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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1680e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1680.k4 | 1680e1 | \([0, 1, 0, 9, 84]\) | \(4499456/180075\) | \(-2881200\) | \([2]\) | \(256\) | \(-0.080674\) | \(\Gamma_0(N)\)-optimal |
| 1680.k3 | 1680e2 | \([0, 1, 0, -236, 1260]\) | \(5702413264/275625\) | \(70560000\) | \([2, 2]\) | \(512\) | \(0.26590\) | |
| 1680.k2 | 1680e3 | \([0, 1, 0, -656, -4956]\) | \(30534944836/8203125\) | \(8400000000\) | \([2]\) | \(1024\) | \(0.61247\) | |
| 1680.k1 | 1680e4 | \([0, 1, 0, -3736, 86660]\) | \(5633270409316/14175\) | \(14515200\) | \([4]\) | \(1024\) | \(0.61247\) |
Rank
sage: E.rank()
The elliptic curves in class 1680e have rank \(0\).
Complex multiplication
The elliptic curves in class 1680e do not have complex multiplication.Modular form 1680.2.a.e
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
