Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 485520q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.q4 | 485520q1 | \([0, -1, 0, 687724, -75522240]\) | \(5821462825904/3765234375\) | \(-23266202759100000000\) | \([2]\) | \(9437184\) | \(2.4050\) | \(\Gamma_0(N)\)-optimal* |
485520.q3 | 485520q2 | \([0, -1, 0, -2924776, -618842240]\) | \(111945903743524/58068950625\) | \(1435282741727775360000\) | \([2, 2]\) | \(18874368\) | \(2.7516\) | \(\Gamma_0(N)\)-optimal* |
485520.q2 | 485520q3 | \([0, -1, 0, -26333776, 51573864160]\) | \(40854477373889762/406081190025\) | \(20074112499364964812800\) | \([2]\) | \(37748736\) | \(3.0982\) | \(\Gamma_0(N)\)-optimal* |
485520.q1 | 485520q4 | \([0, -1, 0, -37315776, -87641828640]\) | \(116245908353453762/128063994975\) | \(6330682398975417907200\) | \([2]\) | \(37748736\) | \(3.0982\) |
Rank
sage: E.rank()
The elliptic curves in class 485520q have rank \(1\).
Complex multiplication
The elliptic curves in class 485520q do not have complex multiplication.Modular form 485520.2.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.