Show commands:
SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 169050gp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.ds2 | 169050gp1 | \([1, 0, 1, 164124, 26388898]\) | \(265971760991/317400000\) | \(-583465509375000000\) | \([2]\) | \(2211840\) | \(2.0952\) | \(\Gamma_0(N)\)-optimal |
169050.ds1 | 169050gp2 | \([1, 0, 1, -962876, 251788898]\) | \(53706380371489/16171875000\) | \(29728201904296875000\) | \([2]\) | \(4423680\) | \(2.4418\) |
Rank
sage: E.rank()
The elliptic curves in class 169050gp have rank \(1\).
Complex multiplication
The elliptic curves in class 169050gp do not have complex multiplication.Modular form 169050.2.a.gp
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.