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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 10780h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10780.l4 | 10780h1 | \([0, -1, 0, -121781, 14882750]\) | \(106110329552896/10850811125\) | \(20425393248722000\) | \([2]\) | \(82944\) | \(1.8653\) | \(\Gamma_0(N)\)-optimal |
| 10780.l3 | 10780h2 | \([0, -1, 0, -447876, -98989624]\) | \(329890530231376/49933296875\) | \(1503898225676000000\) | \([2]\) | \(165888\) | \(2.2119\) | |
| 10780.l2 | 10780h3 | \([0, -1, 0, -9608181, 11466509090]\) | \(52112158467655991296/71177645\) | \(133983660105680\) | \([2]\) | \(248832\) | \(2.4146\) | |
| 10780.l1 | 10780h4 | \([0, -1, 0, -9610876, 11459757576]\) | \(3259751350395879376/3806353980275\) | \(114640317292895609600\) | \([2]\) | \(497664\) | \(2.7612\) |
Rank
sage: E.rank()
The elliptic curves in class 10780h have rank \(0\).
Complex multiplication
The elliptic curves in class 10780h do not have complex multiplication.Modular form 10780.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
