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: | 24.384.5.2822 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&3\\4&11\end{bmatrix}$, $\begin{bmatrix}7&18\\12&11\end{bmatrix}$, $\begin{bmatrix}19&15\\20&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_6:\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.ez.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: | $2^{28}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 144.2.a.b, 192.2.a.b, 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} $ |
| $=$ | $x^{2} - 2 x y - x z - 2 y^{2} + 2 y z - z^{2} + w^{2}$ | |
| $=$ | $4 x^{2} + x y - x z + y^{2} - y z - z^{2} - w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 625 x^{8} - 9000 x^{7} z - 522 x^{6} y^{2} + 56100 x^{6} z^{2} + 4980 x^{5} y^{2} z - 197640 x^{5} z^{3} + \cdots + 50625 z^{8} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{(w^{2}+t^{2})^{3}(546z^{2}w^{16}-204z^{2}w^{14}t^{2}-1470z^{2}w^{12}t^{4}-23208z^{2}w^{10}t^{6}-50574z^{2}w^{8}t^{8}-51324z^{2}w^{6}t^{10}-28938z^{2}w^{4}t^{12}-8736z^{2}w^{2}t^{14}-1092z^{2}t^{16}+547w^{18}+159w^{16}t^{2}+1242w^{14}t^{4}+7854w^{12}t^{6}+13296w^{10}t^{8}+10404w^{8}t^{10}+3906w^{6}t^{12}+582w^{4}t^{14}+9w^{2}t^{16}+t^{18})}{t^{2}w^{8}(2w^{2}+t^{2})(12z^{2}w^{10}-12z^{2}w^{8}t^{2}-132z^{2}w^{6}t^{4}-204z^{2}w^{4}t^{6}-120z^{2}w^{2}t^{8}-24z^{2}t^{10}-4w^{12}+6w^{10}t^{2}-141w^{8}t^{4}-272w^{6}t^{6}-228w^{4}t^{8}-96w^{2}t^{10}-16t^{12})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.ez.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+2y-z+3t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}w+\frac{5}{3}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 625X^{8}-522X^{6}Y^{2}-243X^{4}Y^{4}-9000X^{7}Z+4980X^{5}Y^{2}Z+1620X^{3}Y^{4}Z+56100X^{6}Z^{2}-20350X^{4}Y^{2}Z^{2}-4050X^{2}Y^{4}Z^{2}-197640X^{5}Z^{3}+45720X^{3}Y^{2}Z^{3}+4500XY^{4}Z^{3}+430326X^{4}Z^{4}-59598X^{2}Y^{2}Z^{4}-1875Y^{4}Z^{4}-592920X^{3}Z^{5}+42660XY^{2}Z^{5}+504900X^{2}Z^{6}-13050Y^{2}Z^{6}-243000XZ^{7}+50625Z^{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.da.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.da.1.8 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.df.3.7 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.df.3.11 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dn.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dn.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.fa.1.2 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.fa.1.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.fr.1.4 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.fr.1.12 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.gm.3.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gm.3.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gu.3.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gu.3.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
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.ep.1.2 | $24$ | $3$ | $3$ | $25$ | $3$ | $1^{10}\cdot2^{5}$ |