Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| 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 | $20^{6}$ | 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: | 20A8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.8.4 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&1\\8&17\end{bmatrix}$, $\begin{bmatrix}13&12\\28&5\end{bmatrix}$, $\begin{bmatrix}29&7\\36&11\end{bmatrix}$, $\begin{bmatrix}37&24\\36&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.120.8.f.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{22}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 50.2.a.b$^{3}$, 200.2.a.e, 400.2.a.a$^{2}$, 400.2.a.f$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x y - x z + x w - x u - x v + y^{2} - y w - y t + z^{2} - z w + z r - u v - v^{2} - v r $ |
| $=$ | $x z - x t + x u + x v + x r + y z + y t + y v + y r - z v + v^{2} + v r$ | |
| $=$ | $x y - x z + x w + x t + x v + y^{2} - y z - y w - y v - y r + z v + z r + w v + w r - u v - v^{2} + r^{2}$ | |
| $=$ | $x y + x w + x t - 2 x u + 2 x v + 2 x r - 2 y z - y t + 2 y v - z t - u v - u r - v^{2} - v r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 31 x^{11} + 142 x^{10} y + 66 x^{10} z - 292 x^{9} y^{2} - 230 x^{9} y z - 32 x^{9} z^{2} + \cdots + 32 y^{5} z^{6} $ |
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 |
|---|
| $(0:0:1:1:0:0:0:0)$, $(0:0:0:-1:0:0:0:1)$ |
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 20.60.4.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle u+v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t$ |
| $\displaystyle W$ | $=$ | $\displaystyle -2y-z+w+t+v+r$ |
Equation of the image curve:
| $0$ | $=$ | $ 7X^{2}+Y^{2}+2ZW+W^{2} $ |
| $=$ | $ X^{3}-XY^{2}+XZ^{2}+YZ^{2}+YZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.120.8.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ -31X^{11}+142X^{10}Y+66X^{10}Z-292X^{9}Y^{2}-230X^{9}YZ-32X^{9}Z^{2}+540X^{8}Y^{3}+666X^{8}Y^{2}Z+396X^{8}YZ^{2}+130X^{8}Z^{3}-487X^{7}Y^{4}-908X^{7}Y^{3}Z-750X^{7}Y^{2}Z^{2}-368X^{7}YZ^{3}-34X^{7}Z^{4}+530X^{6}Y^{5}+1188X^{6}Y^{4}Z+1406X^{6}Y^{3}Z^{2}+1024X^{6}Y^{2}Z^{3}+402X^{6}YZ^{4}+78X^{6}Z^{5}-142X^{5}Y^{6}-726X^{5}Y^{5}Z-1222X^{5}Y^{4}Z^{2}-1226X^{5}Y^{3}Z^{3}-762X^{5}Y^{2}Z^{4}-282X^{5}YZ^{5}-40X^{5}Z^{6}+164X^{4}Y^{7}+436X^{4}Y^{6}Z+726X^{4}Y^{5}Z^{2}+948X^{4}Y^{4}Z^{3}+818X^{4}Y^{3}Z^{4}+410X^{4}Y^{2}Z^{5}+116X^{4}YZ^{6}+14X^{4}Z^{7}+44X^{3}Y^{8}-72X^{3}Y^{7}Z-302X^{3}Y^{6}Z^{2}-510X^{3}Y^{5}Z^{3}-611X^{3}Y^{4}Z^{4}-454X^{3}Y^{3}Z^{5}-232X^{3}Y^{2}Z^{6}-72X^{3}YZ^{7}-7X^{3}Z^{8}+48X^{2}Y^{9}-24X^{2}Y^{8}Z-260X^{2}Y^{7}Z^{2}-320X^{2}Y^{6}Z^{3}-100X^{2}Y^{5}Z^{4}+128X^{2}Y^{4}Z^{5}+136X^{2}Y^{3}Z^{6}+48X^{2}Y^{2}Z^{7}+8X^{2}YZ^{8}-96XY^{9}Z-180XY^{8}Z^{2}+56XY^{7}Z^{3}+416XY^{6}Z^{4}+472XY^{5}Z^{5}+252XY^{4}Z^{6}+64XY^{3}Z^{7}+8XY^{2}Z^{8}+48Y^{9}Z^{2}+160Y^{8}Z^{3}+208Y^{7}Z^{4}+128Y^{6}Z^{5}+32Y^{5}Z^{6} $ |
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 |
| 8.48.0-4.c.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 |
|---|---|---|---|---|---|---|
| 8.48.0-4.c.1.1 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.80.2-20.b.1.1 | $40$ | $3$ | $3$ | $2$ | $1$ | $1^{6}$ |
| 40.120.4-20.h.1.2 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-20.h.1.3 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.17-40.fa.1.2 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{7}\cdot2$ |
| 40.480.17-40.fe.1.4 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{7}\cdot2$ |
| 40.480.17-40.gm.1.2 | $40$ | $2$ | $2$ | $17$ | $8$ | $1^{7}\cdot2$ |
| 40.480.17-40.gq.1.2 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{7}\cdot2$ |
| 40.480.17-40.hs.1.2 | $40$ | $2$ | $2$ | $17$ | $6$ | $1^{7}\cdot2$ |
| 40.480.17-40.hw.1.2 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{7}\cdot2$ |
| 40.480.17-40.ii.1.2 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.im.1.3 | $40$ | $2$ | $2$ | $17$ | $8$ | $1^{7}\cdot2$ |
| 40.720.22-20.p.1.1 | $40$ | $3$ | $3$ | $22$ | $4$ | $1^{14}$ |
| 40.960.29-20.w.1.1 | $40$ | $4$ | $4$ | $29$ | $7$ | $1^{21}$ |
| 120.480.17-120.pa.1.7 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.pi.1.8 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ro.1.7 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.rw.1.7 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.tk.1.8 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ts.1.7 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.vg.1.7 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.vo.1.7 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.om.1.8 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ou.1.8 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ps.1.6 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.qa.1.6 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.qy.1.7 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.rg.1.7 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.se.1.6 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.sm.1.4 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |