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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 931.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
931.a1 | 931b3 | \([0, -1, 1, -37697, 2829742]\) | \(-50357871050752/19\) | \(-2235331\) | \([]\) | \(1134\) | \(1.0064\) | |
931.a2 | 931b2 | \([0, -1, 1, -457, 4157]\) | \(-89915392/6859\) | \(-806954491\) | \([]\) | \(378\) | \(0.45709\) | |
931.a3 | 931b1 | \([0, -1, 1, 33, -8]\) | \(32768/19\) | \(-2235331\) | \([]\) | \(126\) | \(-0.092218\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 931.a have rank \(0\).
Complex multiplication
The elliptic curves in class 931.a do not have complex multiplication.Modular form 931.2.a.a
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.