Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.362 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&34\\21&39\end{bmatrix}$, $\begin{bmatrix}33&24\\2&13\end{bmatrix}$, $\begin{bmatrix}37&28\\22&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.1.j.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - y z $ |
| $=$ | $4 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(3z^{2}-w^{2})^{3}}{z^{2}(z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.24.1.j.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+Y^{2}Z^{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 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0-4.c.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
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.r.1.3 | $40$ | $5$ | $5$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.288.9-40.z.1.6 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
| 40.480.17-40.fx.1.7 | $40$ | $10$ | $10$ | $17$ | $4$ | $1^{12}\cdot2^{2}$ |
| 80.96.3-16.s.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-16.s.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-16.u.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-16.u.1.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-80.w.1.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-80.w.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-80.y.1.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3-80.y.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.5-24.bh.1.2 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.192.5-24.v.1.5 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 240.96.3-48.s.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-48.s.1.7 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-48.u.1.3 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-48.u.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-240.w.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-240.w.1.11 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-240.y.1.7 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3-240.y.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.384.13-56.v.1.12 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |