Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20J3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.1083 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&0\\38&33\end{bmatrix}$, $\begin{bmatrix}13&20\\22&11\end{bmatrix}$, $\begin{bmatrix}17&31\\34&29\end{bmatrix}$, $\begin{bmatrix}19&37\\10&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.72.3.n.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{11}\cdot5^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 80.2.a.b, 200.2.a.c, 400.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - 2 z u + w t - w u $ |
| $=$ | $x z + 2 y w$ | |
| $=$ | $x^{2} + 4 y^{2} + 2 z w + w^{2}$ | |
| $=$ | $2 x^{2} - 2 y^{2} - z w + 2 w^{2} + 2 u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{2} + 5 x^{4} z^{2} + 4 x^{2} y^{4} - 10 x^{2} y^{2} z^{2} - 50 x^{2} z^{4} + 25 y^{2} z^{4} + 125 z^{6} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 25 w^{2} $ | $=$ | $ -7 x^{4} + 24 x^{3} y + 23 x^{2} z^{2} + 52 x y z^{2} + 11 z^{4} $ |
| $0$ | $=$ | $x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3125zw^{9}+10000zw^{7}u^{2}+12000zw^{5}u^{4}+25600zw^{3}u^{6}-966400zwu^{8}-6250w^{10}-13750w^{8}u^{2}-8000w^{6}u^{4}+1600w^{4}u^{6}-5120w^{2}u^{8}+32t^{10}-320t^{9}u+1440t^{8}u^{2}-3840t^{7}u^{3}+6720t^{6}u^{4}-6528t^{5}u^{5}+6720t^{4}u^{6}+11520t^{3}u^{7}+62880t^{2}u^{8}-78656tu^{9}+387616u^{10}}{u^{6}(5w^{2}+4u^{2})(5zw-2u^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.72.3.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}u$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{2}+4X^{2}Y^{4}+5X^{4}Z^{2}-10X^{2}Y^{2}Z^{2}-50X^{2}Z^{4}+25Y^{2}Z^{4}+125Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-20.f.1.4 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.72.1-20.b.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.288.5-20.s.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-20.s.2.6 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-20.t.1.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-20.t.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.ew.1.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.ew.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.fd.1.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.fd.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.720.19-20.bq.1.4 | $40$ | $5$ | $5$ | $19$ | $3$ | $1^{16}$ |
| 120.288.5-60.fi.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fi.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fj.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fj.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bmk.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bmk.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bmr.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bmr.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.15-60.cb.1.5 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-140.by.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.by.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.bz.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.bz.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.nm.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.nm.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.nt.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.nt.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |