Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 10766f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10766.h2 | 10766f1 | \([1, -1, 1, -38469, -2894467]\) | \(6295726008614605473/118167616\) | \(118167616\) | \([2]\) | \(15840\) | \(1.0849\) | \(\Gamma_0(N)\)-optimal |
10766.h1 | 10766f2 | \([1, -1, 1, -38509, -2888115]\) | \(6315385451217127713/27272627873288\) | \(27272627873288\) | \([2]\) | \(31680\) | \(1.4315\) |
Rank
sage: E.rank()
The elliptic curves in class 10766f have rank \(1\).
Complex multiplication
The elliptic curves in class 10766f do not have complex multiplication.Modular form 10766.2.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 Cremona 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 Cremona labels.