Show commands:
SageMath
E = EllipticCurve("ha1")
E.isogeny_class()
Elliptic curves in class 485100.ha
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.ha1 | 485100ha2 | \([0, 0, 0, -7309575, 6517428750]\) | \(4662947952/717409\) | \(6645173806852812000000\) | \([2]\) | \(22118400\) | \(2.9109\) | \(\Gamma_0(N)\)-optimal* |
485100.ha2 | 485100ha1 | \([0, 0, 0, 793800, 561448125]\) | \(95551488/290521\) | \(-168188799863526750000\) | \([2]\) | \(11059200\) | \(2.5643\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485100.ha have rank \(0\).
Complex multiplication
The elliptic curves in class 485100.ha do not have complex multiplication.Modular form 485100.2.a.ha
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.