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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 317130.cu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 317130.cu1 | 317130cu5 | \([1, 0, 0, -164412705, 811416498525]\) | \(553808571467029327441/12529687500\) | \(11120143778029687500\) | \([2]\) | \(47185920\) | \(3.1788\) | |
| 317130.cu2 | 317130cu4 | \([1, 0, 0, -11363845, -14713359535]\) | \(182864522286982801/463015182960\) | \(410927679235888475760\) | \([2]\) | \(23592960\) | \(2.8322\) | |
| 317130.cu3 | 317130cu3 | \([1, 0, 0, -10287525, 12647340657]\) | \(135670761487282321/643043610000\) | \(570703570918528410000\) | \([2, 2]\) | \(23592960\) | \(2.8322\) | |
| 317130.cu4 | 317130cu6 | \([1, 0, 0, -5002025, 25627471557]\) | \(-15595206456730321/310672490129100\) | \(-275722978575012415217100\) | \([2]\) | \(47185920\) | \(3.1788\) | |
| 317130.cu5 | 317130cu2 | \([1, 0, 0, -985045, -35660575]\) | \(119102750067601/68309049600\) | \(60624532965611577600\) | \([2, 2]\) | \(11796480\) | \(2.4856\) | |
| 317130.cu6 | 317130cu1 | \([1, 0, 0, 245035, -4416543]\) | \(1833318007919/1070530560\) | \(-950099812622991360\) | \([2]\) | \(5898240\) | \(2.1390\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 317130.cu have rank \(1\).
Complex multiplication
The elliptic curves in class 317130.cu do not have complex multiplication.Modular form 317130.2.a.cu
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
