defined by a polynomial

Reductions of elliptic curves over number fields I have found two elliptic curves over a number field whose reductions a

Reductions of elliptic curves over number fields I have found two elliptic curves over a number field whose reductions a

roots() does not return correct values I'm trying to compute the roots of a polynomial over the splitting field. sage:

@David Coudert Thanks, that method would be good enough to generate few examples. Hopefully, someone fix the issue later

How to check two maximal orders in a quaternion algebra are isomorphic Two maximal orders in a quaternion algebra are is

How to check two maximal orders in a quaternion algebra are isomorphic Two maximal orders in a quaternion algebra are is

How to change the label text size of vertices When I have a graph with over 1000 vertices, vertex label text is too smal

I'm using older version but does it work properly on Sage 9.6? It seems there is no ticket saying it has been fixed yet.

Let Q = QuaternionAlgebra(-2167, -7) be a rational quaternion algebra with standard basis {1, i , j, k} as follows.

sage: Q.<i, j, k> = QuaternionAlgebra(-2161,-7)
sage: B = [1/2 + 2/7*j + 1/14*k, 1/32*i + 13/32*j + 19/8*k, 4/7*j + 1/7*k, 4*k]
sage: O = Q.quaternion_order(B)
sage: O
Order of Quaternion Algebra (-2161, -7) with base ring Rational Field
  with basis (1/2 + 2/7*j + 1/14*k, 1/32*i + 13/32*j + 19/8*k, 4/7*j + 1/7*k, 4*k)

I think this (maximal Z-)order doesn't contain k but I get

sage: k in O
True

The is inconsistent with the result on Magma:

> K := Rationals();
> Q<i, j, k> := QuaternionAlgebra<K|-2161, -7>;
> B := [ 1/2 + 2/7*j + 1/14*k, 1/32*i + 13/32*j + 19/8*k, 4/7*j + 1/7*k, 4*k ];
> O := QuaternionOrder(B);
> O;
Order of Quaternion Algebra with base ring Rational Field, defined by i^2 = -2161, j^2 = -7
with coefficient ring Integer Ring
> k in O;
false
> 2*k in O;
false
> 4*k in O;
true

What is the issue here?

something wrong with the checking if an element is in an order Let Q = (-2167, -7) be a rational quaternion algebra with

something wrong with the checking if an element is in an order Let Q = (-2167, -7) be a rational quaternion algebra with

something wrong with the checking if an element is in an order Let Q = (-2167, -7) be a rational quaternion algebra with

something wrong with the checking if an element is in an order Let Q = (-2167, -7) be a rational quaternion algebra with

something wrong with the checking if an element is in an order Let Q = (-2167, -7) be a rational quaternion algebra with

something wrong with the checking if an element is in an order Let Q be a rational quaternion algebra with standard basi

Thank you for your answer. As long as I can evaluate the map point-wise it would be okay.

Let $E: y^2 = x^3 + x$ be an elliptic curve over a field $K$ of characteristic $p\neq 2, 3$. It is well known that the map $[i]$ defined as below is an endomorphism on $E$ and $[i]^2=-1$

$[i]:(x,y)\mapsto(-x, iy)$

I'm wondering how to construct this isogeny on sage when $K$ is a finite field. If I can define an isogeny by giving its rational maps then I can simply compute a fourth root of unity $i$ in $K$ and let $\phi = (-x, iy)$, but as I know of sage doesn't let you define an isogeny from rational maps?

How to construct isogeny [i] such that [i]^2= -1? Let $E: y^2 = x^3 + x$ be an elliptic curve over a field $K$ of chara