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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 45486.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45486.bf1 | 45486x3 | \([1, -1, 1, -54939575, -111701748977]\) | \(19804628171203875/5638671302656\) | \(5221432607921773675020288\) | \([2]\) | \(9953280\) | \(3.4496\) | |
| 45486.bf2 | 45486x1 | \([1, -1, 1, -50428880, -137824793421]\) | \(11165451838341046875/572244736\) | \(726887459323775232\) | \([2]\) | \(3317760\) | \(2.9003\) | \(\Gamma_0(N)\)-optimal |
| 45486.bf3 | 45486x2 | \([1, -1, 1, -50342240, -138322037709]\) | \(-11108001800138902875/79947274872976\) | \(-101552129458604613320112\) | \([2]\) | \(6635520\) | \(3.2468\) | |
| 45486.bf4 | 45486x4 | \([1, -1, 1, 144678985, -736986926321]\) | \(361682234074684125/462672528510976\) | \(-428436647126229696340635648\) | \([2]\) | \(19906560\) | \(3.7961\) |
Rank
sage: E.rank()
The elliptic curves in class 45486.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 45486.bf do not have complex multiplication.Modular form 45486.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
