Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.443 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&20\\38&37\end{bmatrix}$, $\begin{bmatrix}9&24\\6&39\end{bmatrix}$, $\begin{bmatrix}25&36\\8&23\end{bmatrix}$, $\begin{bmatrix}39&36\\32&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.be.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{5}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 800.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 275x - 1750 $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5^2}\cdot\frac{120x^{2}y^{14}+265491250x^{2}y^{12}z^{2}+79026301875000x^{2}y^{10}z^{4}+5747837250087890625x^{2}y^{8}z^{6}+134200360963593750000000x^{2}y^{6}z^{8}+1276514304000111236572265625x^{2}y^{4}z^{10}+5221768888319999771118164062500x^{2}y^{2}z^{12}+7636775567360000002384185791015625x^{2}z^{14}+25900xy^{14}z+20577225000xy^{12}z^{3}+3967774374609375xy^{10}z^{5}+200163304666113281250xy^{8}z^{7}+3697927782368505859375000xy^{6}z^{9}+29987168255998883056640625000xy^{4}z^{11}+109220770611200002384185791015625xy^{2}z^{13}+146184193638399999976158142089843750xz^{15}+y^{16}+2514000y^{14}z^{2}+1359213562500y^{12}z^{4}+153546889734375000y^{10}z^{6}+4917296730081054687500y^{8}z^{8}+61279076351282226562500000y^{6}z^{10}+340493631487980628967285156250y^{4}z^{12}+832083525632000038146972656250000y^{2}z^{14}+698164379647999999582767486572265625z^{16}}{zy^{4}(7175x^{2}y^{8}z+208750000x^{2}y^{6}z^{3}+896023828125x^{2}y^{4}z^{5}+195312500x^{2}y^{2}z^{7}+6103515625x^{2}z^{9}+xy^{10}+298750xy^{8}z^{2}+5082937500xy^{6}z^{4}+17151781250000xy^{4}z^{6}-1708984375xy^{2}z^{8}-61035156250xz^{10}+120y^{10}z+8153750y^{8}z^{3}+60701250000y^{6}z^{5}+81915449218750y^{4}z^{7}-39062500000y^{2}z^{9}-1068115234375z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.e.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.h.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.h.1.26 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-40.c.1.16 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1-40.c.1.20 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.192.1-40.h.1.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.x.1.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.bd.1.3 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.bh.1.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.bv.1.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.bz.1.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.cb.1.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.cd.1.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.480.17-40.by.2.3 | $40$ | $5$ | $5$ | $17$ | $2$ | $1^{6}\cdot2^{5}$ |
| 40.576.17-40.eu.1.7 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.960.33-40.ix.2.5 | $40$ | $10$ | $10$ | $33$ | $3$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 120.192.1-120.fw.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.gc.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.hd.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.hj.2.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.mf.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ml.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.nl.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.nr.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-120.rz.2.7 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-120.jj.2.4 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.192.1-280.fx.2.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.gb.1.13 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.gn.1.11 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.gr.2.11 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ij.1.13 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.in.2.13 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.iz.2.5 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.jd.1.13 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |