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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 193200.gn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.gn1 | 193200bf8 | \([0, 1, 0, -101559606408, 11688220738387188]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(8023581024169921875000000000000000\) | \([2]\) | \(1146617856\) | \(5.2536\) | |
| 193200.gn2 | 193200bf6 | \([0, 1, 0, -99807558408, 12136405129075188]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(587554443441000000000000000000\) | \([2, 2]\) | \(573308928\) | \(4.9070\) | |
| 193200.gn3 | 193200bf3 | \([0, 1, 0, -99807430408, 12136437814899188]\) | \(1718036403880129446396978632449/49057344000000\) | \(3139670016000000000000\) | \([2]\) | \(286654464\) | \(4.5604\) | |
| 193200.gn4 | 193200bf7 | \([0, 1, 0, -98057558408, 12582497629075188]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-8051783650315178260536000000000000\) | \([2]\) | \(1146617856\) | \(5.2536\) | |
| 193200.gn5 | 193200bf5 | \([0, 1, 0, -18926790408, -998791033228812]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(2970583839989337600000000000000\) | \([2]\) | \(382205952\) | \(4.7043\) | |
| 193200.gn6 | 193200bf2 | \([0, 1, 0, -1757382408, 1120949875188]\) | \(9378698233516887309850369/5418996968417034240000\) | \(346815805978690191360000000000\) | \([2, 2]\) | \(191102976\) | \(4.3577\) | |
| 193200.gn7 | 193200bf1 | \([0, 1, 0, -1233094408, 16622048883188]\) | \(3239908336204082689644289/9880281924658790400\) | \(632338043178162585600000000\) | \([2]\) | \(95551488\) | \(4.0111\) | \(\Gamma_0(N)\)-optimal |
| 193200.gn8 | 193200bf4 | \([0, 1, 0, 7023417592, 8970985075188]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-22207799151824220735897600000000\) | \([2]\) | \(382205952\) | \(4.7043\) |
Rank
sage: E.rank()
The elliptic curves in class 193200.gn have rank \(0\).
Complex multiplication
The elliptic curves in class 193200.gn do not have complex multiplication.Modular form 193200.2.a.gn
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.
