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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 131859bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
131859.bd2 | 131859bc1 | \([1, -1, 0, 7488, -250965]\) | \(541343375/625807\) | \(-53673038884647\) | \([2]\) | \(276480\) | \(1.3208\) | \(\Gamma_0(N)\)-optimal |
131859.bd1 | 131859bc2 | \([1, -1, 0, -43227, -2350566]\) | \(104154702625/32188247\) | \(2760661086979887\) | \([2]\) | \(552960\) | \(1.6674\) |
Rank
sage: E.rank()
The elliptic curves in class 131859bc have rank \(1\).
Complex multiplication
The elliptic curves in class 131859bc do not have complex multiplication.Modular form 131859.2.a.bc
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.