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SageMath
E = EllipticCurve("ee1")
E.isogeny_class()
Elliptic curves in class 203490.ee
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
203490.ee1 | 203490g8 | \([1, -1, 1, -26135392757, -1625317462284019]\) | \(2708215857449597952771459256806409/1815677562935478375000000000\) | \(1323628943379963735375000000000\) | \([6]\) | \(477757440\) | \(4.7183\) | |
203490.ee2 | 203490g5 | \([1, -1, 1, -26131186022, -1625867129640931]\) | \(2706908330196708836642873424493849/816939805815000\) | \(595549118439135000\) | \([2]\) | \(159252480\) | \(4.1690\) | |
203490.ee3 | 203490g6 | \([1, -1, 1, -1956594677, -14632320905971]\) | \(1136315122909965387044499819529/530704359775758422016000000\) | \(386883478276527889649664000000\) | \([2, 6]\) | \(238878720\) | \(4.3717\) | |
203490.ee4 | 203490g2 | \([1, -1, 1, -1633199342, -25403860641859]\) | \(660866552951225193140994678169/363054521201227329600\) | \(264666745955694723278400\) | \([2, 2]\) | \(79626240\) | \(3.8224\) | |
203490.ee5 | 203490g4 | \([1, -1, 1, -1624013942, -25703738232739]\) | \(-649778658927959232413187423769/15498405515425377751317720\) | \(-11298337620745100380710617880\) | \([2]\) | \(159252480\) | \(4.1690\) | |
203490.ee6 | 203490g3 | \([1, -1, 1, -1001079797, 12034570571021]\) | \(152195662006675487969752714249/2254051004206282702848000\) | \(1643203182066380090376192000\) | \([6]\) | \(119439360\) | \(4.0252\) | |
203490.ee7 | 203490g1 | \([1, -1, 1, -102649262, -392223454531]\) | \(164083032511008797673646489/3779535863669623787520\) | \(2755281644615155741102080\) | \([2]\) | \(39813120\) | \(3.4759\) | \(\Gamma_0(N)\)-optimal |
203490.ee8 | 203490g7 | \([1, -1, 1, 6933965323, -110639700233971]\) | \(50575615882668425252678113940471/36522079745400816582633408000\) | \(-26624596134397195288739754432000\) | \([6]\) | \(477757440\) | \(4.7183\) |
Rank
sage: E.rank()
The elliptic curves in class 203490.ee have rank \(0\).
Complex multiplication
The elliptic curves in class 203490.ee do not have complex multiplication.Modular form 203490.2.a.ee
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 6 & 12 & 4 & 12 & 4 \\ 3 & 1 & 6 & 2 & 4 & 12 & 4 & 12 \\ 2 & 6 & 1 & 3 & 6 & 2 & 6 & 2 \\ 6 & 2 & 3 & 1 & 2 & 6 & 2 & 6 \\ 12 & 4 & 6 & 2 & 1 & 12 & 4 & 3 \\ 4 & 12 & 2 & 6 & 12 & 1 & 3 & 4 \\ 12 & 4 & 6 & 2 & 4 & 3 & 1 & 12 \\ 4 & 12 & 2 & 6 & 3 & 4 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.