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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 6864m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6864.f2 | 6864m1 | \([0, -1, 0, -109, -18116]\) | \(-9033613312/8891539371\) | \(-142264629936\) | \([2]\) | \(5376\) | \(0.81910\) | \(\Gamma_0(N)\)-optimal |
6864.f1 | 6864m2 | \([0, -1, 0, -11044, -438020]\) | \(581972233018192/7558011747\) | \(1934851007232\) | \([2]\) | \(10752\) | \(1.1657\) |
Rank
sage: E.rank()
The elliptic curves in class 6864m have rank \(1\).
Complex multiplication
The elliptic curves in class 6864m do not have complex multiplication.Modular form 6864.2.a.m
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.