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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 199410.bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 199410.bo1 | 199410y3 | \([1, 1, 1, -31905606, 69353018919]\) | \(148809678420065817601/20700\) | \(499647678300\) | \([2]\) | \(7077888\) | \(2.5696\) | |
| 199410.bo2 | 199410y6 | \([1, 1, 1, -7464876, -6727521177]\) | \(1905890658841300321/293666194803750\) | \(7088388040042957083750\) | \([2]\) | \(14155776\) | \(2.9162\) | |
| 199410.bo3 | 199410y4 | \([1, 1, 1, -2046126, 1023458823]\) | \(39248884582600321/3935264062500\) | \(94987707841814062500\) | \([2, 2]\) | \(7077888\) | \(2.5696\) | |
| 199410.bo4 | 199410y2 | \([1, 1, 1, -1994106, 1083011319]\) | \(36330796409313601/428490000\) | \(10342706940810000\) | \([2, 2]\) | \(3538944\) | \(2.2230\) | |
| 199410.bo5 | 199410y1 | \([1, 1, 1, -121386, 17808183]\) | \(-8194759433281/965779200\) | \(-23311562078764800\) | \([2]\) | \(1769472\) | \(1.8765\) | \(\Gamma_0(N)\)-optimal |
| 199410.bo6 | 199410y5 | \([1, 1, 1, 2540304, 4964119479]\) | \(75108181893694559/484313964843750\) | \(-11690161744079589843750\) | \([2]\) | \(14155776\) | \(2.9162\) |
Rank
sage: E.rank()
The elliptic curves in class 199410.bo have rank \(1\).
Complex multiplication
The elliptic curves in class 199410.bo do not have complex multiplication.Modular form 199410.2.a.bo
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
