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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 51600cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51600.bg1 | 51600cf1 | \([0, -1, 0, -3208, -115088]\) | \(-2282665/2322\) | \(-3715200000000\) | \([]\) | \(86400\) | \(1.1075\) | \(\Gamma_0(N)\)-optimal |
51600.bg2 | 51600cf2 | \([0, -1, 0, 26792, 2044912]\) | \(1329238535/1908168\) | \(-3053068800000000\) | \([]\) | \(259200\) | \(1.6568\) |
Rank
sage: E.rank()
The elliptic curves in class 51600cf have rank \(1\).
Complex multiplication
The elliptic curves in class 51600cf do not have complex multiplication.Modular form 51600.2.a.cf
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.