Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1680c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1680.e3 | 1680c1 | \([0, -1, 0, -71, 246]\) | \(2508888064/118125\) | \(1890000\) | \([2]\) | \(384\) | \(-0.034864\) | \(\Gamma_0(N)\)-optimal |
| 1680.e2 | 1680c2 | \([0, -1, 0, -196, -704]\) | \(3269383504/893025\) | \(228614400\) | \([2, 2]\) | \(768\) | \(0.31171\) | |
| 1680.e1 | 1680c3 | \([0, -1, 0, -2896, -59024]\) | \(2624033547076/324135\) | \(331914240\) | \([2]\) | \(1536\) | \(0.65828\) | |
| 1680.e4 | 1680c4 | \([0, -1, 0, 504, -5184]\) | \(13799183324/18600435\) | \(-19046845440\) | \([2]\) | \(1536\) | \(0.65828\) |
Rank
sage: E.rank()
The elliptic curves in class 1680c have rank \(0\).
Complex multiplication
The elliptic curves in class 1680c do not have complex multiplication.Modular form 1680.2.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 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.
