Show commands:
SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 42432.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42432.cp1 | 42432bi2 | \([0, 1, 0, -20961, -397953]\) | \(3885442650361/1996623837\) | \(523402959126528\) | \([2]\) | \(294912\) | \(1.5165\) | |
42432.cp2 | 42432bi1 | \([0, 1, 0, -16801, -843073]\) | \(2000852317801/2094417\) | \(549038850048\) | \([2]\) | \(147456\) | \(1.1699\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42432.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 42432.cp do not have complex multiplication.Modular form 42432.2.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.