Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 80850.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80850.a1 | 80850p2 | \([1, 1, 0, -9120150, -10602211500]\) | \(45637459887836881/13417633152\) | \(24665173792182000000\) | \([2]\) | \(5160960\) | \(2.7000\) | |
80850.a2 | 80850p1 | \([1, 1, 0, -496150, -210291500]\) | \(-7347774183121/6119866368\) | \(-11249939973888000000\) | \([2]\) | \(2580480\) | \(2.3534\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 80850.a have rank \(2\).
Complex multiplication
The elliptic curves in class 80850.a do not have complex multiplication.Modular form 80850.2.a.a
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.