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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 494190gg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
494190.gg4 | 494190gg1 | \([1, -1, 1, 32458, 1953861]\) | \(214921799/218880\) | \(-3851475473882880\) | \([2]\) | \(4718592\) | \(1.6775\) | \(\Gamma_0(N)\)-optimal* |
494190.gg3 | 494190gg2 | \([1, -1, 1, -175622, 18100869]\) | \(34043726521/11696400\) | \(205813220635616400\) | \([2, 2]\) | \(9437184\) | \(2.0240\) | \(\Gamma_0(N)\)-optimal* |
494190.gg1 | 494190gg3 | \([1, -1, 1, -2516522, 1536876789]\) | \(100162392144121/23457780\) | \(412769848052541780\) | \([2]\) | \(18874368\) | \(2.3706\) | \(\Gamma_0(N)\)-optimal* |
494190.gg2 | 494190gg4 | \([1, -1, 1, -1164002, -469763499]\) | \(9912050027641/311647500\) | \(5483839102462147500\) | \([2]\) | \(18874368\) | \(2.3706\) |
Rank
sage: E.rank()
The elliptic curves in class 494190gg have rank \(0\).
Complex multiplication
The elliptic curves in class 494190gg do not have complex multiplication.Modular form 494190.2.a.gg
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.