Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 169050.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.r1 | 169050ig2 | \([1, 1, 0, -4875525, -4143049875]\) | \(6972359126281921/5071500000\) | \(9322764117187500000\) | \([2]\) | \(8847360\) | \(2.5745\) | |
169050.r2 | 169050ig1 | \([1, 1, 0, -367525, -36261875]\) | \(2986606123201/1421952000\) | \(2613925482000000000\) | \([2]\) | \(4423680\) | \(2.2279\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 169050.r have rank \(1\).
Complex multiplication
The elliptic curves in class 169050.r do not have complex multiplication.Modular form 169050.2.a.r
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 LMFDB 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 LMFDB labels.