Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $200$ | ||
| 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 | $2^{4}$ | ||
| 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: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.694 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&22\\6&19\end{bmatrix}$, $\begin{bmatrix}5&38\\24&25\end{bmatrix}$, $\begin{bmatrix}9&2\\14&19\end{bmatrix}$, $\begin{bmatrix}21&20\\10&29\end{bmatrix}$, $\begin{bmatrix}27&14\\8&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.120.7.d.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: | $2^{13}\cdot5^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 20.2.a.a, 40.2.a.a, 50.2.a.a, 50.2.a.b$^{2}$, 100.2.a.a, 200.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} + 2 x y - x z - y^{2} - y z + w t - t v $ |
| $=$ | $x^{2} - y^{2} + y z + z^{2} - w v + t^{2} - t u - u v$ | |
| $=$ | $x y + 2 x z - y^{2} - y z - w^{2} - w t - w u - t u$ | |
| $=$ | $2 y^{2} - z^{2} + w^{2} + 2 w t + w v + t^{2} - v^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{10} - 267 x^{8} y^{2} + 29 x^{8} z^{2} + 39 x^{7} y^{3} + 258 x^{7} y z^{2} + 178 x^{6} y^{4} + \cdots + 81 y^{2} 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 :
| $\displaystyle X$ | $=$ | $\displaystyle 2x-3y-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4x+y+2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x-y+3z$ |
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 20.120.7.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{10}-267X^{8}Y^{2}+29X^{8}Z^{2}+39X^{7}Y^{3}+258X^{7}YZ^{2}+178X^{6}Y^{4}-1382X^{6}Y^{2}Z^{2}-125X^{6}Z^{4}-511X^{5}Y^{5}+950X^{5}Y^{3}Z^{2}+813X^{5}YZ^{4}+462X^{4}Y^{6}+1833X^{4}Y^{4}Z^{2}-1756X^{4}Y^{2}Z^{4}-203X^{4}Z^{6}-279X^{3}Y^{7}-482X^{3}Y^{5}Z^{2}+133X^{3}Y^{3}Z^{4}+628X^{3}YZ^{6}+102X^{2}Y^{8}-1256X^{2}Y^{6}Z^{2}+1621X^{2}Y^{4}Z^{4}-386X^{2}Y^{2}Z^{6}-81X^{2}Z^{8}-17XY^{9}+178XY^{7}Z^{2}-224XY^{5}Z^{4}-18XY^{3}Z^{6}+81XYZ^{8}+Y^{10}+78Y^{6}Z^{4}-160Y^{4}Z^{6}+81Y^{2}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-20.b.1.1 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.120.3-10.a.1.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 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 |
|---|---|---|---|---|---|---|
| 40.480.13-20.f.1.4 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-20.h.1.3 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-20.n.1.4 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-20.p.1.3 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-40.q.1.5 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.w.1.1 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.bo.1.5 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.bu.1.1 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.15-20.d.1.5 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.d.1.6 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.e.1.5 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-20.e.1.6 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.g.1.4 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
| 40.480.15-40.g.1.5 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
| 40.480.15-40.j.1.1 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.j.1.8 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-20.n.1.3 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.n.1.4 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.o.1.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.o.1.3 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bu.1.1 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bu.1.8 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bx.1.4 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bx.1.5 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.720.19-20.o.1.1 | $40$ | $3$ | $3$ | $19$ | $1$ | $1^{12}$ |
| 120.480.13-60.r.1.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.t.1.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.bp.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.br.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ca.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cg.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.eu.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.fa.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-60.h.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.h.1.13 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.j.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.j.1.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.s.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.s.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.y.1.8 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.y.1.26 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.ba.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.ba.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bb.1.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bb.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.di.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.di.1.27 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dl.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dl.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-140.v.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.x.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.bd.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.bf.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cm.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cs.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.dk.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.dq.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-140.i.1.5 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.i.1.9 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.j.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.j.1.13 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.s.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.s.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.t.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.t.1.13 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.u.1.15 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.u.1.19 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.x.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.x.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ck.1.13 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ck.1.17 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cn.1.5 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cn.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |