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: | 24.192.3.4164 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&3\\16&5\end{bmatrix}$, $\begin{bmatrix}5&6\\8&17\end{bmatrix}$, $\begin{bmatrix}11&0\\20&5\end{bmatrix}$, $\begin{bmatrix}13&3\\0&23\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2\times D_4):D_{12}$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.gu.3 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 w + x t + z w $ |
| $=$ | $ - x t + 3 y w - y t$ | |
| $=$ | $2 x z + 2 y z + w^{2}$ | |
| $=$ | $x^{2} + 4 x y + x z + y z$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{6} - 9 x^{4} y^{2} - 2 x^{4} z^{2} - 6 x^{2} y^{2} z^{2} - 12 x^{2} z^{4} + 8 z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 3x^{8} - 120x^{4} + 432 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^3\cdot3\,\frac{1257984xyu^{10}-6561w^{12}-52488w^{10}u^{2}-52488w^{8}u^{4}-54432w^{6}u^{6}-703728w^{4}u^{8}+19683w^{2}t^{10}-255908w^{2}t^{8}u^{2}+264672w^{2}t^{6}u^{4}-737376w^{2}t^{4}u^{6}+664272w^{2}t^{2}u^{8}+808704w^{2}u^{10}-13122wt^{11}+150932wt^{9}u^{2}-291032wt^{7}u^{4}+505056wt^{5}u^{6}+250464wt^{3}u^{8}+578304wtu^{10}-6561t^{12}+131220t^{10}u^{2}-459212t^{8}u^{4}+547008t^{6}u^{6}-393648t^{4}u^{8}-282816t^{2}u^{10}-576u^{12}}{u^{2}(31104w^{4}u^{6}+w^{2}t^{8}-48w^{2}t^{6}u^{2}-14076w^{2}t^{4}u^{4}+61992w^{2}t^{2}u^{6}-82944w^{2}u^{8}-wt^{9}+52wt^{7}u^{2}-5784wt^{5}u^{4}+27216wt^{3}u^{6}-55728wtu^{8}-2t^{8}u^{2}+84t^{6}u^{4}-10800t^{4}u^{6}+27216t^{2}u^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.gu.3 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{6}-9X^{4}Y^{2}-2X^{4}Z^{2}-6X^{2}Y^{2}Z^{2}-12X^{2}Z^{4}+8Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.96.3.gu.3 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -24x^{3}u-4xw^{2}u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x$ |
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.4.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.4.27 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.iu.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.1-24.iu.1.18 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.2-24.g.1.5 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.2-24.g.1.8 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
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.3.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.dm.1.8 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.384.5-24.ei.2.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.em.3.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ew.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ez.1.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.384.5-24.fs.3.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.fy.2.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.576.13-24.kz.1.7 | $24$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 120.384.5-120.bhf.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhh.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhv.3.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhx.3.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bjr.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bjt.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bkh.3.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bkj.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhf.1.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhh.2.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhv.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhx.2.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bjr.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bjt.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bkh.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bkj.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhf.3.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhh.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhv.4.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhx.2.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bjr.1.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bjt.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bkh.4.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bkj.1.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhf.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhh.2.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhv.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhx.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bjr.2.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bjt.3.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bkh.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bkj.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |