Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| 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 | $2^{8}\cdot6^{8}\cdot8^{4}\cdot24^{4}$ | Cusp orbits | $2^{6}\cdot4^{3}$ | ||
| 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: | 24Z5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.5.1332 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&9\\4&7\end{bmatrix}$, $\begin{bmatrix}1&15\\4&7\end{bmatrix}$, $\begin{bmatrix}7&15\\8&17\end{bmatrix}$, $\begin{bmatrix}19&3\\8&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4^2:D_6$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.eh.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{20}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 48.2.a.a, 48.2.c.a, 144.2.a.b$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y^{2} + y z + z^{2} - z w $ |
| $=$ | $3 x^{2} + y w$ | |
| $=$ | $2 y^{2} - y z - 2 y w - z^{2} + z w - w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{8} - 132 x^{6} y^{2} + 94 x^{4} y^{4} + 252 x^{4} y^{2} z^{2} - 20 x^{2} y^{6} - 120 x^{2} y^{4} z^{2} + \cdots + 81 y^{2} z^{6} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{396718579712yw^{23}+1636464279552yw^{21}t^{2}+2873104404480yw^{19}t^{4}+2862743316480yw^{17}t^{6}+1845960938496yw^{15}t^{8}+839398173696yw^{13}t^{10}+281035961856yw^{11}t^{12}+69572494080yw^{9}t^{14}+12790485792yw^{7}t^{16}+1619989632yw^{5}t^{18}+135104112yw^{3}t^{20}+4251528ywt^{22}+132239527936w^{24}+644667559936w^{22}t^{2}+1348229210112w^{20}t^{4}+1602666731520w^{18}t^{6}+1222022515968w^{16}t^{8}+645831387648w^{14}t^{10}+249610657536w^{12}t^{12}+72052353792w^{10}t^{14}+15422968944w^{8}t^{16}+2415812688w^{6}t^{18}+253517040w^{4}t^{20}+15588936w^{2}t^{22}+177147t^{24}}{t^{2}w^{4}(64yw^{17}+528yw^{15}t^{2}+32286888yw^{13}t^{4}+96858612yw^{11}t^{6}+115397460yw^{9}t^{8}+68765112yw^{7}t^{10}+21030192yw^{5}t^{12}+2965572yw^{3}t^{14}+131220ywt^{16}-64w^{18}-544w^{16}t^{2}+10759704w^{14}t^{4}+40352373w^{12}t^{6}+61160562w^{10}t^{8}+47794455w^{8}t^{10}+20228292w^{6}t^{12}+4393683w^{4}t^{14}+398034w^{2}t^{16}+6561t^{18})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.eh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{8}-132X^{6}Y^{2}+94X^{4}Y^{4}+252X^{4}Y^{2}Z^{2}-20X^{2}Y^{6}-120X^{2}Y^{4}Z^{2}-216X^{2}Y^{2}Z^{4}+Y^{8}+12Y^{6}Z^{2}+36Y^{4}Z^{4}+81Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.192.1-12.e.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-12.e.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dn.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dn.2.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dn.4.2 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dn.4.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.ed.2.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ed.2.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.em.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.em.1.20 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gs.2.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gs.2.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gs.4.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gs.4.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.768.17-24.ta.1.4 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2\cdot4$ |
| 24.768.17-24.tc.1.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 24.768.17-24.te.1.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 24.768.17-24.tf.1.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 24.1152.25-24.cw.1.3 | $24$ | $3$ | $3$ | $25$ | $0$ | $1^{10}\cdot2^{5}$ |