Show commands:
SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 22050dy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.da2 | 22050dy1 | \([1, -1, 1, -783005, 264932637]\) | \(505318200625/4251528\) | \(446680404481537800\) | \([]\) | \(483840\) | \(2.2122\) | \(\Gamma_0(N)\)-optimal |
22050.da1 | 22050dy2 | \([1, -1, 1, -63294755, 193836317217]\) | \(266916252066900625/162\) | \(17020286712450\) | \([]\) | \(1451520\) | \(2.7615\) |
Rank
sage: E.rank()
The elliptic curves in class 22050dy have rank \(1\).
Complex multiplication
The elliptic curves in class 22050dy do not have complex multiplication.Modular form 22050.2.a.dy
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.