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SageMath
E = EllipticCurve("kf1")
E.isogeny_class()
Elliptic curves in class 286650.kf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.kf1 | 286650kf6 | \([1, -1, 1, -99390605, -381362306853]\) | \(81025909800741361/11088090\) | \(14859101071857656250\) | \([2]\) | \(37748736\) | \(3.0911\) | |
| 286650.kf2 | 286650kf3 | \([1, -1, 1, -9316355, 10938724647]\) | \(66730743078481/60937500\) | \(81662078100585937500\) | \([2]\) | \(18874368\) | \(2.7445\) | |
| 286650.kf3 | 286650kf4 | \([1, -1, 1, -6229355, -5922469353]\) | \(19948814692561/231344100\) | \(310023219894314062500\) | \([2, 2]\) | \(18874368\) | \(2.7445\) | |
| 286650.kf4 | 286650kf5 | \([1, -1, 1, -1268105, -15100781853]\) | \(-168288035761/73415764890\) | \(-98384146482238932656250\) | \([2]\) | \(37748736\) | \(3.0911\) | |
| 286650.kf5 | 286650kf2 | \([1, -1, 1, -716855, 86155647]\) | \(30400540561/15210000\) | \(20382854693906250000\) | \([2, 2]\) | \(9437184\) | \(2.3979\) | |
| 286650.kf6 | 286650kf1 | \([1, -1, 1, 165145, 10303647]\) | \(371694959/249600\) | \(-334487871900000000\) | \([2]\) | \(4718592\) | \(2.0513\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 286650.kf have rank \(1\).
Complex multiplication
The elliptic curves in class 286650.kf do not have complex multiplication.Modular form 286650.2.a.kf
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
