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 | $2^{4}\cdot4^{2}$ | ||
| 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: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.680 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&0\\12&5\end{bmatrix}$, $\begin{bmatrix}13&4\\16&21\end{bmatrix}$, $\begin{bmatrix}19&20\\12&13\end{bmatrix}$, $\begin{bmatrix}21&16\\16&21\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.f.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| 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 x^{2} - 3 y^{2} - w^{2} $ |
| $=$ | $3 x^{2} + 3 y^{2} - z^{2}$ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(z^{4}-2z^{3}w+2z^{2}w^{2}+2zw^{3}+w^{4})^{3}(z^{4}+2z^{3}w+2z^{2}w^{2}-2zw^{3}+w^{4})^{3}}{w^{4}z^{4}(z-w)^{4}(z+w)^{4}(z^{2}+w^{2})^{4}}$ |
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.h.1.2 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.0-24.a.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.a.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.b.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.b.1.23 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.ba.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.ba.2.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bb.2.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bb.2.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-8.h.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.m.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.m.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.q.2.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.q.2.13 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.384.5-24.r.1.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.r.3.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.s.1.7 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.s.3.7 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.576.17-24.ku.1.22 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.dq.2.9 | $24$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 48.384.5-48.y.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.y.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.z.1.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.z.2.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.dn.3.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.dn.4.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.do.3.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.do.4.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.bo.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bo.3.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bp.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bp.4.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bo.1.15 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bo.2.15 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bp.1.15 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bp.2.15 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.gg.3.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.gg.4.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.gh.3.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.gh.4.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.9-240.sj.1.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.sj.2.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.sk.1.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.sk.2.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.5-264.bo.1.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bo.2.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bp.1.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bp.2.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bo.1.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bo.2.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bp.1.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bp.2.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |