Show commands:
SageMath
E = EllipticCurve("hg1")
E.isogeny_class()
Elliptic curves in class 286650.hg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
286650.hg1 | 286650hg2 | \([1, -1, 0, -14235867, -20669871209]\) | \(653316772092011/19958562\) | \(9747223735183593750\) | \([2]\) | \(13107200\) | \(2.7412\) | |
286650.hg2 | 286650hg1 | \([1, -1, 0, -927117, -294174959]\) | \(180457909451/27761292\) | \(13557866759226562500\) | \([2]\) | \(6553600\) | \(2.3946\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 286650.hg have rank \(0\).
Complex multiplication
The elliptic curves in class 286650.hg do not have complex multiplication.Modular form 286650.2.a.hg
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.