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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 126960z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 126960.p3 | 126960z1 | \([0, -1, 0, -105976, 7271536]\) | \(217081801/88320\) | \(53553273718702080\) | \([2]\) | \(811008\) | \(1.9077\) | \(\Gamma_0(N)\)-optimal |
| 126960.p2 | 126960z2 | \([0, -1, 0, -783096, -261409680]\) | \(87587538121/1904400\) | \(1154742464559513600\) | \([2, 2]\) | \(1622016\) | \(2.2542\) | |
| 126960.p4 | 126960z3 | \([0, -1, 0, 63304, -797688720]\) | \(46268279/453342420\) | \(-274886443688392212480\) | \([2]\) | \(3244032\) | \(2.6008\) | |
| 126960.p1 | 126960z4 | \([0, -1, 0, -12463416, -16931562384]\) | \(353108405631241/172500\) | \(104596237731840000\) | \([2]\) | \(3244032\) | \(2.6008\) |
Rank
sage: E.rank()
The elliptic curves in class 126960z have rank \(1\).
Complex multiplication
The elliptic curves in class 126960z do not have complex multiplication.Modular form 126960.2.a.z
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.
