Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 550b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
550.f2 | 550b1 | \([1, 0, 1, 249, -6102]\) | \(109902239/1100000\) | \(-17187500000\) | \([]\) | \(480\) | \(0.64508\) | \(\Gamma_0(N)\)-optimal |
550.f1 | 550b2 | \([1, 0, 1, -148501, -22038602]\) | \(-23178622194826561/1610510\) | \(-25164218750\) | \([]\) | \(2400\) | \(1.4498\) |
Rank
sage: E.rank()
The elliptic curves in class 550b have rank \(0\).
Complex multiplication
The elliptic curves in class 550b do not have complex multiplication.Modular form 550.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.