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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 141610bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
141610.cw2 | 141610bc1 | \([1, 1, 1, -20525, 1322467]\) | \(-115501303/25600\) | \(-211947165875200\) | \([2]\) | \(665600\) | \(1.4694\) | \(\Gamma_0(N)\)-optimal |
141610.cw1 | 141610bc2 | \([1, 1, 1, -344205, 77581475]\) | \(544737993463/20000\) | \(165583723340000\) | \([2]\) | \(1331200\) | \(1.8160\) |
Rank
sage: E.rank()
The elliptic curves in class 141610bc have rank \(0\).
Complex multiplication
The elliptic curves in class 141610bc do not have complex multiplication.Modular form 141610.2.a.bc
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.