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: | 24.192.1.225 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&14\\16&17\end{bmatrix}$, $\begin{bmatrix}13&20\\8&17\end{bmatrix}$, $\begin{bmatrix}19&16\\0&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3:\GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.cc.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: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 2 y^{2} - z w + w^{2} $ |
| $=$ | $2 x^{2} - 2 y^{2} + z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 4 x^{2} y^{2} + 2 x^{2} z^{2} + y^{4} + 4 y^{2} z^{2} + 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{(z^{4}-2z^{3}w+6z^{2}w^{2}+4zw^{3}+4w^{4})^{3}(13z^{4}-38z^{3}w+42z^{2}w^{2}-20zw^{3}+4w^{4})^{3}}{z^{4}(z-2w)^{4}(z^{2}-zw+w^{2})^{4}(z^{2}+2zw-2w^{2})^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.cc.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}+4X^{2}Y^{2}+Y^{4}+2X^{2}Z^{2}+4Y^{2}Z^{2}+Z^{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.h.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.h.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.j.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.j.2.15 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.k.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bc.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bc.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.bd.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bd.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bf.2.11 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bf.2.13 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bu.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bu.1.14 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.ble.1.10 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.oc.2.5 | $24$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.384.5-48.bj.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.bz.1.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.ed.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.ej.1.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 240.384.5-240.sw.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.te.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.xe.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.xm.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |