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: | 24C9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.9.58 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&8\\16&5\end{bmatrix}$, $\begin{bmatrix}11&6\\0&13\end{bmatrix}$, $\begin{bmatrix}15&4\\8&21\end{bmatrix}$, $\begin{bmatrix}17&6\\0&13\end{bmatrix}$, $\begin{bmatrix}23&12\\0&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.et.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{40}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 36.2.a.a$^{3}$, 64.2.a.a, 144.2.a.a, 576.2.a.a, 576.2.a.e, 576.2.a.f, 576.2.a.i |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ u s + v r $ |
| $=$ | $w^{2} + u v$ | |
| $=$ | $x v - z s$ | |
| $=$ | $x u + z r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{6} z^{2} - y^{8} + y^{4} z^{4} $ |
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 |
|---|
| $(0:0:0:1:0:-1:1:0:0)$, $(0:0:0:-1:0:-1:1:0:0)$ |
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.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
| $\displaystyle W$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}+Z^{2}-W^{2} $ |
| $=$ | $ 8Y^{3}-XZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.9.et.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{6}Z^{2}-Y^{8}+Y^{4}Z^{4} $ |
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.p.1.2 | $8$ | $3$ | $3$ | $1$ | $0$ | $1^{8}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.p.1.2 | $8$ | $3$ | $3$ | $1$ | $0$ | $1^{8}$ |
| 24.144.4-24.c.1.11 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 24.144.4-24.c.1.33 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 24.144.4-24.ch.1.31 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 24.144.5-24.d.1.11 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5-24.d.1.32 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
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.gq.1.11 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.hb.1.9 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 24.576.17-24.jy.1.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 24.576.17-24.kd.1.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 24.576.17-24.tg.1.10 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tg.2.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.th.1.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.th.2.9 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.ti.1.12 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.ti.2.3 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tj.1.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tj.2.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tk.1.16 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tk.2.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tl.1.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tl.2.12 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tm.1.9 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tm.2.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tn.1.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.tn.2.10 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 24.576.17-24.wc.1.15 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 24.576.17-24.wh.1.13 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 24.576.17-24.yd.1.10 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.yp.1.13 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.19-48.ix.1.28 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.je.1.22 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.jr.1.26 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.jw.1.21 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{10}$ |
| 48.576.19-48.kc.1.22 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.kc.2.27 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.kd.1.22 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.kd.2.27 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ke.1.21 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ke.2.25 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.kf.1.21 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.kf.2.25 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.ki.1.26 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.kj.1.21 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.kk.1.28 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.kl.1.22 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |