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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 333270.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333270.q1 | 333270q7 | \([1, -1, 0, -1208813215275, -479961801250248339]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(13529533210312291118144989013671875000\) | \([2]\) | \(8408530944\) | \(5.8728\) | |
| 333270.q2 | 333270q6 | \([1, -1, 0, -1187959463955, -498365858231937675]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(990746816347375450026890625000000\) | \([2, 2]\) | \(4204265472\) | \(5.5262\) | |
| 333270.q3 | 333270q3 | \([1, -1, 0, -1187957940435, -498367200428986059]\) | \(1718036403880129446396978632449/49057344000000\) | \(5294178450112716864000000\) | \([2]\) | \(2102132736\) | \(5.1796\) | |
| 333270.q4 | 333270q8 | \([1, -1, 0, -1167130088955, -516684014620812675]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-13577089079181920071407968038615875000\) | \([2]\) | \(8408530944\) | \(5.8728\) | |
| 333270.q5 | 333270q4 | \([1, -1, 0, -225276122835, 41013777693359925]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(5009061738902383506530705400000000\) | \([2]\) | \(2802843648\) | \(5.3235\) | |
| 333270.q6 | 333270q2 | \([1, -1, 0, -20917244115, -46049090838219]\) | \(9378698233516887309850369/5418996968417034240000\) | \(584808198573074107530680893440000\) | \([2, 2]\) | \(1401421824\) | \(4.9769\) | |
| 333270.q7 | 333270q1 | \([1, -1, 0, -14676906195, -682574791286475]\) | \(3239908336204082689644289/9880281924658790400\) | \(1066261876031583897667397222400\) | \([2]\) | \(700710912\) | \(4.6303\) | \(\Gamma_0(N)\)-optimal |
| 333270.q8 | 333270q5 | \([1, -1, 0, 83596227885, -368305930403019]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-37447263914635308312849047308070400\) | \([2]\) | \(2802843648\) | \(5.3235\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.q have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.q do not have complex multiplication.Modular form 333270.2.a.q
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
