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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 7728.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7728.c1 | 7728h4 | \([0, -1, 0, -24263848, 45788177008]\) | \(385693937170561837203625/2159357734550274048\) | \(8844729280717922500608\) | \([2]\) | \(691200\) | \(3.0535\) | |
| 7728.c2 | 7728h2 | \([0, -1, 0, -1791928, -879068816]\) | \(155355156733986861625/8291568305839392\) | \(33962263780718149632\) | \([2]\) | \(230400\) | \(2.5042\) | |
| 7728.c3 | 7728h3 | \([0, -1, 0, -670888, 1508909680]\) | \(-8152944444844179625/235342826399858688\) | \(-963964216933821186048\) | \([2]\) | \(345600\) | \(2.7069\) | |
| 7728.c4 | 7728h1 | \([0, -1, 0, 74312, -54937232]\) | \(11079872671250375/324440155855872\) | \(-1328906878385651712\) | \([2]\) | \(115200\) | \(2.1576\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7728.c have rank \(0\).
Complex multiplication
The elliptic curves in class 7728.c do not have complex multiplication.Modular form 7728.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
