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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 54720.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54720.bm1 | 54720bg2 | \([0, 0, 0, -932268, -346461392]\) | \(468898230633769/5540400\) | \(1058786928230400\) | \([2]\) | \(589824\) | \(2.0330\) | |
54720.bm2 | 54720bg1 | \([0, 0, 0, -56748, -5709008]\) | \(-105756712489/12476160\) | \(-2384231305052160\) | \([2]\) | \(294912\) | \(1.6864\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54720.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 54720.bm do not have complex multiplication.Modular form 54720.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.