Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24B8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.8.855 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&8\\8&21\end{bmatrix}$, $\begin{bmatrix}7&14\\16&13\end{bmatrix}$, $\begin{bmatrix}17&12\\0&13\end{bmatrix}$, $\begin{bmatrix}19&14\\8&17\end{bmatrix}$, $\begin{bmatrix}19&20\\16&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.fo.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: | $2^{22}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{3}$, 72.2.d.a$^{2}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
| $ 0 $ | $=$ | $ 2 y u - w t $ |
| $=$ | $2 y t + z u$ | |
| $=$ | $y t - y u - z t + t v + u r$ | |
| $=$ | $x y + x z - x v - z u + w u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{6} y^{2} - 4 x^{4} y^{4} + 2 x^{2} y^{6} + 27 x^{2} z^{6} + 27 y^{2} z^{6} $ |
Rational points
This modular curve has no real points, 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 24.72.4.y.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
| $\displaystyle W$ | $=$ | $\displaystyle y+z-r$ |
Equation of the image curve:
| $0$ | $=$ | $ 12XY+ZW $ |
| $=$ | $ 6X^{3}-48Y^{3}+XZ^{2}-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.8.fo.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{6}Y^{2}-4X^{4}Y^{4}+2X^{2}Y^{6}+27X^{2}Z^{6}+27Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-24.bc.1.3 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.144.4-24.y.1.36 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.y.1.42 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.ch.1.41 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.gk.1.5 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.gk.1.28 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.576.15-24.ky.1.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.le.1.4 | $24$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.lw.2.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.mc.1.2 | $24$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nr.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ny.2.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.oq.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ow.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.17-24.ol.1.17 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.tm.1.9 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.ble.1.10 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.blm.1.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bru.2.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bsc.2.6 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btm.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btu.2.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.v.1.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.y.1.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ci.1.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cr.1.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.di.1.18 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ed.1.18 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ej.1.17 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ey.1.17 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jd.1.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ka.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ma.1.9 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mz.1.9 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.oq.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pa.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.qb.1.2 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.qf.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |