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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 485520dq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.dq4 | 485520dq1 | \([0, -1, 0, 2505, 397482]\) | \(4499456/180075\) | \(-69545163802800\) | \([2]\) | \(1179648\) | \(1.3359\) | \(\Gamma_0(N)\)-optimal* |
485520.dq3 | 485520dq2 | \([0, -1, 0, -68300, 6600000]\) | \(5702413264/275625\) | \(1703146868640000\) | \([2, 2]\) | \(2359296\) | \(1.6825\) | \(\Gamma_0(N)\)-optimal* |
485520.dq1 | 485520dq3 | \([0, -1, 0, -1079800, 432239200]\) | \(5633270409316/14175\) | \(350361641548800\) | \([2]\) | \(4718592\) | \(2.0291\) | \(\Gamma_0(N)\)-optimal* |
485520.dq2 | 485520dq4 | \([0, -1, 0, -189680, -23210928]\) | \(30534944836/8203125\) | \(202755579600000000\) | \([2]\) | \(4718592\) | \(2.0291\) |
Rank
sage: E.rank()
The elliptic curves in class 485520dq have rank \(1\).
Complex multiplication
The elliptic curves in class 485520dq do not have complex multiplication.Modular form 485520.2.a.dq
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.