Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 40.120.4.239 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&38\\4&9\end{bmatrix}$, $\begin{bmatrix}17&37\\16&3\end{bmatrix}$, $\begin{bmatrix}31&24\\12&27\end{bmatrix}$, $\begin{bmatrix}33&11\\0&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.60.4.l.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: | $2^{10}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b$^{2}$, 400.2.a.d, 400.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 35 x^{2} + 5 y^{2} - z^{2} + w^{2} $ |
| $=$ | $5 x^{3} - 5 x y^{2} + x z^{2} - y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 100 x^{6} - 20 x^{4} z^{2} + 35 x^{2} y^{2} z^{2} - x^{2} z^{4} + 20 y^{4} z^{2} + 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^3\,\frac{148982400xyz^{7}w-3222883195xyz^{5}w^{3}+4033242850xyz^{3}w^{5}-568793015xyzw^{7}+5529600y^{2}z^{8}-605077400y^{2}z^{6}w^{2}+2303974375y^{2}z^{4}w^{4}-1125211610y^{2}z^{2}w^{6}+83706395y^{2}w^{8}-1009152z^{10}+65496440z^{8}w^{2}-182910399z^{6}w^{4}+148461757z^{4}w^{6}-32329237z^{2}w^{8}+2061215w^{10}}{12600xyz^{7}w-6790xyz^{5}w^{3}-700xyz^{3}w^{5}+490xyzw^{7}-1600y^{2}z^{8}+4125y^{2}z^{6}w^{2}-1125y^{2}z^{4}w^{4}+35y^{2}z^{2}w^{6}+5y^{2}w^{8}-128z^{10}-200z^{8}w^{2}+707z^{6}w^{4}-477z^{4}w^{6}+97z^{2}w^{8}+w^{10}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.60.4.l.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -100X^{6}-20X^{4}Z^{2}+35X^{2}Y^{2}Z^{2}-X^{2}Z^{4}+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.h.1.2 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.60.2-20.c.1.1 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 40.60.2-20.c.1.11 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.360.10-20.t.1.3 | $40$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 40.480.13-20.bu.1.2 | $40$ | $4$ | $4$ | $13$ | $2$ | $1^{9}$ |
| 40.240.8-40.cm.1.4 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.cm.1.5 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.cn.1.5 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.cn.1.10 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.cw.1.3 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.cw.1.6 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.cx.1.3 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.cx.1.6 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 120.360.14-60.bn.1.8 | $120$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.480.17-60.x.1.2 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 120.240.8-120.eo.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.eo.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ep.1.5 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ep.1.16 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ew.1.5 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ew.1.16 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ex.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ex.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.di.1.12 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.di.1.13 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dj.1.10 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dj.1.15 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dm.1.10 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dm.1.15 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dn.1.12 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dn.1.13 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |