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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 9702.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9702.bm1 | 9702be4 | \([1, -1, 1, -64910, -1574639]\) | \(13060888875/7086244\) | \(16409510829167148\) | \([2]\) | \(69120\) | \(1.8023\) | |
9702.bm2 | 9702be2 | \([1, -1, 1, -50210, -4317855]\) | \(4406910829875/7744\) | \(24598994112\) | \([2]\) | \(23040\) | \(1.2530\) | |
9702.bm3 | 9702be3 | \([1, -1, 1, -38450, 2891809]\) | \(2714704875/21296\) | \(49314833446032\) | \([2]\) | \(34560\) | \(1.4557\) | |
9702.bm4 | 9702be1 | \([1, -1, 1, -3170, -65439]\) | \(1108717875/45056\) | \(143121420288\) | \([2]\) | \(11520\) | \(0.90644\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9702.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 9702.bm do not have complex multiplication.Modular form 9702.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.