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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 91035.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 91035.bm1 | 91035bg6 | \([1, -1, 0, -186595794, 979216195675]\) | \(40832710302042509761/91556816413125\) | \(1611060091748668013788125\) | \([2]\) | \(18874368\) | \(3.5271\) | |
| 91035.bm2 | 91035bg4 | \([1, -1, 0, -15905169, 3173063800]\) | \(25288177725059761/14387797265625\) | \(253171821508378344140625\) | \([2, 2]\) | \(9437184\) | \(3.1805\) | |
| 91035.bm3 | 91035bg2 | \([1, -1, 0, -10169964, -12423252677]\) | \(6610905152742241/35128130625\) | \(618124696388622005625\) | \([2, 2]\) | \(4718592\) | \(2.8340\) | |
| 91035.bm4 | 91035bg1 | \([1, -1, 0, -10156959, -12456761360]\) | \(6585576176607121/187425\) | \(3297984241102425\) | \([2]\) | \(2359296\) | \(2.4874\) | \(\Gamma_0(N)\)-optimal |
| 91035.bm5 | 91035bg3 | \([1, -1, 0, -4642839, -25875169502]\) | \(-629004249876241/16074715228425\) | \(-282855315478483755943425\) | \([2]\) | \(9437184\) | \(3.1805\) | |
| 91035.bm6 | 91035bg5 | \([1, -1, 0, 63022176, 25225363993]\) | \(1573196002879828319/926055908203125\) | \(-16295146280558624267578125\) | \([2]\) | \(18874368\) | \(3.5271\) |
Rank
sage: E.rank()
The elliptic curves in class 91035.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 91035.bm do not have complex multiplication.Modular form 91035.2.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.
