Show commands:
SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 162450.el
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162450.el1 | 162450cj2 | \([1, -1, 1, -32424005, 39482427997]\) | \(260549802603/104256800\) | \(1508472214403713537500000\) | \([2]\) | \(33177600\) | \(3.3375\) | |
162450.el2 | 162450cj1 | \([1, -1, 1, 6563995, 4471203997]\) | \(2161700757/1848320\) | \(-26742997706880240000000\) | \([2]\) | \(16588800\) | \(2.9910\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162450.el have rank \(1\).
Complex multiplication
The elliptic curves in class 162450.el do not have complex multiplication.Modular form 162450.2.a.el
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.