Show commands:
SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 49686cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 49686.bx3 | 49686cp1 | \([1, 1, 1, -22149, 1257927]\) | \(4649101309/6804\) | \(1758662899812\) | \([2]\) | \(115200\) | \(1.2513\) | \(\Gamma_0(N)\)-optimal |
| 49686.bx4 | 49686cp2 | \([1, 1, 1, -15779, 2004491]\) | \(-1680914269/5786802\) | \(-1495742796290106\) | \([2]\) | \(230400\) | \(1.5979\) | |
| 49686.bx1 | 49686cp3 | \([1, 1, 1, -662334, -207675525]\) | \(124318741396429/51631104\) | \(13345342016627712\) | \([2]\) | \(576000\) | \(2.0560\) | |
| 49686.bx2 | 49686cp4 | \([1, 1, 1, -560414, -273638149]\) | \(-75306487574989/81352871712\) | \(-21027671556887058336\) | \([2]\) | \(1152000\) | \(2.4026\) |
Rank
sage: E.rank()
The elliptic curves in class 49686cp have rank \(0\).
Complex multiplication
The elliptic curves in class 49686cp do not have complex multiplication.Modular form 49686.2.a.cp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
