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 | $2^{6}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-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: | 24.384.5.17 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&0\\12&17\end{bmatrix}$, $\begin{bmatrix}7&0\\16&1\end{bmatrix}$, $\begin{bmatrix}17&8\\12&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^2\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.bj.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{28}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 32.2.a.a, 288.2.a.d, 576.2.a.c, 576.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ z w - z t - w^{2} - w t $ |
| $=$ | $2 y^{2} - 2 y z + 2 y t + z w - z t + w t + t^{2}$ | |
| $=$ | $6 x^{2} + 2 y^{2} - z^{2} + z w + z t + w t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} y^{4} - 4 x^{4} y^{3} z - 6 x^{4} y^{2} z^{2} - 4 x^{4} y z^{3} - x^{4} z^{4} + 36 y^{8} + \cdots + 36 z^{8} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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.r.1 :
| $\displaystyle X$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z-2w-t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z+t$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}-2Y^{3}Z-2YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.bj.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{4}Y^{4}-4X^{4}Y^{3}Z-6X^{4}Y^{2}Z^{2}-4X^{4}YZ^{3}-X^{4}Z^{4}+36Y^{8}-144Y^{6}Z^{2}-360Y^{4}Z^{4}-144Y^{2}Z^{6}+36Z^{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.g.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-8.g.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.o.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.o.1.9 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.w.1.1 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.w.1.11 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.3-24.r.1.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.r.1.2 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.s.2.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.s.2.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.t.1.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.t.1.2 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.ba.1.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ba.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
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.rp.1.1 | $24$ | $3$ | $3$ | $37$ | $4$ | $1^{16}\cdot2^{6}\cdot4$ |
| 24.1536.41-24.gl.1.1 | $24$ | $4$ | $4$ | $41$ | $4$ | $1^{18}\cdot2^{7}\cdot4$ |
| 48.768.13-48.c.2.7 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.r.1.3 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.bu.1.3 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.cj.1.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.dh.3.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.dh.4.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.dk.3.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.dk.4.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.eb.2.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.eo.1.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.fn.1.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.ga.1.1 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.17-48.dt.5.3 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{3}$ |
| 48.768.17-48.dt.8.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{3}$ |
| 48.768.17-48.ea.5.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{3}$ |
| 48.768.17-48.ea.8.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{3}$ |