Show commands:
SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 289800.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
289800.cp1 | 289800cp2 | \([0, 0, 0, -109875, -14017250]\) | \(12576878500/1127\) | \(13145328000000\) | \([2]\) | \(1105920\) | \(1.5573\) | |
289800.cp2 | 289800cp1 | \([0, 0, 0, -6375, -251750]\) | \(-9826000/3703\) | \(-10797948000000\) | \([2]\) | \(552960\) | \(1.2107\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 289800.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 289800.cp do not have complex multiplication.Modular form 289800.2.a.cp
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.