Show commands:
SageMath
E = EllipticCurve("cq1")
E.isogeny_class()
Elliptic curves in class 58800.cq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58800.cq1 | 58800fd8 | \([0, -1, 0, -126440008, 344327942512]\) | \(29689921233686449/10380965400750\) | \(78163852699701552000000000\) | \([2]\) | \(15925248\) | \(3.6698\) | |
| 58800.cq2 | 58800fd5 | \([0, -1, 0, -112916008, 461866790512]\) | \(21145699168383889/2593080\) | \(19524689210880000000\) | \([2]\) | \(5308416\) | \(3.1205\) | |
| 58800.cq3 | 58800fd6 | \([0, -1, 0, -52940008, -144300057488]\) | \(2179252305146449/66177562500\) | \(498286339236000000000000\) | \([2, 2]\) | \(7962624\) | \(3.3233\) | |
| 58800.cq4 | 58800fd3 | \([0, -1, 0, -52548008, -146598745488]\) | \(2131200347946769/2058000\) | \(15495785088000000000\) | \([2]\) | \(3981312\) | \(2.9767\) | |
| 58800.cq5 | 58800fd2 | \([0, -1, 0, -7076008, 7178150512]\) | \(5203798902289/57153600\) | \(430340088729600000000\) | \([2, 2]\) | \(2654208\) | \(2.7740\) | |
| 58800.cq6 | 58800fd4 | \([0, -1, 0, -1588008, 18022438512]\) | \(-58818484369/18600435000\) | \(-140052644948160000000000\) | \([2]\) | \(5308416\) | \(3.1205\) | |
| 58800.cq7 | 58800fd1 | \([0, -1, 0, -804008, -97369488]\) | \(7633736209/3870720\) | \(29144725585920000000\) | \([2]\) | \(1327104\) | \(2.4274\) | \(\Gamma_0(N)\)-optimal |
| 58800.cq8 | 58800fd7 | \([0, -1, 0, 14287992, -485818297488]\) | \(42841933504271/13565917968750\) | \(-102145067718750000000000000\) | \([2]\) | \(15925248\) | \(3.6698\) |
Rank
sage: E.rank()
The elliptic curves in class 58800.cq have rank \(1\).
Complex multiplication
The elliptic curves in class 58800.cq do not have complex multiplication.Modular form 58800.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
