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: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24V3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.3.3550 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&22\\0&5\end{bmatrix}$, $\begin{bmatrix}11&4\\0&5\end{bmatrix}$, $\begin{bmatrix}11&18\\12&19\end{bmatrix}$, $\begin{bmatrix}19&17\\12&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_6:\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.eo.1 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: | $2^{16}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 144.2.a.b, 192.2.a.d, 576.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x^{2} - x y + x z - y^{2} $ |
| $=$ | $3 x t - 3 y t - w u$ | |
| $=$ | $x w - 5 y w + z w - t u$ | |
| $=$ | $6 x u + 2 y u - 2 z u + w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 17328 x^{8} + 2960 x^{6} y^{2} - 27360 x^{6} z^{2} + 12 x^{4} y^{4} - 3224 x^{4} y^{2} z^{2} + \cdots + 75 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 9 w^{2} $ | $=$ | $ 252 x^{4} - 144 x^{3} y - 84 x^{2} z^{2} + 24 x y z^{2} + 7 z^{4} $ |
| $0$ | $=$ | $3 x^{2} - y^{2} - z^{2}$ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{11}\cdot\frac{540030976128000yzu^{10}-31975944z^{2}w^{10}+7091568z^{2}w^{8}u^{2}+1865738688z^{2}w^{6}u^{4}+94487783808z^{2}w^{4}u^{6}+1398097422720z^{2}w^{2}u^{8}-85471698312z^{2}t^{10}-512830189872z^{2}t^{8}u^{2}-2925031453344z^{2}t^{6}u^{4}-12941048248128z^{2}t^{4}u^{6}-63042421875840z^{2}t^{2}u^{8}-221800044557184z^{2}u^{10}-53512855w^{12}+2960144w^{10}u^{2}-225276348w^{8}u^{4}-1532605888w^{6}u^{6}+33848178352w^{4}u^{8}-4566748182640w^{2}u^{10}+4696247160t^{12}+85471698312t^{10}u^{2}+657892046592t^{8}u^{4}+3487885372224t^{6}u^{6}+16694799187968t^{4}u^{8}+43677468031680t^{2}u^{10}+12054273167264u^{12}}{u^{2}(156764160yzu^{8}-31944z^{2}w^{8}+342672z^{2}w^{6}u^{2}+4524960z^{2}w^{4}u^{4}+20581056z^{2}w^{2}u^{6}-33732864z^{2}t^{2}u^{6}-64385280z^{2}u^{8}-1331w^{10}+16940w^{8}u^{2}+9658000w^{6}u^{4}+28838992w^{4}u^{6}+104442000w^{2}u^{8}+16866432t^{4}u^{6}-66019200t^{2}u^{8}+3499200u^{10})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.eo.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 12z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2u$ |
Equation of the image curve:
| $0$ | $=$ | $ 17328X^{8}+2960X^{6}Y^{2}+12X^{4}Y^{4}-27360X^{6}Z^{2}-3224X^{4}Y^{2}Z^{2}-12X^{2}Y^{4}Z^{2}+8520X^{4}Z^{4}+1324X^{2}Y^{2}Z^{4}+3Y^{4}Z^{4}+1800X^{2}Z^{6}-226Y^{2}Z^{6}+75Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.0-24.be.1.8 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.es.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.es.1.20 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iu.1.18 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iu.1.32 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iv.1.11 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iv.1.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.5-24.el.1.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.el.2.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.el.3.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.el.4.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.em.1.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.em.2.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.em.3.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.em.4.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.576.13-24.hq.1.12 | $24$ | $3$ | $3$ | $13$ | $2$ | $1^{10}$ |
| 120.384.5-120.vb.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.vb.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.vb.3.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.vb.4.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.vc.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.vc.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.vc.3.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.vc.4.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vb.1.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vb.2.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vb.3.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vb.4.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vc.1.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vc.2.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vc.3.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vc.4.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.vb.1.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.vb.2.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.vb.3.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.vb.4.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.vc.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.vc.2.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.vc.3.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.vc.4.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.vb.1.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.vb.2.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.vb.3.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.vb.4.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.vc.1.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.vc.2.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.vc.3.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.vc.4.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |