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SageMath
E = EllipticCurve("pm1")
E.isogeny_class()
Elliptic curves in class 331200.pm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 331200.pm1 | 331200pm3 | \([0, 0, 0, -433922700, 3477742346000]\) | \(3026030815665395929/1364501953125\) | \(4074381000000000000000000\) | \([2]\) | \(117964800\) | \(3.6800\) | |
| 331200.pm2 | 331200pm4 | \([0, 0, 0, -238514700, -1393169686000]\) | \(502552788401502649/10024505152875\) | \(29933011994402304000000000\) | \([2]\) | \(117964800\) | \(3.6800\) | |
| 331200.pm3 | 331200pm2 | \([0, 0, 0, -31514700, 35544314000]\) | \(1159246431432649/488076890625\) | \(1457389786176000000000000\) | \([2, 2]\) | \(58982400\) | \(3.3334\) | |
| 331200.pm4 | 331200pm1 | \([0, 0, 0, 6573300, 4083626000]\) | \(10519294081031/8500170375\) | \(-25381372737024000000000\) | \([2]\) | \(29491200\) | \(2.9869\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 331200.pm have rank \(0\).
Complex multiplication
The elliptic curves in class 331200.pm do not have complex multiplication.Modular form 331200.2.a.pm
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
