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: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.1087 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&4\\12&7\end{bmatrix}$, $\begin{bmatrix}13&16\\6&3\end{bmatrix}$, $\begin{bmatrix}31&20\\32&37\end{bmatrix}$, $\begin{bmatrix}35&12\\12&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.j.1 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y + y^{2} + 2 w^{2} $ |
| $=$ | $2 x^{2} + 3 x y - y^{2} - z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 2 x^{2} y^{2} - z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(z^{4}+16w^{4})^{3}}{w^{8}z^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.48.1.j.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}+2X^{2}Y^{2}-Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-8.c.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.c.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.h.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.h.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-8.c.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1-8.c.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.c.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-8.c.2.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-8.h.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-8.h.2.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.k.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.k.2.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.v.1.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.v.2.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.bd.1.7 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.576.17-40.cf.1.17 | $40$ | $6$ | $6$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.960.33-40.fl.1.23 | $40$ | $10$ | $10$ | $33$ | $11$ | $1^{28}\cdot2^{2}$ |
| 80.192.5-16.o.1.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.r.1.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.r.2.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.s.1.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.bn.1.8 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.bw.1.16 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.bw.2.10 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.by.1.8 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.1-24.k.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.k.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.v.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.v.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.bi.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.bi.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ch.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ch.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-24.dj.1.6 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-24.bw.1.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.192.5-48.bn.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.bw.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.bw.2.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.by.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ek.1.22 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.fl.1.20 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.fl.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.fq.1.22 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.1-56.k.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.k.2.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.v.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.v.2.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.bi.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.bi.2.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ch.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ch.2.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |