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SageMath
E = EllipticCurve("cq1")
E.isogeny_class()
Elliptic curves in class 157170cq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 157170.x6 | 157170cq1 | \([1, 0, 1, 10136, -2224714]\) | \(23862997439/457113600\) | \(-2206400038502400\) | \([2]\) | \(983040\) | \(1.6246\) | \(\Gamma_0(N)\)-optimal |
| 157170.x5 | 157170cq2 | \([1, 0, 1, -206184, -34067018]\) | \(200828550012481/12454560000\) | \(60115782299040000\) | \([2, 2]\) | \(1966080\) | \(1.9712\) | |
| 157170.x4 | 157170cq3 | \([1, 0, 1, -625304, 148334006]\) | \(5601911201812801/1271193750000\) | \(6135809433243750000\) | \([2]\) | \(3932160\) | \(2.3177\) | |
| 157170.x2 | 157170cq4 | \([1, 0, 1, -3248184, -2253510218]\) | \(785209010066844481/3324675600\) | \(16047574108160400\) | \([2, 2]\) | \(3932160\) | \(2.3177\) | |
| 157170.x3 | 157170cq5 | \([1, 0, 1, -3197484, -2327248298]\) | \(-749011598724977281/51173462246460\) | \(-247004528132373346140\) | \([2]\) | \(7864320\) | \(2.6643\) | |
| 157170.x1 | 157170cq6 | \([1, 0, 1, -51970884, -144211968938]\) | \(3216206300355197383681/57660\) | \(278313806940\) | \([2]\) | \(7864320\) | \(2.6643\) |
Rank
sage: E.rank()
The elliptic curves in class 157170cq have rank \(1\).
Complex multiplication
The elliptic curves in class 157170cq do not have complex multiplication.Modular form 157170.2.a.cq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
