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SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 346560.hc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 346560.hc1 | 346560hc3 | \([0, 1, 0, -43898081, 111933388575]\) | \(3034301922374404/1425\) | \(4393558371532800\) | \([2]\) | \(11796480\) | \(2.7752\) | |
| 346560.hc2 | 346560hc4 | \([0, 1, 0, -3292801, 998227199]\) | \(1280615525284/601171875\) | \(1853532437990400000000\) | \([2]\) | \(11796480\) | \(2.7752\) | |
| 346560.hc3 | 346560hc2 | \([0, 1, 0, -2744081, 1747668975]\) | \(2964647793616/2030625\) | \(1565205169858560000\) | \([2, 2]\) | \(5898240\) | \(2.4287\) | |
| 346560.hc4 | 346560hc1 | \([0, 1, 0, -137661, 38378739]\) | \(-5988775936/9774075\) | \(-470865888599116800\) | \([2]\) | \(2949120\) | \(2.0821\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 346560.hc have rank \(2\).
Complex multiplication
The elliptic curves in class 346560.hc do not have complex multiplication.Modular form 346560.2.a.hc
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.
