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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 53040bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53040.s7 | 53040bp1 | \([0, -1, 0, -33538336, -71975108864]\) | \(1018563973439611524445729/42904970360310988800\) | \(175738758595833810124800\) | \([2]\) | \(6635520\) | \(3.2249\) | \(\Gamma_0(N)\)-optimal |
| 53040.s6 | 53040bp2 | \([0, -1, 0, -88916256, 227021356800]\) | \(18980483520595353274840609/5549773448629762560000\) | \(22731872045587507445760000\) | \([2, 2]\) | \(13271040\) | \(3.5715\) | |
| 53040.s5 | 53040bp3 | \([0, -1, 0, -412684576, 3205946556160]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(58402521476949491712000000\) | \([2]\) | \(19906560\) | \(3.7742\) | |
| 53040.s8 | 53040bp4 | \([0, -1, 0, 238093024, 1508636126976]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-1847141587507710873600000000\) | \([2]\) | \(26542080\) | \(3.9180\) | |
| 53040.s4 | 53040bp5 | \([0, -1, 0, -1301972256, 18080294342400]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(36185774110975229932339200\) | \([4]\) | \(26542080\) | \(3.9180\) | |
| 53040.s2 | 53040bp6 | \([0, -1, 0, -6591000096, 205958491312896]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(1807852350096000000000000\) | \([2, 2]\) | \(39813120\) | \(4.1208\) | |
| 53040.s3 | 53040bp7 | \([0, -1, 0, -6579048416, 206742626695680]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-239303504882812500000000000000\) | \([2]\) | \(79626240\) | \(4.4673\) | |
| 53040.s1 | 53040bp8 | \([0, -1, 0, -105456000096, 13181238367312896]\) | \(31664865542564944883878115208137569/103216295812500\) | \(422773947648000000\) | \([4]\) | \(79626240\) | \(4.4673\) |
Rank
sage: E.rank()
The elliptic curves in class 53040bp have rank \(0\).
Complex multiplication
The elliptic curves in class 53040bp do not have complex multiplication.Modular form 53040.2.a.bp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
