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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 113715t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 113715.bd3 | 113715t1 | \([1, -1, 0, -8190, 272335]\) | \(1771561/105\) | \(3601126961145\) | \([2]\) | \(193536\) | \(1.1624\) | \(\Gamma_0(N)\)-optimal |
| 113715.bd2 | 113715t2 | \([1, -1, 0, -24435, -1127984]\) | \(47045881/11025\) | \(378118330920225\) | \([2, 2]\) | \(387072\) | \(1.5090\) | |
| 113715.bd4 | 113715t3 | \([1, -1, 0, 56790, -7089899]\) | \(590589719/972405\) | \(-33350036787163845\) | \([2]\) | \(774144\) | \(1.8556\) | |
| 113715.bd1 | 113715t4 | \([1, -1, 0, -365580, -84981425]\) | \(157551496201/13125\) | \(450140870143125\) | \([2]\) | \(774144\) | \(1.8556\) |
Rank
sage: E.rank()
The elliptic curves in class 113715t have rank \(0\).
Complex multiplication
The elliptic curves in class 113715t do not have complex multiplication.Modular form 113715.2.a.t
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.
