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: | $2$ | ||||||
| $\Q$-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: | 24.288.9.1619 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&8\\16&13\end{bmatrix}$, $\begin{bmatrix}3&14\\20&9\end{bmatrix}$, $\begin{bmatrix}13&8\\16&1\end{bmatrix}$, $\begin{bmatrix}15&8\\20&9\end{bmatrix}$, $\begin{bmatrix}23&22\\20&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.jf.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: | $2^{40}\cdot3^{18}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a, 576.2.a.a, 576.2.a.c, 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^{2} - r s $ |
| $=$ | $x r + w t$ | |
| $=$ | $z r - w s$ | |
| $=$ | $x s + z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} - x^{4} y^{4} + 54 y^{2} 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 |
|---|
| $(0:0:0:0:0:1:0:1:1)$, $(0:0:0:0:0:-1:0:1:1)$ |
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.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -r$ |
| $\displaystyle W$ | $=$ | $\displaystyle -s$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{2}-Z^{2}+W^{2} $ |
| $=$ | $ 24Y^{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.jf.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{8}-X^{4}Y^{4}+54Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-24.bv.1.1 | $24$ | $3$ | $3$ | $1$ | $1$ | $1^{8}$ |
| 24.144.4-24.h.1.11 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 24.144.4-24.h.1.33 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 24.144.4-24.ch.1.15 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 24.144.5-24.h.1.11 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5-24.h.1.32 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
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.bea.1.11 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.beb.1.11 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 24.576.17-24.beq.1.6 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.ber.1.7 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.blg.1.9 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blg.2.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blh.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blh.2.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bli.1.12 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bli.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blj.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blj.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blk.1.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blk.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bll.1.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bll.2.12 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blm.1.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.blm.2.2 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bln.1.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bln.2.9 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bms.1.15 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 24.576.17-24.bmt.1.9 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.bni.1.11 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 24.576.17-24.bnj.1.9 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.19-48.jy.1.26 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.jy.2.26 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.lt.1.28 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.lt.2.28 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.ly.1.12 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.ly.2.6 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.mn.1.21 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.mn.2.9 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.na.1.21 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.na.2.9 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.nb.1.12 | $48$ | $2$ | $2$ | $19$ | $4$ | $1^{10}$ |
| 48.576.19-48.nb.2.7 | $48$ | $2$ | $2$ | $19$ | $4$ | $1^{10}$ |
| 48.576.19-48.nt.1.22 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.nt.2.22 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.ob.1.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.ob.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |