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SageMath
sage: E = EllipticCurve("z1")
sage: E.isogeny_class()
Elliptic curves in class 97020.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 97020.z1 | 97020bu4 | \([0, 0, 0, -21871983, 20758643318]\) | \(52702650535889104/22020583921875\) | \(483486736674351564000000\) | \([2]\) | \(11943936\) | \(3.2417\) | |
| 97020.z2 | 97020bu2 | \([0, 0, 0, -18855543, 31514220542]\) | \(33766427105425744/9823275\) | \(215681073220166400\) | \([2]\) | \(3981312\) | \(2.6924\) | |
| 97020.z3 | 97020bu1 | \([0, 0, 0, -1173648, 496640333]\) | \(-130287139815424/2250652635\) | \(-3088475939558061360\) | \([2]\) | \(1990656\) | \(2.3458\) | \(\Gamma_0(N)\)-optimal |
| 97020.z4 | 97020bu3 | \([0, 0, 0, 4541712, 2379994337]\) | \(7549996227362816/6152409907875\) | \(-8442693321606497646000\) | \([2]\) | \(5971968\) | \(2.8951\) |
Rank
sage: E.rank()
The elliptic curves in class 97020.z have rank \(1\).
Complex multiplication
The elliptic curves in class 97020.z do not have complex multiplication.Modular form 97020.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
