Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}\cdot12^{4}\cdot24^{4}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AF9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.9.2307 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&0\\0&5\end{bmatrix}$, $\begin{bmatrix}5&2\\0&1\end{bmatrix}$, $\begin{bmatrix}5&16\\0&19\end{bmatrix}$, $\begin{bmatrix}13&12\\0&11\end{bmatrix}$, $\begin{bmatrix}19&4\\12&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_6:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.192.9.ca.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: | $2^{46}\cdot3^{11}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 24.2.a.a, 64.2.a.a$^{2}$, 72.2.a.a, 144.2.a.b, 192.2.a.a, 192.2.a.c, 576.2.a.b, 576.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x t + y z $ |
| $=$ | $y v + z r - t r$ | |
| $=$ | $x r - y r + 2 z v + t v$ | |
| $=$ | $y w - y r + 2 z w - z v - w t - w s + t r + v s - r s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 606528 x^{12} + 101088 x^{11} y + 16848 x^{10} y^{2} - 46656 x^{10} z^{2} - 23328 x^{9} y z^{2} + \cdots + z^{12} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=41$, and therefore no rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.5.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x+y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle w$ |
| $\displaystyle T$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ XZ+YZ+XT $ |
| $=$ | $ YZ-2W^{2}-XT $ | |
| $=$ | $ XY+Y^{2}+2Z^{2}-ZT-T^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.9.ca.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 6s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2r$ |
Equation of the image curve:
| $0$ | $=$ | $ 606528X^{12}+101088X^{11}Y+16848X^{10}Y^{2}-46656X^{10}Z^{2}-23328X^{9}YZ^{2}-3888X^{8}Y^{2}Z^{2}-432X^{7}Y^{3}Z^{2}-47952X^{8}Z^{4}-15120X^{7}YZ^{4}-5112X^{6}Y^{2}Z^{4}-864X^{5}Y^{3}Z^{4}-72X^{4}Y^{4}Z^{4}-2376X^{6}Z^{6}-2304X^{5}YZ^{6}-756X^{4}Y^{2}Z^{6}-84X^{3}Y^{3}Z^{6}+252X^{4}Z^{8}-138X^{3}YZ^{8}-23X^{2}Y^{2}Z^{8}+6X^{2}Z^{10}+2XYZ^{10}+Z^{12} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-24.u.1.3 | $24$ | $4$ | $4$ | $1$ | $0$ | $1^{8}$ |
| 24.192.3-12.e.1.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{6}$ |
| 24.192.3-12.e.1.21 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{6}$ |
| 24.192.3-24.cg.1.3 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{6}$ |
| 24.192.3-24.cg.1.12 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{6}$ |
| 24.192.5-24.d.1.8 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.d.1.15 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.768.17-24.bv.1.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.bv.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.bv.3.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.bv.4.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.gc.1.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.gc.2.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.gg.1.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.gg.2.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.gi.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{2}\cdot4$ |
| 24.768.17-24.gi.2.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{2}\cdot4$ |
| 24.768.17-24.gi.3.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{2}\cdot4$ |
| 24.768.17-24.gi.4.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{2}\cdot4$ |
| 24.768.17-24.gj.1.5 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{2}\cdot4$ |
| 24.768.17-24.gj.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{2}\cdot4$ |
| 24.768.17-24.gj.3.9 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{2}\cdot4$ |
| 24.768.17-24.gj.4.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{2}\cdot4$ |
| 24.768.17-24.gm.1.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.gm.2.2 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.go.1.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.go.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.iz.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.iz.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.iz.3.12 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.768.17-24.iz.4.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.1152.33-24.ev.1.6 | $24$ | $3$ | $3$ | $33$ | $6$ | $1^{24}$ |