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 | $2^{2}\cdot4^{5}$ | ||
| 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: | 24.384.5.2829 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&0\\4&1\end{bmatrix}$, $\begin{bmatrix}13&3\\12&19\end{bmatrix}$, $\begin{bmatrix}19&0\\4&23\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_4:D_{12}$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.fw.4 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{24}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 48.2.c.a, 144.2.a.b, 192.2.a.d, 576.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x y + z^{2} $ |
| $=$ | $3 x^{2} - y^{2} + z^{2} - t^{2}$ | |
| $=$ | $2 x y - 3 z^{2} + 3 w^{2} + 2 t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2601 x^{8} - 1224 x^{7} z + 1692 x^{6} y^{2} + 348 x^{6} z^{2} + 2448 x^{5} y^{2} z + 1992 x^{5} z^{3} + \cdots + 9 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^3}{3}\cdot\frac{(3w^{2}-2t^{2})^{3}(9552816y^{2}w^{16}+55707264y^{2}w^{14}t^{2}+95364864y^{2}w^{12}t^{4}+982513152y^{2}w^{10}t^{6}+288354816y^{2}w^{8}t^{8}+436672512y^{2}w^{6}t^{10}+18837504y^{2}w^{4}t^{12}+4890624y^{2}w^{2}t^{14}+372736y^{2}t^{16}-10766601w^{18}-60426810w^{16}t^{2}-193470768w^{14}t^{4}-52418016w^{12}t^{6}+34268832w^{10}t^{8}-22845888w^{8}t^{10}+15531264w^{6}t^{12}+25477632w^{4}t^{14}+3536640w^{2}t^{16}+280064t^{18})}{t^{2}w^{2}(3w^{2}+2t^{2})^{4}(1944y^{2}w^{10}+9072y^{2}w^{8}t^{2}-22464y^{2}w^{6}t^{4}+14976y^{2}w^{4}t^{6}-2688y^{2}w^{2}t^{8}-256y^{2}t^{10}+729w^{12}+4374w^{10}t^{2}+55404w^{8}t^{4}+15120w^{6}t^{6}+24624w^{4}t^{8}+864w^{2}t^{10}+64t^{12})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.fw.4 :
| $\displaystyle X$ | $=$ | $\displaystyle x+\frac{1}{2}t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4z+4w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y-\frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 2601X^{8}+1692X^{6}Y^{2}+36X^{4}Y^{4}-1224X^{7}Z+2448X^{5}Y^{2}Z+348X^{6}Z^{2}+744X^{4}Y^{2}Z^{2}+1992X^{5}Z^{3}+528X^{3}Y^{2}Z^{3}-170X^{4}Z^{4}-36X^{2}Y^{2}Z^{4}+8X^{3}Z^{5}+412X^{2}Z^{6}+120XZ^{7}+9Z^{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.1.14 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dl.3.8 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dl.3.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dq.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dq.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.fy.2.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.fy.2.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gd.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gd.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gs.4.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gs.4.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gx.2.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gx.2.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
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.fa.1.5 | $24$ | $3$ | $3$ | $25$ | $2$ | $1^{10}\cdot2^{5}$ |