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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 240669p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
240669.p4 | 240669p1 | \([1, -1, 1, -586994, 173247000]\) | \(17319700013617/25857\) | \(33393487274433\) | \([2]\) | \(1310720\) | \(1.8637\) | \(\Gamma_0(N)\)-optimal |
240669.p3 | 240669p2 | \([1, -1, 1, -592439, 169873278]\) | \(17806161424897/668584449\) | \(863455400455014081\) | \([2, 2]\) | \(2621440\) | \(2.2103\) | |
240669.p5 | 240669p3 | \([1, -1, 1, 240646, 609075690]\) | \(1193377118543/124806800313\) | \(-161183984917618674297\) | \([2]\) | \(5242880\) | \(2.5568\) | |
240669.p2 | 240669p4 | \([1, -1, 1, -1512644, -485312682]\) | \(296380748763217/92608836489\) | \(119601345971901920841\) | \([2, 2]\) | \(5242880\) | \(2.5568\) | |
240669.p6 | 240669p5 | \([1, -1, 1, 4220941, -3290182464]\) | \(6439735268725823/7345472585373\) | \(-9486442561176847417437\) | \([2]\) | \(10485760\) | \(2.9034\) | |
240669.p1 | 240669p6 | \([1, -1, 1, -21969509, -39623386800]\) | \(908031902324522977/161726530797\) | \(208864634261817541293\) | \([2]\) | \(10485760\) | \(2.9034\) |
Rank
sage: E.rank()
The elliptic curves in class 240669p have rank \(0\).
Complex multiplication
The elliptic curves in class 240669p do not have complex multiplication.Modular form 240669.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.