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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 30912bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30912.e2 | 30912bi1 | \([0, -1, 0, -943649, -431338047]\) | \(-354499561600764553/101902222098432\) | \(-26713056109771358208\) | \([2]\) | \(811008\) | \(2.4426\) | \(\Gamma_0(N)\)-optimal |
30912.e1 | 30912bi2 | \([0, -1, 0, -16016929, -24666157631]\) | \(1733490909744055732873/99355964553216\) | \(26045569971838255104\) | \([2]\) | \(1622016\) | \(2.7892\) |
Rank
sage: E.rank()
The elliptic curves in class 30912bi have rank \(0\).
Complex multiplication
The elliptic curves in class 30912bi do not have complex multiplication.Modular form 30912.2.a.bi
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 Cremona 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 Cremona labels.