Show commands:
SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 8619m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8619.k2 | 8619m1 | \([1, 0, 1, 9, -11]\) | \(42875/51\) | \(-112047\) | \([2]\) | \(768\) | \(-0.34488\) | \(\Gamma_0(N)\)-optimal |
8619.k1 | 8619m2 | \([1, 0, 1, -56, -115]\) | \(8615125/2601\) | \(5714397\) | \([2]\) | \(1536\) | \(0.0016982\) |
Rank
sage: E.rank()
The elliptic curves in class 8619m have rank \(0\).
Complex multiplication
The elliptic curves in class 8619m do not have complex multiplication.Modular form 8619.2.a.m
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.