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: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.8.2111 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&4\\16&5\end{bmatrix}$, $\begin{bmatrix}3&20\\4&9\end{bmatrix}$, $\begin{bmatrix}17&8\\16&17\end{bmatrix}$, $\begin{bmatrix}21&10\\16&9\end{bmatrix}$, $\begin{bmatrix}23&16\\20&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.fg.2 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.b$^{2}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x t - x v - 2 y v - w r $ |
| $=$ | $x t - x v - 2 y t + w u$ | |
| $=$ | $2 x t + 2 y t + z r + w u - w r$ | |
| $=$ | $x u + x r - 2 y r - z v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{14} - 16 x^{12} y^{2} + 36 x^{10} y^{2} z^{2} + 16 x^{8} y^{6} - 36 x^{8} y^{4} z^{2} + \cdots + 27 y^{8} z^{6} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=47$, 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.u.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
| $\displaystyle W$ | $=$ | $\displaystyle -v$ |
Equation of the image curve:
| $0$ | $=$ | $ 12X^{2}+36XY+48Y^{2}-ZW+W^{2} $ |
| $=$ | $ 6X^{3}-12XY^{2}-YZ^{2}-XZW+YZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.8.fg.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 8X^{14}-16X^{12}Y^{2}+36X^{10}Y^{2}Z^{2}+16X^{8}Y^{6}-36X^{8}Y^{4}Z^{2}-6X^{6}Y^{8}-36X^{6}Y^{6}Z^{2}+54X^{6}Y^{4}Z^{4}-4X^{4}Y^{10}+36X^{4}Y^{8}Z^{2}+2X^{2}Y^{12}-54X^{2}Y^{8}Z^{4}+27X^{2}Y^{6}Z^{6}+27Y^{8}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.4-24.u.2.14 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.u.2.29 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.ch.1.14 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.gg.1.10 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.gg.1.23 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
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.kz.2.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.la.1.1 | $24$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ls.1.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.md.1.3 | $24$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nr.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nv.2.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.on.2.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ow.2.2 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.17-24.lj.2.12 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.ti.2.3 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bla.2.12 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bli.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.brq.1.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bry.2.3 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bti.1.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btq.2.3 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.k.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bf.1.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cd.1.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cs.2.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dp.2.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ds.1.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ek.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.et.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jf.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.js.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mk.2.4 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mo.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ok.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pc.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pm.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.qn.2.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |