Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
| 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: | 24.288.9.4600 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&10\\20&1\end{bmatrix}$, $\begin{bmatrix}1&18\\12&23\end{bmatrix}$, $\begin{bmatrix}3&2\\16&9\end{bmatrix}$, $\begin{bmatrix}13&16\\16&23\end{bmatrix}$, $\begin{bmatrix}17&8\\16&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.kr.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: | $2^{31}\cdot3^{18}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{2}$, 72.2.d.a, 72.2.d.b, 288.2.a.b, 288.2.a.c, 288.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x u + y w $ |
| $=$ | $x s - t v - u v$ | |
| $=$ | $2 y r + t v$ | |
| $=$ | $2 x t + z r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{12} - 32 x^{10} y^{2} + 32 x^{8} y^{4} - 72 x^{8} y^{2} z^{2} - 16 x^{6} y^{6} + \cdots + 27 y^{6} 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 |
|---|
| $(0:0:0:0:1/2:-1/2:0:1:1)$, $(0:0:0:0:-1/2:1/2:0:1:1)$ |
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.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -r$ |
| $\displaystyle W$ | $=$ | $\displaystyle -s$ |
Equation of the image curve:
| $0$ | $=$ | $ 12X^{2}+36XY+48Y^{2}-ZW+W^{2} $ |
| $=$ | $ 6X^{3}-12XY^{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.kr.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{12}-32X^{10}Y^{2}+32X^{8}Y^{4}-72X^{8}Y^{2}Z^{2}-16X^{6}Y^{6}+108X^{6}Y^{4}Z^{2}+4X^{4}Y^{8}-36X^{4}Y^{6}Z^{2}+54X^{4}Y^{2}Z^{6}-81X^{2}Y^{4}Z^{6}+27Y^{6}Z^{6} $ |
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.2.40 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.u.2.62 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.11 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.47 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.5-24.p.1.21 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 24.144.5-24.p.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.fo.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.gu.2.9 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.lo.1.13 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.pb.1.5 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.sf.1.6 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.sp.1.6 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.xh.2.2 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.yk.1.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bgo.2.22 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bha.2.12 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bhy.1.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bie.1.10 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bjf.1.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bjl.1.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bju.2.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bkd.2.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bpi.2.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bpo.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bqo.1.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bqu.2.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bsu.1.14 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.bte.2.14 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.btq.2.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.btv.1.13 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |