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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 68544.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68544.s1 | 68544dr2 | \([0, 0, 0, -12396, -503984]\) | \(2204605874/127449\) | \(12177941594112\) | \([2]\) | \(163840\) | \(1.2648\) | |
68544.s2 | 68544dr1 | \([0, 0, 0, 564, -32240]\) | \(415292/9639\) | \(-460510396416\) | \([2]\) | \(81920\) | \(0.91820\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68544.s have rank \(1\).
Complex multiplication
The elliptic curves in class 68544.s do not have complex multiplication.Modular form 68544.2.a.s
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.