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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 21840bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.a3 | 21840bf1 | \([0, -1, 0, -12136, 235120]\) | \(48264326765929/22299191460\) | \(91337488220160\) | \([2]\) | \(82944\) | \(1.3735\) | \(\Gamma_0(N)\)-optimal |
| 21840.a4 | 21840bf2 | \([0, -1, 0, 42744, 1727856]\) | \(2108526614950391/1540302022350\) | \(-6309077083545600\) | \([2]\) | \(165888\) | \(1.7200\) | |
| 21840.a1 | 21840bf3 | \([0, -1, 0, -823576, 287950576]\) | \(15082569606665230489/7751016000\) | \(31748161536000\) | \([2]\) | \(248832\) | \(1.9228\) | |
| 21840.a2 | 21840bf4 | \([0, -1, 0, -819096, 291233520]\) | \(-14837772556740428569/342100087875000\) | \(-1401241959936000000\) | \([2]\) | \(497664\) | \(2.2694\) |
Rank
sage: E.rank()
The elliptic curves in class 21840bf have rank \(1\).
Complex multiplication
The elliptic curves in class 21840bf do not have complex multiplication.Modular form 21840.2.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.
