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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 429429bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
429429.bc2 | 429429bc1 | \([1, 0, 0, -567564, 164533095]\) | \(-6914490272402831/141776649\) | \(-414584269442343\) | \([2]\) | \(3041280\) | \(1.9243\) | \(\Gamma_0(N)\)-optimal* |
429429.bc1 | 429429bc2 | \([1, 0, 0, -9081069, 10532279484]\) | \(28322170269781231871/11907\) | \(34818532749\) | \([2]\) | \(6082560\) | \(2.2709\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 429429bc have rank \(1\).
Complex multiplication
The elliptic curves in class 429429bc do not have complex multiplication.Modular form 429429.2.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.