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SageMath
E = EllipticCurve("fq1")
E.isogeny_class()
Elliptic curves in class 330330.fq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330330.fq1 | 330330fq7 | \([1, 0, 0, -12796061396, 557121937109640]\) | \(130796627670002750950880364889/4007004103295286093000\) | \(7098652196237900326201173000\) | \([2]\) | \(716636160\) | \(4.4416\) | |
| 330330.fq2 | 330330fq6 | \([1, 0, 0, -833396396, 7932734690640]\) | \(36134533748915083453404889/5565686539253841000000\) | \(9859953211167073815801000000\) | \([2, 2]\) | \(358318080\) | \(4.0950\) | |
| 330330.fq3 | 330330fq4 | \([1, 0, 0, -279982931, -568386994449]\) | \(1370131553911340548947529/714126686285699857170\) | \(1265118986482980724667942370\) | \([2]\) | \(238878720\) | \(3.8923\) | |
| 330330.fq4 | 330330fq3 | \([1, 0, 0, -228396396, -1207968309360]\) | \(743764321292317933404889/74603529000000000000\) | \(132164702438769000000000000\) | \([2]\) | \(179159040\) | \(3.7484\) | |
| 330330.fq5 | 330330fq2 | \([1, 0, 0, -222647081, -1277482385739]\) | \(688999042618248810121129/779639711718968100\) | \(1381179307332566846204100\) | \([2, 2]\) | \(119439360\) | \(3.5457\) | |
| 330330.fq6 | 330330fq1 | \([1, 0, 0, -222586581, -1278211979439]\) | \(688437529087783927489129/882972090000\) | \(1564238918732490000\) | \([2]\) | \(59719680\) | \(3.1991\) | \(\Gamma_0(N)\)-optimal |
| 330330.fq7 | 330330fq5 | \([1, 0, 0, -166279231, -1939883538229]\) | \(-286999819333751016766729/751553009101890965970\) | \(-1331422000357555061564779170\) | \([2]\) | \(238878720\) | \(3.8923\) | |
| 330330.fq8 | 330330fq8 | \([1, 0, 0, 1449268604, 43750944271640]\) | \(190026536708029086053555111/576736012771479654093000\) | \(-1021723027521455267484649173000\) | \([2]\) | \(716636160\) | \(4.4416\) |
Rank
sage: E.rank()
The elliptic curves in class 330330.fq have rank \(0\).
Complex multiplication
The elliptic curves in class 330330.fq do not have complex multiplication.Modular form 330330.2.a.fq
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
