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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 28665bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28665.bg1 | 28665bc1 | \([1, -1, 0, -450, 895]\) | \(117649/65\) | \(5574797865\) | \([2]\) | \(13824\) | \(0.56065\) | \(\Gamma_0(N)\)-optimal |
28665.bg2 | 28665bc2 | \([1, -1, 0, 1755, 5746]\) | \(6967871/4225\) | \(-362361861225\) | \([2]\) | \(27648\) | \(0.90722\) |
Rank
sage: E.rank()
The elliptic curves in class 28665bc have rank \(0\).
Complex multiplication
The elliptic curves in class 28665bc do not have complex multiplication.Modular form 28665.2.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.