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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 6864a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6864.i2 | 6864a1 | \([0, -1, 0, 2852, 212608]\) | \(10017976862000/82759712607\) | \(-21186486427392\) | \([2]\) | \(10752\) | \(1.2384\) | \(\Gamma_0(N)\)-optimal |
6864.i1 | 6864a2 | \([0, -1, 0, -40888, 2941984]\) | \(7382814913718500/654774260283\) | \(670488842529792\) | \([2]\) | \(21504\) | \(1.5850\) |
Rank
sage: E.rank()
The elliptic curves in class 6864a have rank \(1\).
Complex multiplication
The elliptic curves in class 6864a do not have complex multiplication.Modular form 6864.2.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 Cremona 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 Cremona labels.