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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 61050cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61050.cz4 | 61050cj1 | \([1, 0, 0, -569663, 142673817]\) | \(1308451928740468777/194033737531392\) | \(3031777148928000000\) | \([2]\) | \(1966080\) | \(2.2709\) | \(\Gamma_0(N)\)-optimal |
| 61050.cz2 | 61050cj2 | \([1, 0, 0, -8761663, 9981265817]\) | \(4760617885089919932457/133756441657344\) | \(2089944400896000000\) | \([2, 2]\) | \(3932160\) | \(2.6175\) | |
| 61050.cz3 | 61050cj3 | \([1, 0, 0, -8409663, 10820081817]\) | \(-4209586785160189454377/801182513521564416\) | \(-12518476773774444000000\) | \([2]\) | \(7864320\) | \(2.9640\) | |
| 61050.cz1 | 61050cj4 | \([1, 0, 0, -140185663, 638845105817]\) | \(19499096390516434897995817/15393430272\) | \(240522348000000\) | \([2]\) | \(7864320\) | \(2.9640\) |
Rank
sage: E.rank()
The elliptic curves in class 61050cj have rank \(0\).
Complex multiplication
The elliptic curves in class 61050cj do not have complex multiplication.Modular form 61050.2.a.cj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
