Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| 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: | 24.192.1.226 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&22\\8&17\end{bmatrix}$, $\begin{bmatrix}7&2\\4&13\end{bmatrix}$, $\begin{bmatrix}21&16\\8&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3:\GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.cb.1 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: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - 2 y^{2} + w^{2} $ |
| $=$ | $2 x^{2} + y^{2} - z^{2} + z w$ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{(4z^{4}-20z^{3}w+42z^{2}w^{2}-38zw^{3}+13w^{4})^{3}(4z^{4}+4z^{3}w+6z^{2}w^{2}-2zw^{3}+w^{4})^{3}}{w^{4}(2z-w)^{4}(z^{2}-zw+w^{2})^{4}(2z^{2}-2zw-w^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.k.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.g.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.g.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.i.2.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.i.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.k.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.ba.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.ba.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.bc.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bc.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.be.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.be.2.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bu.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bu.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.bkz.2.6 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.nz.1.9 | $24$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.384.5-48.bb.2.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.bx.2.7 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.eb.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.ei.2.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 240.384.5-240.st.2.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.tb.2.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.xb.2.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.xj.2.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |