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SageMath
sage: E = EllipticCurve("a1")
sage: E.isogeny_class()
Elliptic curves in class 72.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72.a1 | 72a5 | \([0, 0, 0, -3459, 78302]\) | \(3065617154/9\) | \(13436928\) | \([2]\) | \(32\) | \(0.59710\) | |
| 72.a2 | 72a3 | \([0, 0, 0, -579, -5362]\) | \(28756228/3\) | \(2239488\) | \([2]\) | \(16\) | \(0.25053\) | |
| 72.a3 | 72a4 | \([0, 0, 0, -219, 1190]\) | \(1556068/81\) | \(60466176\) | \([2, 2]\) | \(16\) | \(0.25053\) | |
| 72.a4 | 72a2 | \([0, 0, 0, -39, -70]\) | \(35152/9\) | \(1679616\) | \([2, 2]\) | \(8\) | \(-0.096046\) | |
| 72.a5 | 72a1 | \([0, 0, 0, 6, -7]\) | \(2048/3\) | \(-34992\) | \([4]\) | \(4\) | \(-0.44262\) | \(\Gamma_0(N)\)-optimal |
| 72.a6 | 72a6 | \([0, 0, 0, 141, 4718]\) | \(207646/6561\) | \(-9795520512\) | \([2]\) | \(32\) | \(0.59710\) |
Rank
sage: E.rank()
The elliptic curves in class 72.a have rank \(0\).
Complex multiplication
The elliptic curves in class 72.a do not have complex multiplication.Modular form 72.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
