Show commands:
SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 121296.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121296.cs1 | 121296cm4 | \([0, 1, 0, -660028, -206611384]\) | \(2640279346000/3087\) | \(37179042469632\) | \([2]\) | \(995328\) | \(1.8870\) | |
| 121296.cs2 | 121296cm3 | \([0, 1, 0, -40913, -3294018]\) | \(-10061824000/352947\) | \(-265675240980912\) | \([2]\) | \(497664\) | \(1.5405\) | |
| 121296.cs3 | 121296cm2 | \([0, 1, 0, -10228, -130936]\) | \(9826000/5103\) | \(61459233470208\) | \([2]\) | \(331776\) | \(1.3377\) | |
| 121296.cs4 | 121296cm1 | \([0, 1, 0, 2407, -14694]\) | \(2048000/1323\) | \(-995867209008\) | \([2]\) | \(165888\) | \(0.99115\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121296.cs have rank \(0\).
Complex multiplication
The elliptic curves in class 121296.cs do not have complex multiplication.Modular form 121296.2.a.cs
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.
