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 | $2^{2}\cdot4^{3}$ | ||
| 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: | 40.192.1.310 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}13&0\\6&3\end{bmatrix}$, $\begin{bmatrix}19&8\\6&33\end{bmatrix}$, $\begin{bmatrix}25&36\\18&19\end{bmatrix}$, $\begin{bmatrix}37&20\\8&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.1.b.2 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} + z^{2} $ |
| $=$ | $5 x z + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{4} + 2 x^{2} y^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{5^4}\cdot\frac{(625y^{8}-25y^{4}w^{4}+w^{8})^{3}}{w^{8}y^{8}(5y^{2}-w^{2})^{2}(5y^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.b.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 5y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 25X^{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 |
|---|---|---|---|---|---|---|
| 8.96.0-8.a.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.a.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.b.2.10 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.b.2.17 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.u.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.u.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.v.2.6 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.v.2.11 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-40.o.2.5 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.o.2.9 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.bc.1.3 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.bc.1.16 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.bd.1.3 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.bd.1.14 | $40$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 40.384.5-40.b.1.8 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.384.5-40.c.2.6 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.384.5-40.e.2.8 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.384.5-40.f.2.6 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.960.33-40.bu.1.2 | $40$ | $5$ | $5$ | $33$ | $8$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.gy.1.9 | $40$ | $6$ | $6$ | $33$ | $4$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.iw.2.6 | $40$ | $10$ | $10$ | $65$ | $10$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 120.384.5-120.m.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.o.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.w.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.y.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.j.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.k.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.p.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.q.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |