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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 82810n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82810.bf2 | 82810n1 | \([1, 1, 0, -8453, 11236373]\) | \(-49/40\) | \(-54538162966017640\) | \([]\) | \(861840\) | \(1.8905\) | \(\Gamma_0(N)\)-optimal |
82810.bf1 | 82810n2 | \([1, 1, 0, -4066143, 3154323047]\) | \(-5452947409/250\) | \(-340863518537610250\) | \([]\) | \(2585520\) | \(2.4398\) |
Rank
sage: E.rank()
The elliptic curves in class 82810n have rank \(0\).
Complex multiplication
The elliptic curves in class 82810n do not have complex multiplication.Modular form 82810.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.