Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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: | 24.192.1.1139 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&14\\16&7\end{bmatrix}$, $\begin{bmatrix}11&10\\20&5\end{bmatrix}$, $\begin{bmatrix}15&20\\16&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.bo.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: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 y z - y w + 2 z^{2} - 2 z w + 2 w^{2} $ |
| $=$ | $6 x^{2} + y^{2} + 2 y z - y w + 2 z^{2} - 2 z w - 4 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} - 4 x^{3} z + 12 x^{2} y^{2} - 6 x^{2} z^{2} - 12 x y^{2} z + 8 x z^{3} + 3 y^{2} z^{2} - z^{4} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^4}\cdot\frac{(y^{8}-12y^{6}w^{2}+45y^{4}w^{4}-54y^{2}w^{6}+81w^{8})^{3}}{w^{8}y^{4}(y^{2}-6w^{2})^{2}(y^{2}-3w^{2})^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.bo.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}+12X^{2}Y^{2}-4X^{3}Z-12XY^{2}Z-6X^{2}Z^{2}+3Y^{2}Z^{2}+8XZ^{3}-Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.n.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.0-24.m.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.m.2.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.n.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.n.1.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.v.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.v.2.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.w.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.w.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-8.n.1.8 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.x.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.x.1.11 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ba.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ba.1.13 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.576.17-24.te.1.6 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.hr.2.7 | $24$ | $4$ | $4$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |