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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 249690.cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 249690.cj1 | 249690cj8 | \([1, 0, 0, -2661640801120, -1671368986468560610]\) | \(2085316829313324338342454357175715767150081/499057650210\) | \(499057650210\) | \([2]\) | \(1526726656\) | \(5.0735\) | |
| 249690.cj2 | 249690cj6 | \([1, 0, 0, -166352550070, -26115150810599200]\) | \(509110554031752646938166145889438134881/249058538233126713044100\) | \(249058538233126713044100\) | \([2, 2]\) | \(763363328\) | \(4.7269\) | |
| 249690.cj3 | 249690cj7 | \([1, 0, 0, -166351667020, -26115441927282190]\) | \(-509102446533164296117678747246704511681/11260482449836875928123468727010\) | \(-11260482449836875928123468727010\) | \([2]\) | \(1526726656\) | \(5.0735\) | |
| 249690.cj4 | 249690cj4 | \([1, 0, 0, -10397089570, -408045332529100]\) | \(124296548246895698958145872007982881/2749138304449961117564010000\) | \(2749138304449961117564010000\) | \([2, 4]\) | \(381681664\) | \(4.3803\) | |
| 249690.cj5 | 249690cj5 | \([1, 0, 0, -10026429070, -438485381299000]\) | \(-111471172596350616707438857829030881/18551296611518877695051385524100\) | \(-18551296611518877695051385524100\) | \([4]\) | \(763363328\) | \(4.7269\) | |
| 249690.cj6 | 249690cj2 | \([1, 0, 0, -673039570, -5895575919100]\) | \(33716734605576776462761920782881/4496908623986020832100000000\) | \(4496908623986020832100000000\) | \([2, 8]\) | \(190840832\) | \(4.0338\) | |
| 249690.cj7 | 249690cj1 | \([1, 0, 0, -173039570, 782524080900]\) | \(573007087724441463649920782881/67058993610000000000000000\) | \(67058993610000000000000000\) | \([8]\) | \(95420416\) | \(3.6872\) | \(\Gamma_0(N)\)-optimal |
| 249690.cj8 | 249690cj3 | \([1, 0, 0, 1051010430, -31142219309100]\) | \(128394017933147245660796166417119/493274390082338961058364010000\) | \(-493274390082338961058364010000\) | \([8]\) | \(381681664\) | \(4.3803\) |
Rank
sage: E.rank()
The elliptic curves in class 249690.cj have rank \(1\).
Complex multiplication
The elliptic curves in class 249690.cj do not have complex multiplication.Modular form 249690.2.a.cj
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
