Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $200$ | ||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $10^{2}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.120.4.221 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&31\\12&19\end{bmatrix}$, $\begin{bmatrix}5&9\\32&7\end{bmatrix}$, $\begin{bmatrix}23&26\\8&13\end{bmatrix}$, $\begin{bmatrix}29&1\\0&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.60.4.k.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.a, 50.2.a.b$^{2}$, 200.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 9 x^{2} - x z + y^{2} - y w + z^{2} - w^{2} $ |
| $=$ | $2 x^{3} + 2 x^{2} z - 2 x y^{2} + x y w - 2 x z^{2} - y^{2} z - 2 y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{6} - 2 x^{5} z + 19 x^{4} y^{2} + x^{4} z^{2} - 27 x^{3} y^{2} z + 2 x^{3} z^{3} + \cdots + 4 y^{2} z^{4} $ |
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 |
|---|
| $(0:0:-1:1)$, $(0:0:1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^{14}}{5^5}\cdot\frac{114470795868099xyz^{7}w-916060272059371xyz^{5}w^{3}+673169576903640xyz^{3}w^{5}-54611080203632xyzw^{7}-23237199384160xz^{9}+387165229553113xz^{7}w^{2}-387923529196492xz^{5}w^{4}+41401686125835xz^{3}w^{6}-1459702950064xzw^{8}+82021158760260y^{3}z^{6}w-296479838203700y^{3}z^{4}w^{3}+119762018603280y^{3}z^{2}w^{5}-5911384076320y^{3}w^{7}-30126140675456y^{2}z^{8}+503093575634309y^{2}z^{6}w^{2}-560320961989785y^{2}z^{4}w^{4}+71264242322388y^{2}z^{2}w^{6}-428593124320y^{2}w^{8}-1674886005622yz^{8}w+297096511917158yz^{6}w^{3}-278498393074820yz^{4}w^{5}-3564641215444yz^{2}w^{7}+2583392526960yw^{9}-9166639243820z^{10}+136788053158285z^{8}w^{2}-243529252916890z^{6}w^{4}+130374165700685z^{4}w^{6}-14830641398900z^{2}w^{8}+363414700640w^{10}}{4677309xyz^{7}w-4222061xyz^{5}w^{3}+561615xyz^{3}w^{5}+221113xyzw^{7}+1839440xz^{9}+337758xz^{7}w^{2}-2910622xz^{5}w^{4}+1253610xz^{3}w^{6}-140074xzw^{8}+1113660y^{3}z^{6}w-156700y^{3}z^{4}w^{3}-103020y^{3}z^{2}w^{5}+6380y^{3}w^{7}+1940929y^{2}z^{8}-3956181y^{2}z^{6}w^{2}+756815y^{2}z^{4}w^{4}+122433y^{2}z^{2}w^{6}-11620y^{2}w^{8}+2807498yz^{8}w-2248222yz^{6}w^{3}-417870yz^{4}w^{5}+239846yz^{2}w^{7}-1140yw^{9}-66620z^{10}-635440z^{8}w^{2}+1455760z^{6}w^{4}-972040z^{4}w^{6}+213100z^{2}w^{8}+5240w^{10}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.60.4.k.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{6}-2X^{5}Z+19X^{4}Y^{2}+X^{4}Z^{2}-27X^{3}Y^{2}Z+2X^{3}Z^{3}-20X^{2}Y^{4}+31X^{2}Y^{2}Z^{2}-X^{2}Z^{4}-20XY^{4}Z-8XY^{2}Z^{3}+20Y^{4}Z^{2}+4Y^{2}Z^{4} $ |
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.g.1.3 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.60.2-20.c.1.6 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 40.60.2-20.c.1.9 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.240.8-40.ci.1.2 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 40.240.8-40.ci.1.8 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 40.240.8-40.cj.1.6 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.cj.1.9 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.cs.1.5 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.cs.1.8 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.ct.1.2 | $40$ | $2$ | $2$ | $8$ | $4$ | $1^{4}$ |
| 40.240.8-40.ct.1.8 | $40$ | $2$ | $2$ | $8$ | $4$ | $1^{4}$ |
| 40.360.10-20.s.1.9 | $40$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 40.480.13-20.bt.1.4 | $40$ | $4$ | $4$ | $13$ | $1$ | $1^{9}$ |
| 120.240.8-120.ek.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ek.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.el.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.el.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.es.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.es.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.et.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.et.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.360.14-60.bm.1.16 | $120$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.480.17-60.w.1.28 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 280.240.8-280.da.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.da.1.16 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.db.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.db.1.16 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.de.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.de.1.16 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.df.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.df.1.16 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |