Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | 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 | $4^{12}\cdot12^{12}$ | Cusp orbits | $2^{2}\cdot4^{5}$ | ||
| 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: | 12E5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.5.1656 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&16\\12&19\end{bmatrix}$, $\begin{bmatrix}11&2\\6&11\end{bmatrix}$, $\begin{bmatrix}17&16\\6&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3:D_{12}$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.bz.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{24}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 48.2.c.a, 144.2.a.b, 192.2.a.b, 576.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x^{2} - 2 y^{2} - z^{2} $ |
| $=$ | $3 z^{2} - 2 w^{2} - t^{2}$ | |
| $=$ | $x^{2} - 6 x y + 2 y^{2} + z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{8} + 120 x^{6} y^{2} + 160 x^{6} z^{2} + 9 x^{4} y^{4} + 24 x^{4} y^{2} z^{2} + 376 x^{4} z^{4} + \cdots + 9 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
$j$-invariant map of degree 192 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{(4w^{4}+2w^{2}t^{2}+t^{4})(559104y^{2}w^{18}+1257984y^{2}w^{16}t^{2}+1092096y^{2}w^{14}t^{4}+443520y^{2}w^{12}t^{6}+53568y^{2}w^{10}t^{8}-26784y^{2}w^{8}t^{10}-55440y^{2}w^{6}t^{12}-34128y^{2}w^{4}t^{14}-9828y^{2}w^{2}t^{16}-1092y^{2}t^{18}+248832w^{20}+622080w^{18}t^{2}+641536w^{16}t^{4}+349952w^{14}t^{6}+110464w^{12}t^{8}+22784w^{10}t^{10}+4672w^{8}t^{12}+1256w^{6}t^{14}+1060w^{4}t^{16}+428w^{2}t^{18}+61t^{20})}{t^{4}w^{4}(2w^{2}+t^{2})^{2}(192y^{2}w^{10}+240y^{2}w^{8}t^{2}+48y^{2}w^{6}t^{4}-24y^{2}w^{4}t^{6}-30y^{2}w^{2}t^{8}-6y^{2}t^{10}-4w^{8}t^{4}-4w^{6}t^{6}-15w^{4}t^{8}-7w^{2}t^{10}-t^{12})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.bz.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{4}{3}z+\frac{4}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{8}+120X^{6}Y^{2}+9X^{4}Y^{4}+160X^{6}Z^{2}+24X^{4}Y^{2}Z^{2}+376X^{4}Z^{4}-90X^{2}Y^{2}Z^{4}-120X^{2}Z^{6}+9Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.192.3-12.h.1.2 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.1-24.cl.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.cl.1.9 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.cm.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.cm.2.9 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.cp.4.5 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.cp.4.9 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.3-12.h.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.bg.1.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.bg.1.13 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.bx.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.bx.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.cd.2.3 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.cd.2.9 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.1152.25-24.bu.1.4 | $24$ | $3$ | $3$ | $25$ | $3$ | $1^{10}\cdot2^{5}$ |