Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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 $4$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.657 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&4\\0&27\end{bmatrix}$, $\begin{bmatrix}35&9\\24&29\end{bmatrix}$, $\begin{bmatrix}37&15\\32&19\end{bmatrix}$, $\begin{bmatrix}39&24\\16&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.bb.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(1:0:1)$, $(0:0:1)$, $(-1:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{9970x^{2}y^{12}z^{2}+97971x^{2}y^{8}z^{6}-18255x^{2}y^{4}z^{10}+4095x^{2}z^{14}-172xy^{14}z+161439xy^{10}z^{5}-112816xy^{6}z^{9}+20481xy^{2}z^{13}+y^{16}-200932y^{12}z^{4}-71428y^{8}z^{8}+16206y^{4}z^{12}+z^{16}}{zy^{4}(13x^{2}y^{8}z+501x^{2}y^{4}z^{5}+255x^{2}z^{9}+xy^{10}+268xy^{6}z^{4}+769xy^{2}z^{8}+70y^{8}z^{3}+522y^{4}z^{7}+z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-4.c.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-4.c.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.q.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-8.m.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1-8.m.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.192.1-8.l.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-8.l.2.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.cj.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.cj.2.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.fe.1.8 | $40$ | $5$ | $5$ | $17$ | $3$ | $1^{14}\cdot2$ |
| 40.576.17-40.mt.1.4 | $40$ | $6$ | $6$ | $17$ | $1$ | $1^{14}\cdot2$ |
| 40.960.33-40.xg.1.5 | $40$ | $10$ | $10$ | $33$ | $7$ | $1^{28}\cdot2^{2}$ |
| 80.192.3-16.ck.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.ck.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cn.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cn.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cu.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cu.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gn.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gn.1.12 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gr.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gr.1.13 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hg.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hg.1.12 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.1-24.cs.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cs.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qi.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qi.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-24.uu.1.14 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-24.hw.1.6 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.192.3-48.fl.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fl.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fp.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fp.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.ge.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.ge.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.qv.1.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.qv.1.18 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rd.1.11 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rd.1.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sq.1.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sq.1.19 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.192.1-56.cj.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.cj.2.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.pn.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.pn.2.5 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |