Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 25872q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25872.br4 | 25872q1 | \([0, 1, 0, 1356, -760068]\) | \(9148592/8301447\) | \(-250024176154368\) | \([2]\) | \(98304\) | \(1.4417\) | \(\Gamma_0(N)\)-optimal |
25872.br3 | 25872q2 | \([0, 1, 0, -117224, -15131964]\) | \(1478729816932/38900169\) | \(4686403566265344\) | \([2, 2]\) | \(196608\) | \(1.7883\) | |
25872.br2 | 25872q3 | \([0, 1, 0, -268144, 31653236]\) | \(8849350367426/3314597517\) | \(798636202552387584\) | \([4]\) | \(393216\) | \(2.1349\) | |
25872.br1 | 25872q4 | \([0, 1, 0, -1863584, -979821228]\) | \(2970658109581346/2139291\) | \(515451795167232\) | \([2]\) | \(393216\) | \(2.1349\) |
Rank
sage: E.rank()
The elliptic curves in class 25872q have rank \(1\).
Complex multiplication
The elliptic curves in class 25872q do not have complex multiplication.Modular form 25872.2.a.q
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.