Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 65025.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
65025.f1 | 65025ck1 | \([0, 0, 1, -21675, 1258956]\) | \(-102400/3\) | \(-32993039626875\) | \([]\) | \(245760\) | \(1.3728\) | \(\Gamma_0(N)\)-optimal |
65025.f2 | 65025ck2 | \([0, 0, 1, 108375, -60644844]\) | \(20480/243\) | \(-1670272631110546875\) | \([]\) | \(1228800\) | \(2.1775\) |
Rank
sage: E.rank()
The elliptic curves in class 65025.f have rank \(1\).
Complex multiplication
The elliptic curves in class 65025.f do not have complex multiplication.Modular form 65025.2.a.f
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.