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: | $8$ (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: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.279 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}21&0\\12&11\end{bmatrix}$, $\begin{bmatrix}25&36\\8&35\end{bmatrix}$, $\begin{bmatrix}31&16\\24&3\end{bmatrix}$, $\begin{bmatrix}37&0\\2&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.bc.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $7680$ |
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 $ | $=$ | $ 2 x y - 2 y^{2} - z^{2} $ |
| $=$ | $5 x^{2} + 3 x y + 2 y^{2} + z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 5 x^{2} y^{2} + 3 x^{2} z^{2} + 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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{5^3}\cdot\frac{1968750y^{2}z^{10}+4725000y^{2}z^{8}w^{2}+360000y^{2}z^{6}w^{4}-288000y^{2}z^{4}w^{6}-2419200y^{2}z^{2}w^{8}-645120y^{2}w^{10}+484375z^{12}+750000z^{10}w^{2}-480000z^{8}w^{4}-512000z^{6}w^{6}-1632000z^{4}w^{8}-983040z^{2}w^{10}-131072w^{12}}{w^{4}z^{4}(10y^{2}z^{2}-8y^{2}w^{2}+5z^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.48.1.bc.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+5X^{2}Y^{2}+3X^{2}Z^{2}+2Z^{4} $ |
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.d.2.10 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.d.2.5 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.h.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.h.1.18 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-40.c.1.4 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1-40.c.1.12 | $40$ | $2$ | $2$ | $1$ | $1$ | 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.b.2.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.h.1.5 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.bb.1.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.bd.2.7 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.bt.1.3 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.bv.2.8 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.ca.2.4 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.192.1-40.cb.1.6 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.480.17-40.bw.1.5 | $40$ | $5$ | $5$ | $17$ | $4$ | $1^{6}\cdot2^{5}$ |
| 40.576.17-40.em.2.2 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.960.33-40.is.2.6 | $40$ | $10$ | $10$ | $33$ | $5$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 120.192.1-120.fq.2.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.fu.2.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.gx.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.hb.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lz.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.md.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.nf.2.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.nj.2.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-120.ro.1.17 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-120.je.1.25 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.192.1-280.fr.2.11 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ft.2.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.gh.2.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.gj.2.6 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.id.2.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.if.2.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.it.2.10 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.iv.2.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |