Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 2640.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2640.g1 | 2640f5 | \([0, -1, 0, -77440, 8320480]\) | \(25078144523224322/1815\) | \(3717120\) | \([4]\) | \(4096\) | \(1.1583\) | |
| 2640.g2 | 2640f4 | \([0, -1, 0, -4840, 131200]\) | \(12247559771044/3294225\) | \(3373286400\) | \([2, 4]\) | \(2048\) | \(0.81174\) | |
| 2640.g3 | 2640f6 | \([0, -1, 0, -4240, 164320]\) | \(-4117122162722/3215383215\) | \(-6585104824320\) | \([4]\) | \(4096\) | \(1.1583\) | |
| 2640.g4 | 2640f3 | \([0, -1, 0, -2320, -41168]\) | \(1349195526724/38671875\) | \(39600000000\) | \([2]\) | \(2048\) | \(0.81174\) | |
| 2640.g5 | 2640f2 | \([0, -1, 0, -340, 1600]\) | \(17029316176/6125625\) | \(1568160000\) | \([2, 2]\) | \(1024\) | \(0.46517\) | |
| 2640.g6 | 2640f1 | \([0, -1, 0, 65, 142]\) | \(1869154304/1804275\) | \(-28868400\) | \([2]\) | \(512\) | \(0.11859\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2640.g have rank \(1\).
Complex multiplication
The elliptic curves in class 2640.g do not have complex multiplication.Modular form 2640.2.a.g
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
