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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 25410n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25410.n6 | 25410n1 | \([1, 1, 0, 1208, 23104]\) | \(109902239/188160\) | \(-333336917760\) | \([2]\) | \(40960\) | \(0.89598\) | \(\Gamma_0(N)\)-optimal |
| 25410.n5 | 25410n2 | \([1, 1, 0, -8472, 230256]\) | \(37966934881/8643600\) | \(15312664659600\) | \([2, 2]\) | \(81920\) | \(1.2426\) | |
| 25410.n4 | 25410n3 | \([1, 1, 0, -44772, -3465084]\) | \(5602762882081/345888060\) | \(612761797461660\) | \([2]\) | \(163840\) | \(1.5891\) | |
| 25410.n2 | 25410n4 | \([1, 1, 0, -127052, 17376924]\) | \(128031684631201/9922500\) | \(17578314022500\) | \([2, 2]\) | \(163840\) | \(1.5891\) | |
| 25410.n3 | 25410n5 | \([1, 1, 0, -118582, 19804426]\) | \(-104094944089921/35880468750\) | \(-63564439099218750\) | \([2]\) | \(327680\) | \(1.9357\) | |
| 25410.n1 | 25410n6 | \([1, 1, 0, -2032802, 1114707774]\) | \(524388516989299201/3150\) | \(5580417150\) | \([2]\) | \(327680\) | \(1.9357\) |
Rank
sage: E.rank()
The elliptic curves in class 25410n have rank \(1\).
Complex multiplication
The elliptic curves in class 25410n do not have complex multiplication.Modular form 25410.2.a.n
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.
