Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $8^{24}$ | Cusp orbits | $2^{2}\cdot4^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8A5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.5.30 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&8\\16&5\end{bmatrix}$, $\begin{bmatrix}13&12\\12&5\end{bmatrix}$, $\begin{bmatrix}13&20\\8&3\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^2\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.bc.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{28}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 64.2.a.a, 288.2.a.d$^{2}$, 576.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y^{2} + w t $ |
| $=$ | $2 z^{2} - w^{2} + t^{2}$ | |
| $=$ | $3 x^{2} + z^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} z^{4} + 12 x^{2} y^{4} z^{2} - 6 x^{2} z^{6} + y^{8} - 4 y^{4} z^{4} + 4 z^{8} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=5,29$, and therefore no rational points.
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 24.96.3.q.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-X^{3}Y-3X^{2}Y^{2}-4XY^{3}-2Y^{4}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.bc.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}Z^{4}+12X^{2}Y^{4}Z^{2}-6X^{2}Z^{6}+Y^{8}-4Y^{4}Z^{4}+4Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.192.1-8.d.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-8.d.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.s.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.s.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.s.2.7 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.s.2.10 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.q.2.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.q.2.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.u.1.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.u.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.u.2.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.u.2.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.x.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.x.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.1152.37-24.qi.1.9 | $24$ | $3$ | $3$ | $37$ | $3$ | $1^{16}\cdot2^{6}\cdot4$ |
| 24.1536.41-24.fq.1.2 | $24$ | $4$ | $4$ | $41$ | $3$ | $1^{18}\cdot2^{7}\cdot4$ |
| 48.768.17-48.dc.1.2 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.dc.2.2 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.dk.1.2 | $48$ | $2$ | $2$ | $17$ | $6$ | $1^{4}\cdot2^{4}$ |
| 48.768.17-48.dk.2.2 | $48$ | $2$ | $2$ | $17$ | $6$ | $1^{4}\cdot2^{4}$ |