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: | $0$ | ||||||
| $\Q$-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: | 24.288.9.53 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&20\\20&5\end{bmatrix}$, $\begin{bmatrix}5&10\\8&17\end{bmatrix}$, $\begin{bmatrix}9&16\\16&9\end{bmatrix}$, $\begin{bmatrix}13&0\\0&7\end{bmatrix}$, $\begin{bmatrix}13&10\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.ee.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{38}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{2}$, 64.2.a.a, 72.2.d.a, 288.2.d.a, 576.2.a.a, 576.2.a.i |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ t v - u s $ |
| $=$ | $x s - w v$ | |
| $=$ | $x t - w u$ | |
| $=$ | $x u - z r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{6} z^{2} - 2 x^{4} z^{4} + x^{2} y^{6} + 2 y^{6} 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 |
|---|
| $(0:0:0:1:1:0:0:0:0)$, $(0:0:0:-1:1:0:0:0:0)$ |
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.s.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
| $\displaystyle W$ | $=$ | $\displaystyle -r$ |
Equation of the image curve:
| $0$ | $=$ | $ 2XY+ZW $ |
| $=$ | $ X^{3}+8Y^{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.ee.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{6}Z^{2}-2X^{4}Z^{4}+X^{2}Y^{6}+2Y^{6}Z^{2} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.m.2.2 | $8$ | $3$ | $3$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.m.2.2 | $8$ | $3$ | $3$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.144.4-24.s.2.4 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.s.2.48 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.2 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.48 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.5-24.d.1.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5-24.d.1.31 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.576.17-24.gj.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.gu.2.2 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.jh.2.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.jr.2.2 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.ke.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.oz.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.pq.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.qa.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.qi.1.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.qo.2.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.ra.1.2 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.rc.1.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.rq.1.2 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.rs.2.2 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.se.1.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.sk.1.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.ss.2.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.tc.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.tg.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.tn.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.vp.2.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.vy.2.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.xw.2.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.yi.2.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |