Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| 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: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.1076 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&22\\8&9\end{bmatrix}$, $\begin{bmatrix}11&16\\16&15\end{bmatrix}$, $\begin{bmatrix}23&20\\20&21\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.j.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 y^{2} + 2 z^{2} - 2 w^{2} $ |
| $=$ | $6 x^{2} - 2 z^{2} + w^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(16z^{8}-32z^{6}w^{2}+20z^{4}w^{4}-4z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(z-w)^{2}(z+w)^{2}(2z^{2}-w^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.0-24.k.2.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.k.2.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.m.2.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.m.2.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.p.2.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.p.2.10 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.r.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.r.1.10 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-8.i.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.m.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.m.1.9 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.r.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.r.1.14 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.576.17-24.lg.1.10 | $24$ | $3$ | $3$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.dy.2.1 | $24$ | $4$ | $4$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.384.9-48.dt.2.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.du.2.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.dz.2.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.ea.2.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 240.384.9-240.tj.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.tk.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.tv.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.tw.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |