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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 286110.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286110.bm1 | 286110bm4 | \([1, -1, 0, -15000390150, -707130619050700]\) | \(785681552361835673854227/2604236800\) | \(1237272306338392473600\) | \([2]\) | \(278691840\) | \(4.1533\) | |
| 286110.bm2 | 286110bm3 | \([1, -1, 0, -937511430, -11049060743884]\) | \(-191808834096148160787/11043434659840\) | \(-5246733273824245768519680\) | \([2]\) | \(139345920\) | \(3.8067\) | |
| 286110.bm3 | 286110bm2 | \([1, -1, 0, -185852775, -962666691875]\) | \(1089365384367428097483/16063552169500000\) | \(10468847669662960528500000\) | \([2]\) | \(92897280\) | \(3.6040\) | |
| 286110.bm4 | 286110bm1 | \([1, -1, 0, -1216455, -40999109699]\) | \(-305460292990923/1114070936704000\) | \(-726056030850860679552000\) | \([2]\) | \(46448640\) | \(3.2574\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 286110.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 286110.bm do not have complex multiplication.Modular form 286110.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
