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 | $2^{2}\cdot4^{3}$ | ||
| 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: | 24.192.1.1219 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&10\\20&7\end{bmatrix}$, $\begin{bmatrix}15&16\\8&17\end{bmatrix}$, $\begin{bmatrix}21&2\\8&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3:\GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.be.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: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 y^{2} - 2 y z + 2 z^{2} + w^{2} $ |
| $=$ | $6 x^{2} - y^{2} - 2 y z + 2 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 2 x^{2} y^{2} + 24 x^{2} z^{2} + y^{4} + 6 y^{2} z^{2} + 9 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{70543872yz^{15}w^{8}+164602368yz^{13}w^{10}+152845056yz^{11}w^{12}+71850240yz^{9}w^{14}+17196624yz^{7}w^{16}+1520208yz^{5}w^{18}-85752yz^{3}w^{20}-12168yzw^{22}-2176782336z^{24}-8707129344z^{22}w^{2}-15237476352z^{20}w^{4}-15318097920z^{18}w^{6}-9765287424z^{16}w^{8}-4099942656z^{14}w^{10}-1129355136z^{12}w^{12}-191942784z^{10}w^{14}-15272064z^{8}w^{16}+1068984z^{6}w^{18}+447876z^{4}w^{20}+41652z^{2}w^{22}-2197w^{24}}{w^{8}(186624yz^{15}+435456yz^{13}w^{2}+404352yz^{11}w^{4}+190080yz^{9}w^{6}+48096yz^{7}w^{8}+6624yz^{5}w^{10}+496yz^{3}w^{12}+16yzw^{14}+31104z^{14}w^{2}+67392z^{12}w^{4}+57024z^{10}w^{6}+23184z^{8}w^{8}+4368z^{6}w^{10}+264z^{4}w^{12}-8z^{2}w^{14}-w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.be.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}+2X^{2}Y^{2}+Y^{4}+24X^{2}Z^{2}+6Y^{2}Z^{2}+9Z^{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.m.2.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.0-24.h.2.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.h.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.j.2.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.j.2.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.q.2.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.q.2.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.s.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.s.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-8.m.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.u.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.u.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.y.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.y.1.9 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.qa.2.2 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.gm.2.2 | $24$ | $4$ | $4$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |