Show commands:
SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 25200.cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25200.cf1 | 25200a2 | \([0, 0, 0, -19575, -560250]\) | \(10536048/4375\) | \(344452500000000\) | \([2]\) | \(73728\) | \(1.4866\) | |
25200.cf2 | 25200a1 | \([0, 0, 0, 4050, -64125]\) | \(1492992/1225\) | \(-6027918750000\) | \([2]\) | \(36864\) | \(1.1401\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25200.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 25200.cf do not have complex multiplication.Modular form 25200.2.a.cf
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.