Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | 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 | $4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.936 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&5\\28&23\end{bmatrix}$, $\begin{bmatrix}7&23\\32&39\end{bmatrix}$, $\begin{bmatrix}27&32\\10&13\end{bmatrix}$, $\begin{bmatrix}33&24\\6&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.7.bz.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{33}\cdot5^{12}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b, 320.2.a.d, 320.2.a.f, 400.2.a.c, 400.2.a.f, 1600.2.a.b, 1600.2.a.j |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w + 3 x t - y u + z v $ |
| $=$ | $3 x w - x t - y u + y v - z u$ | |
| $=$ | $2 x^{2} + 2 y z - 2 z^{2} + 2 w^{2} + 2 w t + u^{2} - v^{2}$ | |
| $=$ | $2 x y + 4 x z + w u + w v - t u + t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 324 x^{12} + 432 x^{10} y^{2} - 288 x^{10} z^{2} + 216 x^{8} y^{4} - 1876 x^{8} y^{2} z^{2} + \cdots + 4 y^{4} z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no 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 20.60.3.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -5x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y+2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2y+z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-14X^{2}Y^{2}-Y^{4}+X^{2}YZ-7Y^{3}Z+9X^{2}Z^{2}-19Y^{2}Z^{2}-8YZ^{3}+14Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.7.bz.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 324X^{12}+432X^{10}Y^{2}-288X^{10}Z^{2}+216X^{8}Y^{4}-1876X^{8}Y^{2}Z^{2}+136X^{8}Z^{4}+48X^{6}Y^{6}-614X^{6}Y^{4}Z^{2}+1480X^{6}Y^{2}Z^{4}-32X^{6}Z^{6}+4X^{4}Y^{8}-160X^{4}Y^{6}Z^{2}-543X^{4}Y^{4}Z^{4}-484X^{4}Y^{2}Z^{6}+4X^{4}Z^{8}-14X^{2}Y^{8}Z^{2}-30X^{2}Y^{6}Z^{4}+32X^{2}Y^{4}Z^{6}+72X^{2}Y^{2}Z^{8}+Y^{8}Z^{4}+4Y^{6}Z^{6}+4Y^{4}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-40.p.1.4 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.120.3-20.b.1.3 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.120.3-20.b.1.6 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
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.jf.1.3 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 40.480.13-40.jg.1.3 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.jm.1.3 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.jn.1.1 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.lj.1.4 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.lk.1.3 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.lq.1.3 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.lr.1.2 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.720.19-40.mh.1.6 | $40$ | $3$ | $3$ | $19$ | $5$ | $1^{12}$ |
| 80.480.15-80.y.1.10 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.y.1.12 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.z.1.6 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.z.1.8 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.ba.1.6 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.ba.1.14 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.bb.1.4 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.bb.1.12 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.bc.1.4 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.bc.1.12 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.bd.1.6 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.bd.1.14 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.be.1.7 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.be.1.8 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.bf.1.11 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 80.480.15-80.bf.1.12 | $80$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.13-120.bjb.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bjc.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bji.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bjj.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bpn.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bpo.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bpu.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bpv.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.480.15-240.y.1.12 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.y.1.26 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.z.1.20 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.z.1.26 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.ba.1.14 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.ba.1.26 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.bb.1.14 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.bb.1.26 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.bc.1.5 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.bc.1.29 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.bd.1.9 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.bd.1.29 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.be.1.17 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.be.1.27 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.bf.1.17 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.480.15-240.bf.1.27 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-280.bal.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bam.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bas.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bat.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bcp.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bcq.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bcw.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bcx.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |