Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 910.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
910.a1 | 910c4 | \([1, 0, 1, -14669, -685008]\) | \(349046010201856969/7245875000\) | \(7245875000\) | \([2]\) | \(1728\) | \(1.0107\) | |
910.a2 | 910c3 | \([1, 0, 1, -949, -9984]\) | \(94376601570889/12235496000\) | \(12235496000\) | \([2]\) | \(864\) | \(0.66414\) | |
910.a3 | 910c2 | \([1, 0, 1, -304, 456]\) | \(3092354182009/1689383150\) | \(1689383150\) | \([6]\) | \(576\) | \(0.46141\) | |
910.a4 | 910c1 | \([1, 0, 1, -234, 1352]\) | \(1408317602329/2153060\) | \(2153060\) | \([6]\) | \(288\) | \(0.11484\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 910.a have rank \(1\).
Complex multiplication
The elliptic curves in class 910.a do not have complex multiplication.Modular form 910.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.