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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 321552.er
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 321552.er1 | 321552er3 | \([0, 0, 0, -7683219, 8196251922]\) | \(16798320881842096017/2132227789307\) | \(6366798063226073088\) | \([2]\) | \(11010048\) | \(2.6300\) | |
| 321552.er2 | 321552er4 | \([0, 0, 0, -3047859, -1964382030]\) | \(1048626554636928177/48569076788309\) | \(145026486184662061056\) | \([2]\) | \(11010048\) | \(2.6300\) | |
| 321552.er3 | 321552er2 | \([0, 0, 0, -521379, 104805090]\) | \(5249244962308257/1448621666569\) | \(4325561118428368896\) | \([2, 2]\) | \(5505024\) | \(2.2834\) | |
| 321552.er4 | 321552er1 | \([0, 0, 0, 84141, 10707282]\) | \(22062729659823/29354283343\) | \(-87651420393664512\) | \([2]\) | \(2752512\) | \(1.9368\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 321552.er have rank \(0\).
Complex multiplication
The elliptic curves in class 321552.er do not have complex multiplication.Modular form 321552.2.a.er
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.
