Show commands:
SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 22050ei
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.dj2 | 22050ei1 | \([1, -1, 1, 1975, 27857]\) | \(397535/392\) | \(-840507985800\) | \([]\) | \(34560\) | \(0.97509\) | \(\Gamma_0(N)\)-optimal |
| 22050.dj1 | 22050ei2 | \([1, -1, 1, -20075, -1533283]\) | \(-417267265/235298\) | \(-504514918476450\) | \([]\) | \(103680\) | \(1.5244\) |
Rank
sage: E.rank()
The elliptic curves in class 22050ei have rank \(0\).
Complex multiplication
The elliptic curves in class 22050ei do not have complex multiplication.Modular form 22050.2.a.ei
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
