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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 86490bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86490.bj4 | 86490bf1 | \([1, -1, 0, 12013281, -17984202035]\) | \(296354077829711/387386634240\) | \(-250635349552628264693760\) | \([2]\) | \(11059200\) | \(3.1757\) | \(\Gamma_0(N)\)-optimal |
| 86490.bj3 | 86490bf2 | \([1, -1, 0, -73784799, -175286401907]\) | \(68663623745397169/19216056254400\) | \(12432599761200559813425600\) | \([2]\) | \(22118400\) | \(3.5223\) | |
| 86490.bj2 | 86490bf3 | \([1, -1, 0, -342941679, -2458987043747]\) | \(-6894246873502147249/47925198774000\) | \(-31007133146620049891526000\) | \([2]\) | \(33177600\) | \(3.7250\) | |
| 86490.bj1 | 86490bf4 | \([1, -1, 0, -5496188859, -156832751516135]\) | \(28379906689597370652529/1357352437500\) | \(878193702543072306937500\) | \([2]\) | \(66355200\) | \(4.0716\) |
Rank
sage: E.rank()
The elliptic curves in class 86490bf have rank \(0\).
Complex multiplication
The elliptic curves in class 86490bf do not have complex multiplication.Modular form 86490.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
