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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 63525.t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63525.t1 | 63525bj6 | \([1, 0, 0, -40081313, 97661942742]\) | \(257260669489908001/14267882475\) | \(394944127270210546875\) | \([2]\) | \(5898240\) | \(3.0175\) | |
| 63525.t2 | 63525bj4 | \([1, 0, 0, -2646938, 1343295867]\) | \(74093292126001/14707625625\) | \(407116499372666015625\) | \([2, 2]\) | \(2949120\) | \(2.6710\) | |
| 63525.t3 | 63525bj2 | \([1, 0, 0, -816813, -265384008]\) | \(2177286259681/161417025\) | \(4468126659781640625\) | \([2, 2]\) | \(1474560\) | \(2.3244\) | |
| 63525.t4 | 63525bj1 | \([1, 0, 0, -801688, -276349633]\) | \(2058561081361/12705\) | \(351682539140625\) | \([2]\) | \(737280\) | \(1.9778\) | \(\Gamma_0(N)\)-optimal |
| 63525.t5 | 63525bj3 | \([1, 0, 0, 771312, -1172203383]\) | \(1833318007919/22507682505\) | \(-623027070722504765625\) | \([2]\) | \(2949120\) | \(2.6710\) | |
| 63525.t6 | 63525bj5 | \([1, 0, 0, 5505437, 7987481492]\) | \(666688497209279/1381398046875\) | \(-38237982895623779296875\) | \([2]\) | \(5898240\) | \(3.0175\) |
Rank
sage: E.rank()
The elliptic curves in class 63525.t have rank \(1\).
Complex multiplication
The elliptic curves in class 63525.t do not have complex multiplication.Modular form 63525.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
