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: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.241 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&14\\36&27\end{bmatrix}$, $\begin{bmatrix}5&4\\26&1\end{bmatrix}$, $\begin{bmatrix}5&6\\16&5\end{bmatrix}$, $\begin{bmatrix}27&4\\2&23\end{bmatrix}$, $\begin{bmatrix}27&20\\8&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.7.b.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: | $2^{28}\cdot5^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 1600.2.a.i, 1600.2.a.k, 1600.2.a.w, 1600.2.a.y |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t - x u + x v + y w + y t + y u - 2 y v - z w + z t $ |
| $=$ | $x w + x u + 2 y t + y u - 2 y v + z w + z t + z u - 2 z v$ | |
| $=$ | $2 x^{2} + 2 y^{2} - 2 y z - 2 z^{2} - w t + w v + u v - v^{2}$ | |
| $=$ | $2 x^{2} - 2 x z + 2 y z - 4 z^{2} + w u - t^{2} + t u + t v + u^{2} - u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 256 x^{10} + 384 x^{9} y - 176 x^{8} y^{2} + 40 x^{8} z^{2} - 752 x^{7} y^{3} + 456 x^{7} y z^{2} + \cdots + y^{2} z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, 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 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.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 256X^{10}+384X^{9}Y-176X^{8}Y^{2}+40X^{8}Z^{2}-752X^{7}Y^{3}+456X^{7}YZ^{2}-476X^{6}Y^{4}+328X^{6}Y^{2}Z^{2}-29X^{6}Z^{4}+176X^{5}Y^{5}-230X^{5}Y^{3}Z^{2}+145X^{5}YZ^{4}+376X^{4}Y^{6}-240X^{4}Y^{4}Z^{2}+190X^{4}Y^{2}Z^{4}-30X^{4}Z^{6}+192X^{3}Y^{7}+98X^{3}Y^{5}Z^{2}-83X^{3}Y^{3}Z^{4}-12X^{3}YZ^{6}+36X^{2}Y^{8}+196X^{2}Y^{6}Z^{2}-111X^{2}Y^{4}Z^{4}+52X^{2}Y^{2}Z^{6}-X^{2}Z^{8}+72XY^{7}Z^{2}-4XY^{5}Z^{4}-12XY^{3}Z^{6}-XYZ^{8}+36Y^{6}Z^{4}-8Y^{4}Z^{6}+Y^{2}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.1.3 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.24.0-8.b.1.1 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 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.d.1.8 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.f.1.7 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.j.1.1 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.l.1.3 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-40.bb.1.4 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.bd.1.3 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-40.bh.1.8 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.bj.1.3 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.15-40.b.1.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.b.1.3 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.c.1.4 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.c.1.6 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.i.1.3 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.i.1.4 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.j.1.3 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.j.1.4 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.bf.1.3 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bf.1.4 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bg.1.3 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bg.1.4 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bm.1.2 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.bm.1.3 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.bn.1.2 | $40$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 40.480.15-40.bn.1.3 | $40$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 40.720.19-40.bf.1.4 | $40$ | $3$ | $3$ | $19$ | $3$ | $1^{12}$ |
| 120.480.13-120.p.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.r.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.v.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.x.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dj.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dl.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dp.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dr.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-120.f.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.f.1.8 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.g.1.8 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.g.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.m.1.20 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.m.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.n.1.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.n.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bv.1.25 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bv.1.27 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bw.1.25 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bw.1.27 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.co.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.co.1.23 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cp.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cp.1.23 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-280.bz.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cb.1.12 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cf.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ch.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cx.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cz.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.dd.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.df.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-280.f.1.5 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.f.1.17 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.g.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.g.1.13 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.m.1.9 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.m.1.13 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.n.1.9 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.n.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bv.1.23 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bv.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bw.1.29 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bw.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cc.1.25 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cc.1.27 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cd.1.21 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cd.1.29 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |