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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 259350.ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 259350.ek1 | 259350ek8 | \([1, 1, 1, -19205200088, -1024424037374719]\) | \(50137213659805457275731367898809/4113897879000\) | \(64279654359375000\) | \([2]\) | \(191102976\) | \(4.0770\) | |
| 259350.ek2 | 259350ek6 | \([1, 1, 1, -1200325088, -16006998374719]\) | \(12240533203187013248735018809/3506282465049000000\) | \(54785663516390625000000\) | \([2, 2]\) | \(95551488\) | \(3.7304\) | |
| 259350.ek3 | 259350ek7 | \([1, 1, 1, -1195450088, -16143459374719]\) | \(-12091997009671629064982138809/207252595706436249879000\) | \(-3238321807913066404359375000\) | \([2]\) | \(191102976\) | \(4.0770\) | |
| 259350.ek4 | 259350ek5 | \([1, 1, 1, -237125963, -1405025655469]\) | \(94371532824107026279203049/40995077600666342790\) | \(640548087510411606093750\) | \([2]\) | \(63700992\) | \(3.5277\) | |
| 259350.ek5 | 259350ek3 | \([1, 1, 1, -75325088, -247998374719]\) | \(3024980849878413455018809/50557689000000000000\) | \(789963890625000000000000\) | \([2]\) | \(47775744\) | \(3.3838\) | |
| 259350.ek6 | 259350ek2 | \([1, 1, 1, -17177213, -14509657969]\) | \(35872512095393194378249/14944558319037792900\) | \(233508723734965514062500\) | \([2, 2]\) | \(31850496\) | \(3.1811\) | |
| 259350.ek7 | 259350ek1 | \([1, 1, 1, -8064713, 8654317031]\) | \(3712533999213317890249/76090919904090000\) | \(1188920623501406250000\) | \([2]\) | \(15925248\) | \(2.8345\) | \(\Gamma_0(N)\)-optimal |
| 259350.ek8 | 259350ek4 | \([1, 1, 1, 56971537, -106305810469]\) | \(1308812680909424992398551/1070002284841633041990\) | \(-16718785700650516281093750\) | \([2]\) | \(63700992\) | \(3.5277\) |
Rank
sage: E.rank()
The elliptic curves in class 259350.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 259350.ek do not have complex multiplication.Modular form 259350.2.a.ek
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
