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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 453299q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
453299.q4 | 453299q1 | \([1, -1, 0, 24078934, -51841081041]\) | \(22062729659823/29354283343\) | \(-2054223576913088525175847\) | \([2]\) | \(54190080\) | \(3.3510\) | \(\Gamma_0(N)\)-optimal* |
453299.q3 | 453299q2 | \([1, -1, 0, -149204911, -507334996008]\) | \(5249244962308257/1448621666569\) | \(101375078611919040499849201\) | \([2, 2]\) | \(108380160\) | \(3.6976\) | \(\Gamma_0(N)\)-optimal* |
453299.q2 | 453299q3 | \([1, -1, 0, -872216816, 9509994947767]\) | \(1048626554636928177/48569076788309\) | \(3398881910405717639146981661\) | \([2]\) | \(216760320\) | \(4.0441\) | \(\Gamma_0(N)\)-optimal* |
453299.q1 | 453299q4 | \([1, -1, 0, -2198734526, -39678354903811]\) | \(16798320881842096017/2132227789307\) | \(149214087258179015980526003\) | \([2]\) | \(216760320\) | \(4.0441\) |
Rank
sage: E.rank()
The elliptic curves in class 453299q have rank \(1\).
Complex multiplication
The elliptic curves in class 453299q do not have complex multiplication.Modular form 453299.2.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.