Show commands:
SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 42350.co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42350.co1 | 42350db1 | \([1, -1, 1, -1178805, 492790697]\) | \(52355598021/15092\) | \(52219528539062500\) | \([2]\) | \(691200\) | \(2.1879\) | \(\Gamma_0(N)\)-optimal |
42350.co2 | 42350db2 | \([1, -1, 1, -1027555, 623773197]\) | \(-34677868581/28471058\) | \(-98512140588941406250\) | \([2]\) | \(1382400\) | \(2.5344\) |
Rank
sage: E.rank()
The elliptic curves in class 42350.co have rank \(1\).
Complex multiplication
The elliptic curves in class 42350.co do not have complex multiplication.Modular form 42350.2.a.co
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.