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: | $1$ | ||||||
| $\Q$-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: | 24.384.5.2818 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&12\\4&11\end{bmatrix}$, $\begin{bmatrix}13&6\\8&11\end{bmatrix}$, $\begin{bmatrix}19&15\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_6:D_8$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.fq.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.d, 192.2.c.a, 576.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 3 x z - w^{2} $ |
| $=$ | $x^{2} + 2 x y - 2 y^{2} - 2 y z - z^{2}$ | |
| $=$ | $4 x^{2} - x y + y^{2} + y z - z^{2} + w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{8} - 144 x^{6} y^{2} - 36 x^{6} z^{2} + 40 x^{4} y^{4} + 168 x^{4} y^{2} z^{2} + \cdots + 12 y^{2} z^{6} $ |
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 192 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(w-t)^{3}(w+t)^{3}(546z^{2}w^{16}+204z^{2}w^{14}t^{2}-1470z^{2}w^{12}t^{4}+23208z^{2}w^{10}t^{6}-50574z^{2}w^{8}t^{8}+51324z^{2}w^{6}t^{10}-28938z^{2}w^{4}t^{12}+8736z^{2}w^{2}t^{14}-1092z^{2}t^{16}-547w^{18}+159w^{16}t^{2}-1242w^{14}t^{4}+7854w^{12}t^{6}-13296w^{10}t^{8}+10404w^{8}t^{10}-3906w^{6}t^{12}+582w^{4}t^{14}-9w^{2}t^{16}+t^{18})}{t^{2}w^{8}(2w^{2}-t^{2})(12z^{2}w^{10}+12z^{2}w^{8}t^{2}-132z^{2}w^{6}t^{4}+204z^{2}w^{4}t^{6}-120z^{2}w^{2}t^{8}+24z^{2}t^{10}+4w^{12}+6w^{10}t^{2}+141w^{8}t^{4}-272w^{6}t^{6}+228w^{4}t^{8}-96w^{2}t^{10}+16t^{12})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.fq.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{8}-144X^{6}Y^{2}+40X^{4}Y^{4}-16X^{2}Y^{6}+4Y^{8}-36X^{6}Z^{2}+168X^{4}Y^{2}Z^{2}-44X^{2}Y^{4}Z^{2}+16Y^{6}Z^{2}+9X^{4}Z^{4}-72X^{2}Y^{2}Z^{4}+16Y^{4}Z^{4}+12Y^{2}Z^{6} $ |
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.da.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.da.1.7 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.dk.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dk.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.ds.3.7 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.ds.3.11 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.fr.1.4 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.fr.1.5 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.fv.1.3 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.fv.1.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.gr.3.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gr.3.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gz.3.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gz.3.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
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.fg.1.2 | $24$ | $3$ | $3$ | $25$ | $3$ | $1^{10}\cdot2^{5}$ |