Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 24.192.1.304 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&4\\16&21\end{bmatrix}$, $\begin{bmatrix}11&16\\4&7\end{bmatrix}$, $\begin{bmatrix}17&10\\12&7\end{bmatrix}$, $\begin{bmatrix}17&12\\8&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.a.1 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: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y^{2} + 2 z^{2} $ |
| $=$ | $3 x^{2} - 3 y^{2} - w^{2}$ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{(36z^{4}-36z^{3}w+18z^{2}w^{2}-6zw^{3}+w^{4})^{3}(36z^{4}+36z^{3}w+18z^{2}w^{2}+6zw^{3}+w^{4})^{3}}{w^{8}z^{8}(6z^{2}-w^{2})^{2}(6z^{2}+w^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.a.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.a.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.b.2.10 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.b.2.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.p.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.p.1.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.q.2.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.q.2.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.n.2.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.n.2.5 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bg.2.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bg.2.13 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bh.1.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bh.1.13 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 24.384.5-24.i.1.3 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.384.5-24.k.2.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.l.1.2 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.384.5-24.n.4.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.576.17-24.kf.2.1 | $24$ | $3$ | $3$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.dd.2.11 | $24$ | $4$ | $4$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
| 120.384.5-120.p.1.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.q.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.u.1.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.w.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.p.2.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.q.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.u.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.w.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.p.2.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.q.2.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.u.2.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.w.2.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.p.2.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.q.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.u.2.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.w.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |