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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 422331.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422331.bm1 | 422331bm2 | \([1, 1, 0, -811710, -216739557]\) | \(104154702625/24649677\) | \(13997793641042240757\) | \([2]\) | \(7741440\) | \(2.3853\) | |
422331.bm2 | 422331bm1 | \([1, 1, 0, -273445, 52069984]\) | \(3981876625/232713\) | \(132150557250217233\) | \([2]\) | \(3870720\) | \(2.0388\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 422331.bm have rank \(2\).
Complex multiplication
The elliptic curves in class 422331.bm do not have complex multiplication.Modular form 422331.2.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.