Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 8450.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8450.c1 | 8450c3 | \([1, 1, 0, -1941475, -1042037875]\) | \(-10730978619193/6656\) | \(-501988136000000\) | \([]\) | \(108864\) | \(2.1416\) | |
8450.c2 | 8450c2 | \([1, 1, 0, -19100, -2033000]\) | \(-10218313/17576\) | \(-1325562421625000\) | \([]\) | \(36288\) | \(1.5923\) | |
8450.c3 | 8450c1 | \([1, 1, 0, 2025, 58375]\) | \(12167/26\) | \(-1960891156250\) | \([]\) | \(12096\) | \(1.0430\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8450.c have rank \(1\).
Complex multiplication
The elliptic curves in class 8450.c do not have complex multiplication.Modular form 8450.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.