Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $800$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $5^{2}\cdot10\cdot20\cdot40^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40C8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.8.157 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&12\\21&23\end{bmatrix}$, $\begin{bmatrix}21&20\\10&7\end{bmatrix}$, $\begin{bmatrix}25&24\\2&5\end{bmatrix}$, $\begin{bmatrix}39&16\\27&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.da.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $48$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{26}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 50.2.a.b$^{3}$, 200.2.a.e, 800.2.d.a, 800.2.d.c |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ y w - y t + z^{2} - z v - u^{2} + u v $ |
| $=$ | $2 x t - y w - z^{2} + z u + z r - u r$ | |
| $=$ | $x^{2} - x t + y w + 2 t^{2}$ | |
| $=$ | $x w + x t - y w + z v + w t - t^{2} - u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 324 x^{14} + 216 x^{13} z - 72 x^{12} y^{2} - 5508 x^{12} z^{2} + 408 x^{11} y^{2} z - 5664 x^{11} z^{3} + \cdots + 179776 z^{14} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:-1:0:0:-1:2:1)$, $(0:0:1:0:0:-1:0:1)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 40.60.4.bl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
| $\displaystyle W$ | $=$ | $\displaystyle -w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}-XZ+2Z^{2}+YW $ |
| $=$ | $ 2X^{2}Z-Y^{2}Z-2Z^{3}+XYW-YZW+2XW^{2}+2ZW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.8.da.2 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2r$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 324X^{14}-72X^{12}Y^{2}-4X^{10}Y^{4}+216X^{13}Z+408X^{11}Y^{2}Z-84X^{9}Y^{4}Z-4X^{7}Y^{6}Z-5508X^{12}Z^{2}+580X^{10}Y^{2}Z^{2}+236X^{8}Y^{4}Z^{2}-4X^{6}Y^{6}Z^{2}-5664X^{11}Z^{3}-3488X^{9}Y^{2}Z^{3}+412X^{7}Y^{4}Z^{3}+4X^{5}Y^{6}Z^{3}+38140X^{10}Z^{4}-1583X^{8}Y^{2}Z^{4}-634X^{6}Y^{4}Z^{4}+3X^{4}Y^{6}Z^{4}+52928X^{9}Z^{5}+9532X^{7}Y^{2}Z^{5}-1350X^{5}Y^{4}Z^{5}-35X^{3}Y^{6}Z^{5}-XY^{8}Z^{5}-131356X^{8}Z^{6}-2486X^{6}Y^{2}Z^{6}+455X^{4}Y^{4}Z^{6}+90X^{2}Y^{6}Z^{6}+Y^{8}Z^{6}-240880X^{7}Z^{7}-4512X^{5}Y^{2}Z^{7}+2688X^{3}Y^{4}Z^{7}-2XY^{6}Z^{7}+210352X^{6}Z^{8}+25845X^{4}Y^{2}Z^{8}-535X^{2}Y^{4}Z^{8}-52Y^{6}Z^{8}+573528X^{5}Z^{9}-15280X^{3}Y^{2}Z^{9}-2020XY^{4}Z^{9}-58028X^{4}Z^{10}-53800X^{2}Y^{2}Z^{10}+852Y^{4}Z^{10}-679048X^{3}Z^{11}+12296XY^{2}Z^{11}-217564X^{2}Z^{12}+31264Y^{2}Z^{12}+313760XZ^{13}+179776Z^{14} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.ba.1.1 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.ba.1.1 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-40.bl.1.1 | $40$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 40.120.4-40.bl.1.19 | $40$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.16-40.bk.2.1 | $40$ | $2$ | $2$ | $16$ | $5$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.br.1.8 | $40$ | $2$ | $2$ | $16$ | $4$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.bt.1.12 | $40$ | $2$ | $2$ | $16$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.bu.1.8 | $40$ | $2$ | $2$ | $16$ | $6$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.by.2.1 | $40$ | $2$ | $2$ | $16$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.ca.1.7 | $40$ | $2$ | $2$ | $16$ | $4$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.cc.1.7 | $40$ | $2$ | $2$ | $16$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.ce.1.7 | $40$ | $2$ | $2$ | $16$ | $4$ | $1^{4}\cdot2^{2}$ |
| 40.720.22-40.gq.2.5 | $40$ | $3$ | $3$ | $22$ | $2$ | $1^{6}\cdot4^{2}$ |
| 40.960.29-40.baa.2.1 | $40$ | $4$ | $4$ | $29$ | $5$ | $1^{9}\cdot2^{2}\cdot4^{2}$ |
| 80.480.16-80.cg.1.4 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.ci.2.2 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.co.2.6 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.cq.2.8 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.cu.1.3 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.da.2.1 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.dc.2.5 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.di.2.7 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.17-80.bw.2.10 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.cc.2.12 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.ce.2.16 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.ck.1.14 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.cu.2.9 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.cw.2.11 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.dc.2.15 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.de.1.13 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.16-120.eo.2.2 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.es.2.4 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.ew.2.2 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.fa.2.4 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.fy.1.13 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.gc.1.13 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.gg.2.9 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.gk.1.13 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.dq.1.31 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.ds.1.29 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.eg.1.29 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.ei.1.31 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.fi.1.28 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.fs.1.26 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.fy.1.22 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.gi.1.24 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.17-240.em.1.21 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.ew.1.23 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.fc.1.15 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.fm.2.11 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.im.2.5 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.io.2.7 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.jc.1.7 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.je.2.3 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.16-280.eu.2.1 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.ey.1.15 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.fc.1.7 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.fg.2.15 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.fk.1.13 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.fo.1.13 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.fs.2.9 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.fw.1.13 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |