Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 38640.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38640.b1 | 38640bm4 | \([0, -1, 0, -2077376, 1153138560]\) | \(242052349717010282689/167676468750\) | \(686802816000000\) | \([2]\) | \(497664\) | \(2.1611\) | |
| 38640.b2 | 38640bm3 | \([0, -1, 0, -130656, 17811456]\) | \(60221998378106209/1554376834500\) | \(6366727514112000\) | \([2]\) | \(248832\) | \(1.8145\) | |
| 38640.b3 | 38640bm2 | \([0, -1, 0, -31136, 865536]\) | \(815016062816929/394524156600\) | \(1615970945433600\) | \([2]\) | \(165888\) | \(1.6118\) | |
| 38640.b4 | 38640bm1 | \([0, -1, 0, -16416, -794880]\) | \(119451676585249/1567702080\) | \(6421307719680\) | \([2]\) | \(82944\) | \(1.2652\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 38640.b have rank \(1\).
Complex multiplication
The elliptic curves in class 38640.b do not have complex multiplication.Modular form 38640.2.a.b
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
