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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 306.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 306.a1 | 306b4 | \([1, -1, 0, -1017, 8883]\) | \(159661140625/48275138\) | \(35192575602\) | \([6]\) | \(288\) | \(0.72879\) | |
| 306.a2 | 306b3 | \([1, -1, 0, -927, 11097]\) | \(120920208625/19652\) | \(14326308\) | \([6]\) | \(144\) | \(0.38222\) | |
| 306.a3 | 306b2 | \([1, -1, 0, -387, -2835]\) | \(8805624625/2312\) | \(1685448\) | \([2]\) | \(96\) | \(0.17949\) | |
| 306.a4 | 306b1 | \([1, -1, 0, -27, -27]\) | \(3048625/1088\) | \(793152\) | \([2]\) | \(48\) | \(-0.16709\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 306.a have rank \(1\).
Complex multiplication
The elliptic curves in class 306.a do not have complex multiplication.Modular form 306.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
