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 $4$ are rational) | Cusp widths | $12^{4}\cdot24^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| 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: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24U9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.9.4602 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&22\\8&15\end{bmatrix}$, $\begin{bmatrix}11&8\\20&1\end{bmatrix}$, $\begin{bmatrix}17&20\\8&7\end{bmatrix}$, $\begin{bmatrix}19&10\\8&1\end{bmatrix}$, $\begin{bmatrix}21&22\\20&15\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.jl.1 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: | $2^{34}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 24.2.d.a, 36.2.a.a$^{2}$, 64.2.a.a, 72.2.d.a, 192.2.a.a, 192.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x v - y t $ |
| $=$ | $x t - x u - z w$ | |
| $=$ | $x u + 2 y u + 2 y v - r s$ | |
| $=$ | $x s - 2 y w + y s + z u - v r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{14} + 8 x^{12} z^{2} + 8 x^{10} y^{2} z^{2} - 4 x^{10} z^{4} + 2 x^{8} y^{4} z^{2} + \cdots + y^{2} z^{12} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1:0:0:-1:-1:1:0)$, $(0:0:1:0:0:1:1:1:0)$, $(0:0:0:0:-1:-1:0:0:1)$, $(0:0:0:0:1:1:0:0:1)$ |
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.v.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y$ |
| $\displaystyle W$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}-6XZ+8Z^{2}-YW-W^{2} $ |
| $=$ | $ X^{3}-Y^{2}Z-2XZ^{2}-XYW-YZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.9.jl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{14}+8X^{12}Z^{2}+8X^{10}Y^{2}Z^{2}-4X^{10}Z^{4}+2X^{8}Y^{4}Z^{2}+X^{8}Y^{2}Z^{4}-8X^{8}Z^{6}-16X^{6}Y^{2}Z^{6}-2X^{6}Z^{8}-10X^{4}Y^{4}Z^{6}+2X^{4}Y^{2}Z^{8}-2X^{2}Y^{6}Z^{6}+Y^{2}Z^{12} $ |
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.v.2.39 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.v.2.63 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.7 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.47 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.5-24.k.1.15 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 24.144.5-24.k.1.40 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
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.im.1.2 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.jk.1.21 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.me.1.21 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.pd.1.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.rb.1.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.rj.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.vd.2.2 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.vp.2.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.za.2.6 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.zf.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bag.1.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bao.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bax.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bbf.1.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bcl.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bco.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bpj.2.12 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bpr.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bpy.1.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bqg.2.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.brf.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.brn.2.15 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.brr.2.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.brv.1.15 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |