Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $12^{4}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.9.1869 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&18\\12&1\end{bmatrix}$, $\begin{bmatrix}11&14\\8&7\end{bmatrix}$, $\begin{bmatrix}13&20\\8&9\end{bmatrix}$, $\begin{bmatrix}15&14\\16&9\end{bmatrix}$, $\begin{bmatrix}23&8\\12&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.hf.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{34}\cdot3^{18}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{2}$, 72.2.d.a$^{2}$, 576.2.a.a, 576.2.a.c, 576.2.a.i |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ w^{2} + t s $ |
| $=$ | $x u - z t$ | |
| $=$ | $x w + y v$ | |
| $=$ | $w^{2} - t s + u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{6} y^{2} + 9 x^{4} y^{4} - 6 x^{2} z^{6} + 54 y^{2} z^{6} $ |
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 |
|---|
| $(0:0:0:0:-1:1:0:0:0)$, $(0:0:0:0:1:1:0:0:0)$ |
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 24.72.4.y.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
| $\displaystyle W$ | $=$ | $\displaystyle -s$ |
Equation of the image curve:
| $0$ | $=$ | $ 6XY-ZW $ |
| $=$ | $ 3X^{3}-24Y^{3}+XZ^{2}-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.9.hf.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{6}Y^{2}+9X^{4}Y^{4}-6X^{2}Z^{6}+54Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-24.bj.2.10 | $24$ | $3$ | $3$ | $1$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.144.4-24.y.2.3 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.y.2.43 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.5 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.47 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.5-24.h.1.5 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 24.144.5-24.h.1.37 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.576.17-24.oq.2.2 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.pa.2.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.td.2.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.tf.2.2 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.zf.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.zj.2.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bbc.1.7 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bbe.1.7 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bbs.1.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bbu.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bcp.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bct.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bgv.2.21 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bhb.1.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.biu.1.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.biw.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bjk.1.3 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bjm.1.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bjz.2.9 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bkd.2.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bku.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bkw.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.blm.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bln.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |