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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 281775.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 281775.bm1 | 281775bm6 | \([1, 1, 0, -145757300, -677276009625]\) | \(908031902324522977/161726530797\) | \(60995082753800195203125\) | \([2]\) | \(37748736\) | \(3.3765\) | |
| 281775.bm2 | 281775bm4 | \([1, 1, 0, -10035675, -8304120000]\) | \(296380748763217/92608836489\) | \(34927377824421175640625\) | \([2, 2]\) | \(18874368\) | \(3.0299\) | |
| 281775.bm3 | 281775bm2 | \([1, 1, 0, -3930550, 2898784375]\) | \(17806161424897/668584449\) | \(252156301094757515625\) | \([2, 2]\) | \(9437184\) | \(2.6833\) | |
| 281775.bm4 | 281775bm1 | \([1, 1, 0, -3894425, 2956476000]\) | \(17319700013617/25857\) | \(9751955025515625\) | \([2]\) | \(4718592\) | \(2.3368\) | \(\Gamma_0(N)\)-optimal |
| 281775.bm5 | 281775bm3 | \([1, 1, 0, 1596575, 10410147250]\) | \(1193377118543/124806800313\) | \(-47070824284754048390625\) | \([2]\) | \(18874368\) | \(3.0299\) | |
| 281775.bm6 | 281775bm5 | \([1, 1, 0, 28003950, -56196007875]\) | \(6439735268725823/7345472585373\) | \(-2770341427610143409953125\) | \([2]\) | \(37748736\) | \(3.3765\) |
Rank
sage: E.rank()
The elliptic curves in class 281775.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 281775.bm do not have complex multiplication.Modular form 281775.2.a.bm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
