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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 663.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
663.b1 | 663c2 | \([1, 0, 0, -98, 279]\) | \(104154702625/24649677\) | \(24649677\) | \([2]\) | \(128\) | \(0.12990\) | |
663.b2 | 663c1 | \([1, 0, 0, -33, -72]\) | \(3981876625/232713\) | \(232713\) | \([2]\) | \(64\) | \(-0.21668\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 663.b have rank \(1\).
Complex multiplication
The elliptic curves in class 663.b do not have complex multiplication.Modular form 663.2.a.b
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.