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: | 24.192.3.4219 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&0\\0&7\end{bmatrix}$, $\begin{bmatrix}11&18\\4&17\end{bmatrix}$, $\begin{bmatrix}13&9\\16&23\end{bmatrix}$, $\begin{bmatrix}23&15\\12&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2\times D_4):D_{12}$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.gu.2 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 - y t $ |
| $=$ | $2 x z + 2 z^{2} - w t$ | |
| $=$ | $2 x w + x t + 2 y t + z w + z t$ | |
| $=$ | $x^{2} + 4 x y + x z + y z$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 144 x^{8} - 40 x^{4} z^{4} - 3 x^{2} y^{2} z^{4} + z^{8} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{4352526360xyt^{10}-87092658048xyt^{8}u^{2}+437181235200xyt^{6}u^{4}-470304301056xyt^{4}u^{6}+26059898880xyt^{2}u^{8}+143130624xyu^{10}-725682685w^{2}t^{10}+15960937232w^{2}t^{8}u^{2}-95147330304w^{2}t^{6}u^{4}+146689214464w^{2}t^{4}u^{6}-26698297344w^{2}t^{2}u^{8}-65536w^{2}u^{10}-181488125wt^{11}+2174298904wt^{9}u^{2}+6913406208wt^{7}u^{4}-79721714688wt^{5}u^{6}+64847228928wt^{3}u^{8}-374702080wtu^{10}-1024t^{12}+1087865870t^{10}u^{2}-22503384208t^{8}u^{4}+118951694336t^{6}u^{6}-139085155328t^{4}u^{8}+8591843328t^{2}u^{10}-65536u^{12}}{u^{2}(12xyt^{8}-480xyt^{6}u^{2}-308736xyt^{4}u^{4}-411648xyt^{2}u^{6}-w^{2}t^{8}+36w^{2}t^{6}u^{2}-26624w^{2}t^{4}u^{4}-104704w^{2}t^{2}u^{6}-8192w^{2}u^{8}-wt^{9}+40wt^{7}u^{2}-26752wt^{5}u^{4}-192256wt^{3}u^{6}-53248wtu^{8}+2t^{8}u^{2}-72t^{6}u^{4}-140352t^{4}u^{6}-150016t^{2}u^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.gu.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{16}{3}u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 144X^{8}-40X^{4}Z^{4}-3X^{2}Y^{2}Z^{4}+Z^{8} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.96.3.gu.2 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -16zt^{2}u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
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.1.24 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.1.26 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.iu.1.18 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.1-24.iu.1.24 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.2-24.g.2.18 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.2-24.g.2.25 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
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.2.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.dm.4.7 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.384.5-24.ei.1.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.em.2.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ew.4.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ez.4.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.384.5-24.fs.2.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.fy.3.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.576.13-24.kz.2.13 | $24$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 120.384.5-120.bhf.4.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhh.4.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhv.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bhx.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bjr.4.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bjt.4.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bkh.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bkj.3.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhf.3.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhh.3.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhv.2.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhx.4.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bjr.3.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bjt.3.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bkh.3.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bkj.3.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhf.2.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhh.4.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhv.1.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bhx.3.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bjr.4.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bjt.4.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bkh.2.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bkj.3.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhf.3.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhh.4.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhv.2.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bhx.3.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bjr.3.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bjt.4.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bkh.2.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bkj.2.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |