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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 25410cl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25410.cj6 | 25410cl1 | \([1, 0, 0, -34670556, -78578780400]\) | \(2601656892010848045529/56330588160\) | \(99793073091317760\) | \([2]\) | \(1658880\) | \(2.7887\) | \(\Gamma_0(N)\)-optimal |
| 25410.cj5 | 25410cl2 | \([1, 0, 0, -34709276, -78394480944]\) | \(2610383204210122997209/12104550027662400\) | \(21443948751555629006400\) | \([2, 2]\) | \(3317760\) | \(3.1353\) | |
| 25410.cj4 | 25410cl3 | \([1, 0, 0, -36995571, -67439389599]\) | \(3160944030998056790089/720291785342976000\) | \(1276040835533987905536000\) | \([2]\) | \(4976640\) | \(3.3380\) | |
| 25410.cj7 | 25410cl4 | \([1, 0, 0, -17067476, -158018981064]\) | \(-310366976336070130009/5909282337130963560\) | \(-10468654126450066935317160\) | \([2]\) | \(6635520\) | \(3.4819\) | |
| 25410.cj3 | 25410cl5 | \([1, 0, 0, -52970596, 13025339240]\) | \(9278380528613437145689/5328033205714065000\) | \(9438935833948014705465000\) | \([2]\) | \(6635520\) | \(3.4819\) | |
| 25410.cj2 | 25410cl6 | \([1, 0, 0, -195592691, 995193033825]\) | \(467116778179943012100169/28800309694464000000\) | \(51021505442634338304000000\) | \([2, 2]\) | \(9953280\) | \(3.6846\) | |
| 25410.cj8 | 25410cl7 | \([1, 0, 0, 152887309, 4155697545825]\) | \(223090928422700449019831/4340371122724101696000\) | \(-7689232206544232324667456000\) | \([2]\) | \(19906560\) | \(4.0312\) | |
| 25410.cj1 | 25410cl8 | \([1, 0, 0, -3081626611, 65843798009441]\) | \(1826870018430810435423307849/7641104625000000000\) | \(13536682950569625000000000\) | \([2]\) | \(19906560\) | \(4.0312\) |
Rank
sage: E.rank()
The elliptic curves in class 25410cl have rank \(0\).
Complex multiplication
The elliptic curves in class 25410cl do not have complex multiplication.Modular form 25410.2.a.cl
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
