Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.1.475 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&8\\2&19\end{bmatrix}$, $\begin{bmatrix}9&12\\30&29\end{bmatrix}$, $\begin{bmatrix}31&36\\10&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.1.bh.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $3840$ |
Jacobian
| Conductor: | $2^{5}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 800.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 2 y^{2} - 2 z^{2} $ |
| $=$ | $3 x^{2} - 4 y^{2} - z^{2} + 2 w^{2}$ |
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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^4}\cdot\frac{(625z^{8}-100z^{4}w^{4}+16w^{8})^{3}}{w^{8}z^{8}(5z^{2}-2w^{2})^{2}(5z^{2}+2w^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.f.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.f.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.e.2.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.e.2.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.s.2.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.s.2.14 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.t.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.t.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-40.v.1.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.v.1.5 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.be.2.3 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.be.2.13 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.bf.1.11 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.bf.1.16 | $40$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 40.960.33-40.dh.1.1 | $40$ | $5$ | $5$ | $33$ | $6$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.kp.2.5 | $40$ | $6$ | $6$ | $33$ | $7$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.op.2.3 | $40$ | $10$ | $10$ | $65$ | $11$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |