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: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24U9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.9.4603 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&0\\12&19\end{bmatrix}$, $\begin{bmatrix}11&2\\16&23\end{bmatrix}$, $\begin{bmatrix}13&12\\0&17\end{bmatrix}$, $\begin{bmatrix}15&2\\20&21\end{bmatrix}$, $\begin{bmatrix}17&22\\16&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.kj.1 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, 72.2.d.b, 576.2.a.c, 576.2.a.g, 576.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x r + y t $ |
| $=$ | $x t - x u - z w$ | |
| $=$ | $2 x s + t v - u v - v r$ | |
| $=$ | $w s + t u - t r - u^{2} + r^{2} - s^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{14} - 8 x^{12} y^{2} + 4 x^{10} y^{4} + 24 x^{10} y^{2} z^{2} + 8 x^{8} y^{6} + 3 x^{8} y^{4} z^{2} + \cdots + 3 y^{12} z^{2} $ |
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 |
|---|
| $(0:0:0:1:0:1:0:-1:1)$, $(0:0:0:1:0:-1:0:1:1)$ |
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.u.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
| $\displaystyle W$ | $=$ | $\displaystyle s$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{2}-18XY+24Y^{2}-ZW+W^{2} $ |
| $=$ | $ 3X^{3}-6XY^{2}+YZ^{2}-XZW-YZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.9.kj.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{14}-8X^{12}Y^{2}+4X^{10}Y^{4}+24X^{10}Y^{2}Z^{2}+8X^{8}Y^{6}+3X^{8}Y^{4}Z^{2}-18X^{8}Y^{2}Z^{4}+2X^{6}Y^{8}-48X^{6}Y^{6}Z^{2}+6X^{4}Y^{8}Z^{2}+90X^{4}Y^{6}Z^{4}-54X^{2}Y^{6}Z^{6}+3Y^{12}Z^{2} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.4-24.u.1.40 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.u.1.61 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.12 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.47 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.5-24.o.1.23 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 24.144.5-24.o.1.33 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
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.fv.2.2 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.gl.1.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.lp.1.20 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.pe.1.23 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.qj.1.15 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.qt.2.9 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.wz.2.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.xw.2.3 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bgc.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bgv.2.21 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bhi.1.13 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bho.2.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bip.2.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.biv.1.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bjr.2.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bjy.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bot.1.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.boz.1.15 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bpz.1.9 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bqf.2.10 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bsv.1.13 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.btf.1.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bti.2.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.btn.1.13 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |