Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| 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: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.1535 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&12\\4&23\end{bmatrix}$, $\begin{bmatrix}7&12\\6&1\end{bmatrix}$, $\begin{bmatrix}17&9\\0&13\end{bmatrix}$, $\begin{bmatrix}23&15\\2&23\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.135352 |
| Contains $-I$: | no $\quad$ (see 24.48.1.dq.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} - z w + 2 w^{2} $ |
| $=$ | $2 x^{2} - y^{2} + 2 z^{2} - 2 z w - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} + 10 x^{2} z^{2} - 2 y^{2} z^{2} + z^{4} $ |
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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{(z-w)^{3}(9z^{3}-27z^{2}w+3zw^{2}+31w^{3})^{3}}{w^{6}(z-2w)^{2}(z+w)(3z-5w)^{3}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.dq.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}+10X^{2}Z^{2}-2Y^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.f.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.f.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.y.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.y.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.es.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1-24.es.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.3-24.dn.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.dn.2.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.do.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.do.2.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.288.5-24.ch.1.6 | $24$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 48.192.3-48.gv.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.gw.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.gx.1.11 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.gy.1.10 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.5-48.ey.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 48.192.5-48.ey.2.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 48.192.5-48.fb.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 48.192.5-48.fb.2.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 72.288.5-72.n.1.12 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.z.1.2 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.bc.1.7 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.3-120.il.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.il.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.im.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.im.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.17-120.nz.1.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.3-168.gp.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gp.2.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gq.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gq.2.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.tt.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.tu.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.tv.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.tw.1.3 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.5-240.ma.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ma.2.20 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.md.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.md.2.18 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.3-264.gp.1.15 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gp.2.15 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gq.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gq.2.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.il.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.il.2.13 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.im.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.im.2.13 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |