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 | $2^{6}\cdot4$ | ||
| 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: | 40.192.1.401 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}23&28\\8&19\end{bmatrix}$, $\begin{bmatrix}23&36\\12&15\end{bmatrix}$, $\begin{bmatrix}31&24\\8&13\end{bmatrix}$, $\begin{bmatrix}39&8\\20&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.1.x.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 x y - z^{2} $ |
| $=$ | $x^{2} + 2 x y - 5 y^{2} + 4 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 5 x^{2} y^{2} - 6 x^{2} z^{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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^2}\cdot\frac{(10000z^{8}+4000z^{6}w^{2}+500z^{4}w^{4}+20z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(5z^{2}+w^{2})^{2}(10z^{2}+w^{2})^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.x.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{5}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-5X^{2}Y^{2}-6X^{2}Z^{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.c.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.b.2.11 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.b.2.21 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.c.1.10 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.w.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.w.1.10 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.x.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.x.1.14 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-40.o.2.3 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.o.2.4 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.be.2.4 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.be.2.13 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.bf.2.9 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1-40.bf.2.14 | $40$ | $2$ | $2$ | $1$ | $1$ | 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.x.1.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.384.5-40.y.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.384.5-40.ba.2.4 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.384.5-40.bb.3.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.960.33-40.cv.2.3 | $40$ | $5$ | $5$ | $33$ | $4$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.jx.2.3 | $40$ | $6$ | $6$ | $33$ | $4$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.nr.1.5 | $40$ | $10$ | $10$ | $65$ | $6$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 80.384.5-80.f.1.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.l.1.6 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.bh.1.8 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.bj.1.6 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.eo.1.6 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.eq.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.fm.1.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.fs.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hi.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hk.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hs.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hu.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bd.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bj.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dt.1.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dv.1.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oe.1.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.og.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qq.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qw.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hf.2.10 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hg.2.10 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hl.2.10 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hm.1.10 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |