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SageMath
E = EllipticCurve("dg1")
E.isogeny_class()
Elliptic curves in class 111090.dg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111090.dg1 | 111090de8 | \([1, 0, 0, -134312579475, 17776363009268457]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(18559030466820701122283935546875000\) | \([2]\) | \(1051066368\) | \(5.3235\) | |
| 111090.dg2 | 111090de6 | \([1, 0, 0, -131995495995, 18457994749331025]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(1359049130792010219515625000000\) | \([2, 2]\) | \(525533184\) | \(4.9769\) | |
| 111090.dg3 | 111090de3 | \([1, 0, 0, -131995326715, 18458044460332817]\) | \(1718036403880129446396978632449/49057344000000\) | \(7262247531018816000000\) | \([4]\) | \(262766592\) | \(4.6303\) | |
| 111090.dg4 | 111090de7 | \([1, 0, 0, -129681120995, 19136444985956025]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-18624264854844883499873755882875000\) | \([2]\) | \(1051066368\) | \(5.3235\) | |
| 111090.dg5 | 111090de5 | \([1, 0, 0, -25030680315, -1519028803457775]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(6871140931278989720892600000000\) | \([2]\) | \(350355456\) | \(4.7742\) | |
| 111090.dg6 | 111090de2 | \([1, 0, 0, -2324138235, 1705521882897]\) | \(9378698233516887309850369/5418996968417034240000\) | \(802206033707920586461839360000\) | \([2, 2]\) | \(175177728\) | \(4.4276\) | |
| 111090.dg7 | 111090de1 | \([1, 0, 0, -1630767355, 25280547825425]\) | \(3239908336204082689644289/9880281924658790400\) | \(1462636318287495058528665600\) | \([4]\) | \(87588864\) | \(4.0810\) | \(\Gamma_0(N)\)-optimal |
| 111090.dg8 | 111090de4 | \([1, 0, 0, 9288469765, 13640960385297]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-51367988908964757630794303577600\) | \([2]\) | \(350355456\) | \(4.7742\) |
Rank
sage: E.rank()
The elliptic curves in class 111090.dg have rank \(1\).
Complex multiplication
The elliptic curves in class 111090.dg do not have complex multiplication.Modular form 111090.2.a.dg
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.
