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: | 24.384.5.988 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&3\\12&17\end{bmatrix}$, $\begin{bmatrix}13&12\\8&23\end{bmatrix}$, $\begin{bmatrix}13&21\\20&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2\times D_{12}):C_4$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.fr.4 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 $ | $=$ | $ x y + x z - y^{2} + z^{2} - w^{2} $ |
| $=$ | $x^{2} - x y - 2 y z - z^{2}$ | |
| $=$ | $2 x^{2} - x z + y^{2} + 2 y z - 3 w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{8} - 48 x^{7} z + 148 x^{6} z^{2} - 72 x^{5} y z^{2} - 464 x^{5} z^{3} + 18 x^{4} y^{2} z^{2} + \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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(2w^{2}+t^{2})^{3}(279552xzw^{16}-52224xzw^{14}t^{2}-188160xzw^{12}t^{4}-1485312xzw^{10}t^{6}-1618368xzw^{8}t^{8}-821184xzw^{6}t^{10}-231504xzw^{4}t^{12}-34944xzw^{2}t^{14}-2184xzt^{16}+279552z^{2}w^{16}-52224z^{2}w^{14}t^{2}-188160z^{2}w^{12}t^{4}-1485312z^{2}w^{10}t^{6}-1618368z^{2}w^{8}t^{8}-821184z^{2}w^{6}t^{10}-231504z^{2}w^{4}t^{12}-34944z^{2}w^{2}t^{14}-2184z^{2}t^{16}+186880w^{18}+58112w^{16}t^{2}+221696w^{14}t^{4}+997760w^{12}t^{6}+964928w^{10}t^{8}+440192w^{8}t^{10}+108416w^{6}t^{12}+13976w^{4}t^{14}+746w^{2}t^{16}+t^{18})}{t^{2}w^{8}(4w^{2}+t^{2})(48xzw^{10}-24xzw^{8}t^{2}-132xzw^{6}t^{4}-102xzw^{4}t^{6}-30xzw^{2}t^{8}-3xzt^{10}+48z^{2}w^{10}-24z^{2}w^{8}t^{2}-132z^{2}w^{6}t^{4}-102z^{2}w^{4}t^{6}-30z^{2}w^{2}t^{8}-3z^{2}t^{10}-32w^{12}+20w^{10}t^{2}-97w^{8}t^{4}-102w^{6}t^{6}-47w^{4}t^{8}-11w^{2}t^{10}-t^{12})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.fr.4 :
| $\displaystyle X$ | $=$ | $\displaystyle x+2z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4z+4w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2y+2z-t$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{8}-48X^{7}Z+148X^{6}Z^{2}-72X^{5}YZ^{2}+18X^{4}Y^{2}Z^{2}-464X^{5}Z^{3}+384X^{4}YZ^{3}-96X^{3}Y^{2}Z^{3}+534X^{4}Z^{4}-528X^{3}YZ^{4}+276X^{2}Y^{2}Z^{4}-72XY^{3}Z^{4}+9Y^{4}Z^{4}-464X^{3}Z^{5}+384X^{2}YZ^{5}-96XY^{2}Z^{5}+148X^{2}Z^{6}-72XYZ^{6}+18Y^{2}Z^{6}-48XZ^{7}+9Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.192.1-12.g.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-12.g.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dn.3.4 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dn.3.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dr.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dr.1.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.fw.2.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.fw.2.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ga.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.ga.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gs.4.4 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gs.4.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gw.2.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gw.2.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.1152.25-24.fc.1.5 | $24$ | $3$ | $3$ | $25$ | $0$ | $1^{10}\cdot2^{5}$ |
| 48.768.17-48.yq.1.2 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 48.768.17-48.yt.1.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 48.768.17-48.bnj.3.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2\cdot4$ |
| 48.768.17-48.bno.3.4 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2\cdot4$ |