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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 219849p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 219849.u5 | 219849p1 | \([1, 0, 1, -283032, -62075951]\) | \(-53297461115137/4513839183\) | \(-212357541056555223\) | \([2]\) | \(2654208\) | \(2.0697\) | \(\Gamma_0(N)\)-optimal |
| 219849.u4 | 219849p2 | \([1, 0, 1, -4616837, -3818618125]\) | \(231331938231569617/1472026689\) | \(69252792439518009\) | \([2, 2]\) | \(5308416\) | \(2.4163\) | |
| 219849.u3 | 219849p3 | \([1, 0, 1, -4705282, -3664723825]\) | \(244883173420511137/18418027974129\) | \(866492352325544012649\) | \([2, 2]\) | \(10616832\) | \(2.7629\) | |
| 219849.u1 | 219849p4 | \([1, 0, 1, -73869272, -244373876341]\) | \(947531277805646290177/38367\) | \(1805009316327\) | \([2]\) | \(10616832\) | \(2.7629\) | |
| 219849.u2 | 219849p5 | \([1, 0, 1, -15331317, 18781712509]\) | \(8471112631466271697/1662662681263647\) | \(78221430645870466388007\) | \([2]\) | \(21233664\) | \(3.1095\) | |
| 219849.u6 | 219849p6 | \([1, 0, 1, 4505633, -16261571179]\) | \(215015459663151503/2552757445339983\) | \(-120096722995328844760023\) | \([2]\) | \(21233664\) | \(3.1095\) |
Rank
sage: E.rank()
The elliptic curves in class 219849p have rank \(0\).
Complex multiplication
The elliptic curves in class 219849p do not have complex multiplication.Modular form 219849.2.a.p
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
