Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| 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: | 24.288.7.4365 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&16\\20&21\end{bmatrix}$, $\begin{bmatrix}9&22\\16&9\end{bmatrix}$, $\begin{bmatrix}13&18\\0&13\end{bmatrix}$, $\begin{bmatrix}17&8\\8&13\end{bmatrix}$, $\begin{bmatrix}17&22\\16&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.7.im.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: | $2^{28}\cdot3^{11}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a, 192.2.a.b, 192.2.a.d$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ 2 x t - z u + w v $ |
| $=$ | $x u - 2 y v - w t$ | |
| $=$ | $x v + 2 y u + z t$ | |
| $=$ | $2 z^{2} + 2 w^{2} - u^{2} - v^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{10} - 4 x^{6} y^{4} + 12 x^{4} y^{4} z^{2} - 2 x^{2} y^{8} - 6 x^{2} y^{4} z^{4} + y^{4} z^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,13,19,61$, 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.r.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -u$ |
| $\displaystyle W$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{2}-8Y^{2}-ZW $ |
| $=$ | $ 6X^{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.im.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{10}-4X^{6}Y^{4}+12X^{4}Y^{4}Z^{2}-2X^{2}Y^{8}-6X^{2}Y^{4}Z^{4}+Y^{4}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.3-24.bh.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.bh.1.21 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.ox.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.ox.1.23 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.py.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.py.1.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.4-24.r.1.19 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.r.1.21 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.ch.1.43 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.gn.1.16 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.gn.1.17 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.hg.1.8 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.hg.1.9 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
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.os.1.4 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.os.2.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ot.1.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ot.2.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ou.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ou.2.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ov.1.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ov.2.4 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ow.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ow.2.2 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ox.1.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ox.2.2 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.17-24.ul.1.21 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.576.17-24.wh.1.13 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.576.17-24.bng.1.9 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 24.576.17-24.bnj.1.9 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.15-48.cd.1.10 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cd.2.10 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cv.1.5 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cv.2.5 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.17-48.cy.1.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.17-48.cy.2.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.17-48.da.1.9 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.da.2.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.de.1.9 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.de.2.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.df.1.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.df.2.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.19-48.kr.1.10 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.kr.2.10 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.nv.1.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.nv.2.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |