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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 2112.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2112.bd1 | 2112bd3 | \([0, 1, 0, -644161, 198779327]\) | \(112763292123580561/1932612\) | \(506622640128\) | \([2]\) | \(19200\) | \(1.7876\) | |
| 2112.bd2 | 2112bd4 | \([0, 1, 0, -643521, 199194687]\) | \(-112427521449300721/466873642818\) | \(-122388124222881792\) | \([2]\) | \(38400\) | \(2.1341\) | |
| 2112.bd3 | 2112bd1 | \([0, 1, 0, -2881, -44353]\) | \(10091699281/2737152\) | \(717527973888\) | \([2]\) | \(3840\) | \(0.98285\) | \(\Gamma_0(N)\)-optimal |
| 2112.bd4 | 2112bd2 | \([0, 1, 0, 7359, -279873]\) | \(168105213359/228637728\) | \(-59936008568832\) | \([2]\) | \(7680\) | \(1.3294\) |
Rank
sage: E.rank()
The elliptic curves in class 2112.bd have rank \(0\).
Complex multiplication
The elliptic curves in class 2112.bd do not have complex multiplication.Modular form 2112.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
