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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 471510q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
471510.q3 | 471510q1 | \([1, -1, 0, -161892990, -792807867500]\) | \(133358347042307244649/20922470400\) | \(73620812184707174400\) | \([2]\) | \(49545216\) | \(3.2160\) | \(\Gamma_0(N)\)-optimal* |
471510.q2 | 471510q2 | \([1, -1, 0, -162379710, -787800589484]\) | \(134564764541863949929/1669882841640000\) | \(5875889830621701008040000\) | \([2, 2]\) | \(99090432\) | \(3.5626\) | \(\Gamma_0(N)\)-optimal* |
471510.q1 | 471510q3 | \([1, -1, 0, -303832710, 791012924716]\) | \(881535188079627101929/433344383278146600\) | \(1524827845024371076393362600\) | \([2]\) | \(198180864\) | \(3.9092\) | \(\Gamma_0(N)\)-optimal* |
471510.q4 | 471510q4 | \([1, -1, 0, -28714230, -2046154151300]\) | \(-744093657485624809/513584458115625000\) | \(-1807172107740921285365625000\) | \([2]\) | \(198180864\) | \(3.9092\) |
Rank
sage: E.rank()
The elliptic curves in class 471510q have rank \(0\).
Complex multiplication
The elliptic curves in class 471510q do not have complex multiplication.Modular form 471510.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.