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SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 169050.dj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.dj1 | 169050gi8 | \([1, 0, 1, -311026294626, 62641713532941148]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(230460518533341586589813232421875000\) | \([2]\) | \(2293235712\) | \(5.5334\) | |
| 169050.dj2 | 169050gi6 | \([1, 0, 1, -305660647626, 65043699068961148]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(16876267753024953369140625000000\) | \([2, 2]\) | \(1146617856\) | \(5.1868\) | |
| 169050.dj3 | 169050gi3 | \([1, 0, 1, -305660255626, 65043874244353148]\) | \(1718036403880129446396978632449/49057344000000\) | \(90180429129000000000000\) | \([2]\) | \(573308928\) | \(4.8403\) | |
| 169050.dj4 | 169050gi7 | \([1, 0, 1, -300301272626, 67434473381461148]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-231270579754865822063915982421875000\) | \([2]\) | \(2293235712\) | \(5.5334\) | |
| 169050.dj5 | 169050gi5 | \([1, 0, 1, -57963295626, -5352866712062852]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(85323783738014057446875000000000\) | \([2]\) | \(764411904\) | \(4.9841\) | |
| 169050.dj6 | 169050gi2 | \([1, 0, 1, -5381983626, 6010281729148]\) | \(9378698233516887309850369/5418996968417034240000\) | \(9961555849020244707840000000000\) | \([2, 2]\) | \(382205952\) | \(4.6375\) | |
| 169050.dj7 | 169050gi1 | \([1, 0, 1, -3776351626, 89085681409148]\) | \(3239908336204082689644289/9880281924658790400\) | \(18162582627409094246400000000\) | \([2]\) | \(191102976\) | \(4.2910\) | \(\Gamma_0(N)\)-optimal |
| 169050.dj8 | 169050gi4 | \([1, 0, 1, 21509216374, 48068118529148]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-637872402932853453456449400000000\) | \([2]\) | \(764411904\) | \(4.9841\) |
Rank
sage: E.rank()
The elliptic curves in class 169050.dj have rank \(0\).
Complex multiplication
The elliptic curves in class 169050.dj do not have complex multiplication.Modular form 169050.2.a.dj
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.
