Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.133 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&19\\8&7\end{bmatrix}$, $\begin{bmatrix}15&8\\8&19\end{bmatrix}$, $\begin{bmatrix}23&8\\20&15\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3.\GL(2,\mathbb{Z}/4)$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.ey.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 y z - 3 z^{2} - 2 w^{2} $ |
| $=$ | $2 x^{2} + 2 x y - 4 x z + y^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 6 x^{3} y + 2 x^{2} y^{2} + 12 x^{2} z^{2} + 4 x y z^{2} + 4 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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^2}\cdot\frac{(3y^{2}-4w^{2})^{3}(3y^{2}+4w^{2})^{3}}{w^{8}y^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.ey.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}-6X^{3}Y+2X^{2}Y^{2}+12X^{2}Z^{2}+4XYZ^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-4.c.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-4.c.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bd.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bd.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.n.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1-24.n.1.5 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.288.9-24.bie.1.2 | $24$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
| 24.384.9-24.ky.1.7 | $24$ | $4$ | $4$ | $9$ | $2$ | $1^{8}$ |
| 120.480.17-120.vo.1.2 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |