Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $192$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot8^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{9}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AK7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.7.2917 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\0&7\end{bmatrix}$, $\begin{bmatrix}5&4\\0&7\end{bmatrix}$, $\begin{bmatrix}13&18\\12&17\end{bmatrix}$, $\begin{bmatrix}17&10\\12&7\end{bmatrix}$, $\begin{bmatrix}19&18\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_{12}:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.192.7.bn.4 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: | $1\cdot2^{3}$ |
| Newforms: | 24.2.a.a, 24.2.d.a, 24.2.f.a, 192.2.c.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t - y z $ |
| $=$ | $x u - z v - w u$ | |
| $=$ | $2 z^{2} + t u$ | |
| $=$ | $x y + x v + y w - z t + z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 6 x^{6} y^{2} + 3 x^{6} z^{2} + 8 x^{4} y^{4} + 2 x^{4} y^{2} z^{2} + x^{4} z^{4} + 8 x^{2} y^{6} + \cdots + 4 y^{4} z^{4} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1:0:0:1:0:0)$, $(0:-1:0:0:1:0:0)$ |
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 24.96.3.bm.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y+2X^{2}Y^{2}+XY^{3}-2X^{2}YZ+2XY^{2}Z+2XYZ^{2}-XZ^{3}+YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.7.bn.4 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ -6X^{6}Y^{2}+3X^{6}Z^{2}+8X^{4}Y^{4}+2X^{4}Y^{2}Z^{2}+X^{4}Z^{4}+8X^{2}Y^{6}-12X^{2}Y^{4}Z^{2}+4X^{2}Y^{2}Z^{4}-8Y^{6}Z^{2}+4Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.3-24.bl.2.26 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.bl.2.37 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.bm.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.bm.1.47 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.bq.2.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.bq.2.47 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.768.13-24.cl.1.16 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 24.768.13-24.cp.1.10 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.768.13-24.ct.1.14 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 24.768.13-24.cx.6.9 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.768.13-24.dj.3.14 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.768.13-24.dr.3.9 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.768.13-24.dz.3.10 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.768.13-24.ed.7.9 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.768.17-24.bj.2.13 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 24.768.17-24.bo.2.8 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 24.768.17-24.cp.1.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 24.768.17-24.cq.1.15 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 24.768.17-24.fu.1.22 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 24.768.17-24.fz.4.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 24.768.17-24.hg.4.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 24.768.17-24.hi.3.14 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 24.768.17-24.kf.4.14 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 24.768.17-24.kh.2.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 24.768.17-24.ku.4.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 24.768.17-24.kv.4.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 24.768.17-24.lf.2.12 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 24.768.17-24.lh.6.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 24.768.17-24.lr.8.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 24.768.17-24.ls.4.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 24.1152.29-24.ep.1.29 | $24$ | $3$ | $3$ | $29$ | $0$ | $1^{4}\cdot2^{7}\cdot4$ |