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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 217672f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 217672.k4 | 217672f1 | \([0, 0, 0, -24674, 2603445]\) | \(-21511084032/25465531\) | \(-1966676067529264\) | \([2]\) | \(688128\) | \(1.6285\) | \(\Gamma_0(N)\)-optimal |
| 217672.k3 | 217672f2 | \([0, 0, 0, -471679, 124635810]\) | \(9392111857872/4380649\) | \(5413006340874496\) | \([2, 2]\) | \(1376256\) | \(1.9751\) | |
| 217672.k1 | 217672f3 | \([0, 0, 0, -7546019, 7978568078]\) | \(9614292367656708/2093\) | \(10344971506688\) | \([2]\) | \(2752512\) | \(2.3217\) | |
| 217672.k2 | 217672f4 | \([0, 0, 0, -549419, 80774902]\) | \(3710860803108/1577224103\) | \(7795670523266382848\) | \([2]\) | \(2752512\) | \(2.3217\) |
Rank
sage: E.rank()
The elliptic curves in class 217672f have rank \(0\).
Complex multiplication
The elliptic curves in class 217672f do not have complex multiplication.Modular form 217672.2.a.f
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.
