Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $2^{8}\cdot6^{8}\cdot8^{4}\cdot24^{4}$ | Cusp orbits | $2^{4}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24Z5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.5.2817 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&15\\20&5\end{bmatrix}$, $\begin{bmatrix}13&18\\16&23\end{bmatrix}$, $\begin{bmatrix}19&12\\4&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_6:\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.fh.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{27}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 72.2.a.a, 192.2.a.b, 192.2.c.a, 576.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 3 x z - w^{2} $ |
| $=$ | $x^{2} - 2 x y + x z - 2 y^{2} - 2 y z - z^{2} - w^{2}$ | |
| $=$ | $4 x^{2} + x y + x z + y^{2} + y z - z^{2} + w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 1296 x^{8} + 36 x^{6} y^{2} + 432 x^{6} z^{2} - 72 x^{5} y z^{2} + x^{4} y^{4} + 24 x^{4} y^{2} z^{2} + \cdots + 7 z^{8} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=23$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 192 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{(w^{2}+t^{2})^{3}(546z^{2}w^{16}-204z^{2}w^{14}t^{2}-1470z^{2}w^{12}t^{4}-23208z^{2}w^{10}t^{6}-50574z^{2}w^{8}t^{8}-51324z^{2}w^{6}t^{10}-28938z^{2}w^{4}t^{12}-8736z^{2}w^{2}t^{14}-1092z^{2}t^{16}-547w^{18}-159w^{16}t^{2}-1242w^{14}t^{4}-7854w^{12}t^{6}-13296w^{10}t^{8}-10404w^{8}t^{10}-3906w^{6}t^{12}-582w^{4}t^{14}-9w^{2}t^{16}-t^{18})}{t^{2}w^{8}(2w^{2}+t^{2})(12z^{2}w^{10}-12z^{2}w^{8}t^{2}-132z^{2}w^{6}t^{4}-204z^{2}w^{4}t^{6}-120z^{2}w^{2}t^{8}-24z^{2}t^{10}+4w^{12}-6w^{10}t^{2}+141w^{8}t^{4}+272w^{6}t^{6}+228w^{4}t^{8}+96w^{2}t^{10}+16t^{12})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.fh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-\frac{1}{3}z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4y+4w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 1296X^{8}+36X^{6}Y^{2}+X^{4}Y^{4}+432X^{6}Z^{2}-72X^{5}YZ^{2}+24X^{4}Y^{2}Z^{2}-4X^{3}Y^{3}Z^{2}-144X^{4}Z^{4}-48X^{3}YZ^{4}+3X^{2}Y^{2}Z^{4}-12X^{2}Z^{6}+2XYZ^{6}+7Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.1-24.dc.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dc.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.df.4.6 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.df.4.14 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dk.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dk.2.7 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.fe.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.fe.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.fz.1.4 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.fz.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gm.3.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gm.3.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gr.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gr.1.13 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.1152.25-24.eh.1.2 | $24$ | $3$ | $3$ | $25$ | $2$ | $1^{10}\cdot2^{5}$ |