Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 4774k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4774.h2 | 4774k1 | \([1, 1, 1, -515, 5601]\) | \(-15107691357361/5868735488\) | \(-5868735488\) | \([5]\) | \(4000\) | \(0.58370\) | \(\Gamma_0(N)\)-optimal |
4774.h1 | 4774k2 | \([1, 1, 1, -3735, -555239]\) | \(-5762391987245041/129101095135628\) | \(-129101095135628\) | \([]\) | \(20000\) | \(1.3884\) |
Rank
sage: E.rank()
The elliptic curves in class 4774k have rank \(1\).
Complex multiplication
The elliptic curves in class 4774k do not have complex multiplication.Modular form 4774.2.a.k
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 Cremona 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 Cremona labels.