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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 26775.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 26775.o1 | 26775bf6 | \([1, -1, 1, -16141505, 24916755372]\) | \(40832710302042509761/91556816413125\) | \(1042889361955751953125\) | \([2]\) | \(1572864\) | \(2.9152\) | |
| 26775.o2 | 26775bf4 | \([1, -1, 1, -1375880, 80974122]\) | \(25288177725059761/14387797265625\) | \(163886003228759765625\) | \([2, 2]\) | \(786432\) | \(2.5687\) | |
| 26775.o3 | 26775bf2 | \([1, -1, 1, -879755, -315925878]\) | \(6610905152742241/35128130625\) | \(400131362900390625\) | \([2, 2]\) | \(393216\) | \(2.2221\) | |
| 26775.o4 | 26775bf1 | \([1, -1, 1, -878630, -316778628]\) | \(6585576176607121/187425\) | \(2134887890625\) | \([2]\) | \(196608\) | \(1.8755\) | \(\Gamma_0(N)\)-optimal |
| 26775.o5 | 26775bf3 | \([1, -1, 1, -401630, -658263378]\) | \(-629004249876241/16074715228425\) | \(-183101053148778515625\) | \([2]\) | \(786432\) | \(2.5687\) | |
| 26775.o6 | 26775bf5 | \([1, -1, 1, 5451745, 640839372]\) | \(1573196002879828319/926055908203125\) | \(-10548355579376220703125\) | \([2]\) | \(1572864\) | \(2.9152\) |
Rank
sage: E.rank()
The elliptic curves in class 26775.o have rank \(1\).
Complex multiplication
The elliptic curves in class 26775.o do not have complex multiplication.Modular form 26775.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
