Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12G3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.144.3.1102 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&2\\8&23\end{bmatrix}$, $\begin{bmatrix}1&16\\2&7\end{bmatrix}$, $\begin{bmatrix}5&20\\2&1\end{bmatrix}$, $\begin{bmatrix}19&12\\0&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.3.ba.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $512$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 36.2.a.a, 48.2.a.a, 576.2.a.e |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x y + 2 x t - w u + t u $ |
| $=$ | $y^{2} + y w + 2 u^{2}$ | |
| $=$ | $x y + 2 x w - 2 x t + t u$ | |
| $=$ | $2 x^{2} - y^{2} - z^{2} - w t + t^{2} - u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 49 x^{8} - 126 x^{6} y^{2} + 36 x^{6} z^{2} + 109 x^{4} y^{4} - 224 x^{4} y^{2} z^{2} - 84 x^{4} z^{4} + \cdots + 144 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ w^{2} $ | $=$ | $ -14 x^{4} - 16 x^{2} y z + 8 x^{2} z^{2} + 8 z^{4} $ |
| $0$ | $=$ | $x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^3\,\frac{110592xt^{7}u-801920xt^{5}u^{3}+1230496xt^{3}u^{5}-331088xtu^{7}-6928yt^{8}+107160yt^{6}u^{2}-267692yt^{4}u^{4}+118250yt^{2}u^{6}-1024yu^{8}-16wt^{8}-48304wt^{6}u^{2}+276520wt^{4}u^{4}-288876wt^{2}u^{6}+27743wu^{8}+41376t^{7}u^{2}-193408t^{5}u^{4}+141912t^{3}u^{6}}{512xt^{7}u-1984xt^{5}u^{3}+896xt^{3}u^{5}-36xtu^{7}-32yt^{8}+352yt^{6}u^{2}-280yt^{4}u^{4}+16yt^{2}u^{6}-224wt^{6}u^{2}+560wt^{4}u^{4}-110wt^{2}u^{6}+wu^{8}+192t^{7}u^{2}-320t^{5}u^{4}+28t^{3}u^{6}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.72.3.ba.1 :
| $\displaystyle X$ | $=$ | $\displaystyle u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 49X^{8}-126X^{6}Y^{2}+109X^{4}Y^{4}-36X^{2}Y^{6}+4Y^{8}+36X^{6}Z^{2}-224X^{4}Y^{2}Z^{2}+140X^{2}Y^{4}Z^{2}+48Y^{6}Z^{2}-84X^{4}Z^{4}+336X^{2}Y^{2}Z^{4}+148Y^{4}Z^{4}-144X^{2}Z^{6}+48Y^{2}Z^{6}+144Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.72.2-12.f.1.4 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.72.2-12.f.1.5 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.72.1-24.c.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.72.1-24.c.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.72.2-24.a.1.2 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.72.2-24.a.1.12 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.288.7-24.j.1.5 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 24.288.7-24.j.1.7 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 24.288.7-24.v.1.9 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 24.288.7-24.v.1.11 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 24.288.7-24.dg.1.3 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 24.288.7-24.dg.1.4 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 24.288.7-24.dl.1.6 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 24.288.7-24.dl.1.8 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 72.432.15-72.bg.1.2 | $72$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 120.288.7-120.jz.1.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.jz.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.kj.1.10 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.kj.1.14 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ll.1.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ll.1.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.lv.1.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.lv.1.15 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.iz.1.10 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.iz.1.11 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.jj.1.6 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.jj.1.14 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.kl.1.5 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.kl.1.8 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.kv.1.6 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.kv.1.16 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.iz.1.10 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.iz.1.11 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.jj.1.10 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.jj.1.12 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.kl.1.5 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.kl.1.8 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.kv.1.10 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.kv.1.16 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.iz.1.12 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.iz.1.13 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.jj.1.10 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.jj.1.14 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.kl.1.9 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.kl.1.12 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.kv.1.9 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.kv.1.15 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |