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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 277200bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277200.bf3 | 277200bf1 | \([0, 0, 0, -16050, 667375]\) | \(2508888064/396165\) | \(72201071250000\) | \([2]\) | \(589824\) | \(1.3820\) | \(\Gamma_0(N)\)-optimal |
277200.bf2 | 277200bf2 | \([0, 0, 0, -71175, -6664250]\) | \(13674725584/1334025\) | \(3890016900000000\) | \([2, 2]\) | \(1179648\) | \(1.7285\) | |
277200.bf4 | 277200bf3 | \([0, 0, 0, 86325, -32021750]\) | \(6099383804/41507235\) | \(-484140389040000000\) | \([2]\) | \(2359296\) | \(2.0751\) | |
277200.bf1 | 277200bf4 | \([0, 0, 0, -1110675, -450530750]\) | \(12990838708516/144375\) | \(1683990000000000\) | \([2]\) | \(2359296\) | \(2.0751\) |
Rank
sage: E.rank()
The elliptic curves in class 277200bf have rank \(0\).
Complex multiplication
The elliptic curves in class 277200bf do not have complex multiplication.Modular form 277200.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.