Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24W3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.3.4196 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&21\\16&13\end{bmatrix}$, $\begin{bmatrix}11&9\\12&23\end{bmatrix}$, $\begin{bmatrix}11&12\\16&5\end{bmatrix}$, $\begin{bmatrix}23&0\\4&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2\times D_4):D_{12}$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.gu.4 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{16}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 144.2.a.b, 192.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x t - x u + y t $ |
| $=$ | $3 y z + w^{2}$ | |
| $=$ | $2 x^{2} - 2 z^{2} + 2 w^{2} - t^{2} + t u$ | |
| $=$ | $2 x t - x u - 3 y t - 2 z t + z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 75 x^{8} - 1170 x^{6} y^{2} + 6723 x^{4} y^{4} - 44 x^{4} y^{2} z^{2} - 16848 x^{2} y^{6} + \cdots - 4 y^{4} z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 27x^{8} - 120x^{4} + 48 $ |
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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{167136068736xzw^{10}+188053586496xzw^{8}u^{2}+1828716352128xzw^{6}u^{4}+16643248109280xzw^{4}u^{6}+202531964597784xzw^{2}u^{8}+2876277835575924xzu^{10}+111730155840w^{12}+125335045440w^{10}u^{2}+1226292155568w^{8}u^{4}+11075766284160w^{6}u^{6}+134718193357596w^{4}u^{8}+1913770365019052w^{2}u^{10}-1795355655111t^{12}+5346913351464t^{11}u+6499637346456t^{10}u^{2}+5169837647664t^{9}u^{3}+28638985582764t^{8}u^{4}+111022866117396t^{7}u^{5}+103945384790937t^{6}u^{6}+421384683213468t^{5}u^{7}+260261690860032t^{4}u^{8}+1248030004118560t^{3}u^{9}-2666010965461480t^{2}u^{10}+956885957350504tu^{11}+4782969u^{12}}{51018336xzw^{8}u^{2}+41570496xzw^{6}u^{4}+141587289792xzw^{4}u^{6}+3024810783000xzw^{2}u^{8}+57477563092326xzu^{10}-34012224w^{10}u^{2}+48813840w^{8}u^{4}+94387654080w^{6}u^{6}+2013918755796w^{4}u^{8}+38262372516880w^{2}u^{10}+161243136000t^{12}-494478950400t^{11}u+1185893226720t^{10}u^{2}-1094383932912t^{9}u^{3}+1865000878848t^{8}u^{4}+762809317704t^{7}u^{5}+3255399446814t^{6}u^{6}+7089610957461t^{5}u^{7}+6780053441433t^{4}u^{8}+24065086569470t^{3}u^{9}-53127807368981t^{2}u^{10}+19131167126564tu^{11}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.gu.4 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{3}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 75X^{8}-1170X^{6}Y^{2}+6723X^{4}Y^{4}-16848X^{2}Y^{6}+15552Y^{8}-44X^{4}Y^{2}Z^{2}+108X^{2}Y^{4}Z^{2}+72Y^{6}Z^{2}-4Y^{4}Z^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.96.3.gu.4 :
| $\displaystyle X$ | $=$ | $\displaystyle \frac{2}{15}zt^{2}-\frac{16}{45}ztu+\frac{8}{45}zu^{2}-\frac{1}{9}t^{3}+\frac{14}{45}t^{2}u-\frac{8}{45}tu^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{272}{1215}zwt^{10}+\frac{30592}{18225}zwt^{9}u-\frac{474368}{91125}zwt^{8}u^{2}+\frac{35612416}{4100625}zwt^{7}u^{3}-\frac{1386752}{164025}zwt^{6}u^{4}+\frac{6591488}{1366875}zwt^{5}u^{5}-\frac{6139904}{4100625}zwt^{4}u^{6}+\frac{802816}{4100625}zwt^{3}u^{7}+\frac{64}{243}wt^{11}-\frac{20096}{10935}wt^{10}u+\frac{873728}{164025}wt^{9}u^{2}-\frac{6865408}{820125}wt^{8}u^{3}+\frac{31735808}{4100625}wt^{7}u^{4}-\frac{17317888}{4100625}wt^{6}u^{5}+\frac{5177344}{4100625}wt^{5}u^{6}-\frac{131072}{820125}wt^{4}u^{7}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -\frac{1}{5}zt^{2}+\frac{8}{15}ztu-\frac{4}{15}zu^{2}+\frac{1}{3}t^{3}-\frac{22}{45}t^{2}u+\frac{8}{45}tu^{2}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-24.bu.3.15 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.3.19 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.iu.1.8 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.1-24.iu.1.18 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.2-24.g.2.9 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.2-24.g.2.15 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.5-24.cy.4.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.dm.2.8 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.384.5-24.ei.1.10 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.em.4.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ew.3.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ez.3.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.384.5-24.fs.4.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.fy.4.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.576.13-24.kz.1.7 | $24$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 120.384.5-120.bhf.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhh.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhv.4.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhx.4.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bjr.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bjt.3.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bkh.4.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bkj.4.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhf.4.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhh.4.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhv.3.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhx.3.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bjr.4.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bjt.4.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bkh.4.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bkj.4.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhf.4.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhh.3.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhv.2.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhx.4.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bjr.3.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bjt.3.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bkh.3.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bkj.4.9 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhf.4.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhh.3.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhv.4.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhx.4.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bjr.4.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bjt.2.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bkh.4.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bkj.4.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |