Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| 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: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20H3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.1175 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&24\\24&7\end{bmatrix}$, $\begin{bmatrix}7&26\\6&37\end{bmatrix}$, $\begin{bmatrix}13&6\\36&33\end{bmatrix}$, $\begin{bmatrix}23&23\\34&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.72.3.q.2 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: | $2^{10}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 40.2.c.a, 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x u + z w $ |
| $=$ | $x^{2} - x y + z^{2}$ | |
| $=$ | $x w - y w + z u$ | |
| $=$ | $w^{2} - 2 w t + u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} y^{2} + 5 x^{4} z^{2} + 2 x^{2} y^{2} z^{2} + 6 x^{2} z^{4} + y^{2} z^{4} + z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{8} - 8x^{6} - 22x^{4} - 40x^{2} - 25 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{131328y^{2}z^{8}+597888y^{2}z^{6}u^{2}+456480y^{2}z^{4}u^{4}-27568y^{2}z^{2}u^{6}-69941y^{2}u^{8}-331776z^{10}-380160z^{8}u^{2}+744768z^{6}u^{4}+626128z^{4}u^{6}-11940z^{2}u^{8}-1250wt^{9}-3750wt^{7}u^{2}-3200wt^{5}u^{4}+1450wt^{3}u^{6}+5970wtu^{8}+2500t^{10}+6875t^{8}u^{2}+4375t^{6}u^{4}-5275t^{4}u^{6}-52423t^{2}u^{8}-40000u^{10}}{4864y^{2}z^{8}-4544y^{2}z^{6}u^{2}-528y^{2}z^{4}u^{4}+260y^{2}z^{2}u^{6}-y^{2}u^{8}-12288z^{10}+2816z^{8}u^{2}+384z^{6}u^{4}-112z^{4}u^{6}+32z^{2}u^{8}-10wt^{3}u^{6}-16wtu^{8}-4t^{2}u^{8}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.72.3.q.2 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}Y^{2}+5X^{4}Z^{2}+2X^{2}Y^{2}Z^{2}+6X^{2}Z^{4}+Y^{2}Z^{4}+Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.72.3.q.2 :
| $\displaystyle X$ | $=$ | $\displaystyle u^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 10zw^{4}u^{3}+4zw^{2}u^{5}+2zu^{7}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle wu$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-20.b.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
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.c.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-20.i.1.6 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-20.r.2.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-20.s.2.6 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.bc.2.3 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.ci.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.em.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.ey.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.720.19-20.ce.1.5 | $40$ | $5$ | $5$ | $19$ | $1$ | $1^{6}\cdot2^{5}$ |
| 120.288.5-60.ey.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fa.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fw.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fy.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bju.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bki.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqg.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqu.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.15-60.dk.1.25 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-140.cu.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.cv.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.dc.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.dd.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xe.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xl.1.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zi.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zp.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |