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SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 422331.bq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422331.bq1 | 422331bq5 | \([1, 0, 1, -167061067, 830972919389]\) | \(908031902324522977/161726530797\) | \(91839524078878141606677\) | \([2]\) | \(66060288\) | \(3.4106\) | \(\Gamma_0(N)\)-optimal* |
422331.bq2 | 422331bq3 | \([1, 0, 1, -11502482, 10183601495]\) | \(296380748763217/92608836489\) | \(52589710709395698524049\) | \([2, 2]\) | \(33030144\) | \(3.0640\) | \(\Gamma_0(N)\)-optimal* |
422331.bq3 | 422331bq2 | \([1, 0, 1, -4505037, -3559380485]\) | \(17806161424897/668584449\) | \(379668550979874110409\) | \([2, 2]\) | \(16515072\) | \(2.7174\) | \(\Gamma_0(N)\)-optimal* |
422331.bq4 | 422331bq1 | \([1, 0, 1, -4463632, -3630149911]\) | \(17319700013617/25857\) | \(14683395250024137\) | \([2]\) | \(8257536\) | \(2.3709\) | \(\Gamma_0(N)\)-optimal* |
422331.bq5 | 422331bq4 | \([1, 0, 1, 1829928, -12772953581]\) | \(1193377118543/124806800313\) | \(-70873944343373754688833\) | \([2]\) | \(33030144\) | \(3.0640\) | |
422331.bq6 | 422331bq6 | \([1, 0, 1, 32096983, 68973120101]\) | \(6439735268725823/7345472585373\) | \(-4171268022943436026996293\) | \([2]\) | \(66060288\) | \(3.4106\) |
Rank
sage: E.rank()
The elliptic curves in class 422331.bq have rank \(1\).
Complex multiplication
The elliptic curves in class 422331.bq do not have complex multiplication.Modular form 422331.2.a.bq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.