Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1785.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1785.k1 | 1785e2 | \([1, 1, 0, -77, -294]\) | \(51520374361/212415\) | \(212415\) | \([2]\) | \(256\) | \(-0.12243\) | |
1785.k2 | 1785e1 | \([1, 1, 0, -2, -9]\) | \(-1771561/26775\) | \(-26775\) | \([2]\) | \(128\) | \(-0.46901\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1785.k have rank \(0\).
Complex multiplication
The elliptic curves in class 1785.k do not have complex multiplication.Modular form 1785.2.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 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.