Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 5056.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5056.c1 | 5056f2 | \([0, 1, 0, -577, 4383]\) | \(81182737/12482\) | \(3272081408\) | \([2]\) | \(2304\) | \(0.54926\) | |
5056.c2 | 5056f1 | \([0, 1, 0, 63, 415]\) | \(103823/316\) | \(-82837504\) | \([2]\) | \(1152\) | \(0.20268\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5056.c have rank \(1\).
Complex multiplication
The elliptic curves in class 5056.c do not have complex multiplication.Modular form 5056.2.a.c
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.