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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 6864n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6864.c2 | 6864n1 | \([0, -1, 0, -104, 624]\) | \(-30664297/18876\) | \(-77316096\) | \([2]\) | \(1920\) | \(0.21551\) | \(\Gamma_0(N)\)-optimal |
6864.c1 | 6864n2 | \([0, -1, 0, -1864, 31600]\) | \(174958262857/33462\) | \(137060352\) | \([2]\) | \(3840\) | \(0.56208\) |
Rank
sage: E.rank()
The elliptic curves in class 6864n have rank \(1\).
Complex multiplication
The elliptic curves in class 6864n do not have complex multiplication.Modular form 6864.2.a.n
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.