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SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 31680dc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.d3 | 31680dc1 | \([0, 0, 0, -9948, 381872]\) | \(9115564624/825\) | \(9853747200\) | \([2]\) | \(49152\) | \(0.95708\) | \(\Gamma_0(N)\)-optimal |
| 31680.d2 | 31680dc2 | \([0, 0, 0, -10668, 323408]\) | \(2810381476/680625\) | \(32517365760000\) | \([2, 2]\) | \(98304\) | \(1.3037\) | |
| 31680.d4 | 31680dc3 | \([0, 0, 0, 25332, 2037008]\) | \(18814587262/29648025\) | \(-2832912905011200\) | \([2]\) | \(196608\) | \(1.6502\) | |
| 31680.d1 | 31680dc4 | \([0, 0, 0, -58188, -5131888]\) | \(228027144098/12890625\) | \(1231718400000000\) | \([2]\) | \(196608\) | \(1.6502\) |
Rank
sage: E.rank()
The elliptic curves in class 31680dc have rank \(2\).
Complex multiplication
The elliptic curves in class 31680dc do not have complex multiplication.Modular form 31680.2.a.dc
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.
