Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $6^{8}\cdot24^{4}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| 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: | 24J7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.7.1273 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&10\\8&13\end{bmatrix}$, $\begin{bmatrix}1&22\\8&1\end{bmatrix}$, $\begin{bmatrix}11&0\\0&1\end{bmatrix}$, $\begin{bmatrix}21&4\\8&21\end{bmatrix}$, $\begin{bmatrix}21&10\\8&21\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4^2.C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.144.7.hh.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: | $2^{21}\cdot3^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 36.2.a.a$^{3}$, 72.2.a.a, 144.2.a.a, 144.2.a.b$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ 2 x t + z v - w u $ |
| $=$ | $x v + 2 y u - w t$ | |
| $=$ | $x u - 2 y v + z t$ | |
| $=$ | $3 z^{2} + 3 w^{2} - u^{2} - v^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{10} - 2 x^{6} y^{4} + 36 x^{4} y^{4} z^{2} - x^{2} y^{8} - 108 x^{2} y^{4} z^{4} + 108 y^{4} z^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,19$, 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.m.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -v$ |
| $\displaystyle W$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{2}-12Y^{2}+ZW $ |
| $=$ | $ 9X^{3}+YZ^{2}+2XZW-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.7.hh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{10}-2X^{6}Y^{4}+36X^{4}Y^{4}Z^{2}-X^{2}Y^{8}-108X^{2}Y^{4}Z^{4}+108Y^{4}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.144.3-12.d.1.3 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-12.d.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.np.1.13 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.np.1.20 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.om.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.om.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.4-24.m.1.5 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.m.1.21 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.ch.1.5 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.hv.1.15 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.hv.1.18 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.io.1.7 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.io.1.10 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
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.ls.1.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ls.2.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lt.1.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lt.2.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lu.1.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lu.2.8 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lv.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lv.2.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lw.1.8 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lw.2.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lx.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.lx.2.2 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.17-24.xb.1.18 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.576.17-24.yd.1.10 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 24.576.17-24.beo.1.10 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.576.17-24.beq.1.6 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.15-48.ce.1.19 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.ce.2.18 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cm.1.17 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cm.2.17 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.17-48.bj.1.18 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.bj.2.18 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.bk.1.17 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.bk.2.17 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.bm.1.17 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.17-48.bm.2.17 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.17-48.bn.1.17 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.bn.2.17 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.19-48.ks.1.18 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.ks.2.17 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.lu.1.17 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.lu.2.17 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |