Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $7 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $10^{4}\cdot20^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20B7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.221 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&16\\30&19\end{bmatrix}$, $\begin{bmatrix}7&6\\34&13\end{bmatrix}$, $\begin{bmatrix}7&24\\22&23\end{bmatrix}$, $\begin{bmatrix}27&30\\10&7\end{bmatrix}$, $\begin{bmatrix}31&24\\38&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.7.h.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{28}\cdot5^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 1600.2.a.f, 1600.2.a.j, 1600.2.a.v, 1600.2.a.x |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w - 2 x t + x u - z u - z v $ |
| $=$ | $x t - x u - y w - y t + y u - y v + z w - z t$ | |
| $=$ | $2 x w + x t - 2 x v + y u + y v + z u + z v$ | |
| $=$ | $x t - x u - 2 y w + y t - y v - 2 z w - z u + 2 z v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 7200 x^{12} + 28800 x^{11} y - 40800 x^{10} y^{2} - 1600 x^{10} z^{2} - 60000 x^{9} y^{3} + \cdots + 2 y^{4} z^{8} $ |
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
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 10.60.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle 2x-y-3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4x+2y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x+3y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.7.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 7200X^{12}+28800X^{11}Y-40800X^{10}Y^{2}-1600X^{10}Z^{2}-60000X^{9}Y^{3}-9200X^{9}YZ^{2}+293000X^{8}Y^{4}+9000X^{8}Y^{2}Z^{2}+160X^{8}Z^{4}-457200X^{7}Y^{5}-34050X^{7}Y^{3}Z^{2}+320X^{7}YZ^{4}+368200X^{6}Y^{6}+101825X^{6}Y^{4}Z^{2}+3610X^{6}Y^{2}Z^{4}-145200X^{5}Y^{7}-98150X^{5}Y^{5}Z^{2}-8570X^{5}Y^{3}Z^{4}-80X^{5}YZ^{6}-12000X^{4}Y^{8}+27000X^{4}Y^{6}Z^{2}+6110X^{4}Y^{4}Z^{4}+180X^{4}Y^{2}Z^{6}+30000X^{3}Y^{9}+4300X^{3}Y^{7}Z^{2}-2330X^{3}Y^{5}Z^{4}-220X^{3}Y^{3}Z^{6}-3800X^{2}Y^{10}-325X^{2}Y^{8}Z^{2}+1010X^{2}Y^{6}Z^{4}+160X^{2}Y^{4}Z^{6}+2X^{2}Y^{2}Z^{8}-1200XY^{11}-500XY^{9}Z^{2}-200XY^{7}Z^{4}-60XY^{5}Z^{6}-4XY^{3}Z^{8}+200Y^{12}+100Y^{10}Z^{2}+50Y^{8}Z^{4}+20Y^{6}Z^{6}+2Y^{4}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.120.3-10.a.1.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.120.3-10.a.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.13-40.a.1.8 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.b.1.3 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.j.1.7 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.k.1.20 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{6}$ |
| 40.480.13-40.bk.1.8 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-40.bl.1.8 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-40.bt.1.6 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.bu.1.5 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.15-40.bj.1.1 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bj.1.8 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bk.1.8 | $40$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 40.480.15-40.bk.1.13 | $40$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 40.480.15-40.bl.1.1 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.bl.1.15 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.bm.1.2 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.bm.1.13 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.br.1.2 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.br.1.14 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bs.1.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bs.1.14 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bx.1.4 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bx.1.11 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.by.1.6 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.by.1.16 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.720.19-40.bz.1.8 | $40$ | $3$ | $3$ | $19$ | $5$ | $1^{12}$ |
| 120.480.13-120.y.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.z.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bh.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bi.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.fc.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.fd.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.fl.1.12 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.fm.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-120.ct.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ct.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cu.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cu.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cz.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cz.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.da.1.31 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.da.1.32 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ed.1.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ed.1.14 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ee.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ee.1.31 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ej.1.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ej.1.14 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ek.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ek.1.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-280.dv.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.dw.1.12 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.eb.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ec.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ff.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.fg.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.fl.1.12 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.fm.1.12 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-280.dq.1.28 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dq.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dr.1.5 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dr.1.18 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dt.1.20 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dt.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.du.1.24 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.du.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.eo.1.23 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.eo.1.28 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ep.1.23 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ep.1.32 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.er.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.er.1.18 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.es.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.es.1.26 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |