Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | 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: | $4$ (none of which are rational) | Cusp widths | $10^{2}\cdot20^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.120.4.207 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}17&23\\34&29\end{bmatrix}$, $\begin{bmatrix}29&38\\12&1\end{bmatrix}$, $\begin{bmatrix}37&3\\30&21\end{bmatrix}$, $\begin{bmatrix}37&16\\8&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.60.4.e.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{12}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b, 200.2.a.e, 400.2.a.f$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 7 x^{2} - y^{2} - z^{2} - w^{2} $ |
| $=$ | $x^{3} + x y^{2} + x z^{2} + y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{6} - 4 x^{4} z^{2} - 7 x^{2} y^{2} z^{2} + x^{2} z^{4} + 4 y^{4} z^{2} + 4 y^{2} z^{4} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^3\,\frac{29796480xyz^{7}w+644576639xyz^{5}w^{3}+806648570xyz^{3}w^{5}+113758603xyzw^{7}+1105920y^{2}z^{8}+121015480y^{2}z^{6}w^{2}+460794875y^{2}z^{4}w^{4}+225042322y^{2}z^{2}w^{6}+16741279y^{2}w^{8}+1009152z^{10}+65496440z^{8}w^{2}+182910399z^{6}w^{4}+148461757z^{4}w^{6}+32329237z^{2}w^{8}+2061215w^{10}}{2520xyz^{7}w+1358xyz^{5}w^{3}-140xyz^{3}w^{5}-98xyzw^{7}-320y^{2}z^{8}-825y^{2}z^{6}w^{2}-225y^{2}z^{4}w^{4}-7y^{2}z^{2}w^{6}+y^{2}w^{8}+128z^{10}-200z^{8}w^{2}-707z^{6}w^{4}-477z^{4}w^{6}-97z^{2}w^{8}+w^{10}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.60.4.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{6}-4X^{4}Z^{2}-7X^{2}Y^{2}Z^{2}+X^{2}Z^{4}+4Y^{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 |
|---|---|---|---|---|---|---|
| 40.24.0-4.c.1.2 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.60.2-20.b.1.3 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 40.60.2-20.b.1.6 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.240.9-40.q.1.3 | $40$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 40.240.9-40.r.1.3 | $40$ | $2$ | $2$ | $9$ | $2$ | $1^{3}\cdot2$ |
| 40.240.9-40.s.1.2 | $40$ | $2$ | $2$ | $9$ | $2$ | $1^{3}\cdot2$ |
| 40.240.9-40.t.1.3 | $40$ | $2$ | $2$ | $9$ | $2$ | $1^{3}\cdot2$ |
| 40.360.10-20.i.1.8 | $40$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 40.480.13-20.r.1.5 | $40$ | $4$ | $4$ | $13$ | $1$ | $1^{9}$ |
| 120.240.9-120.q.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.r.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.s.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.t.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.360.14-60.q.1.16 | $120$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.480.17-60.i.1.22 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 280.240.9-280.q.1.5 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.240.9-280.r.1.8 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.240.9-280.s.1.7 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.240.9-280.t.1.6 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |