Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| 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: | 24.96.1.2117 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&1\\12&17\end{bmatrix}$, $\begin{bmatrix}5&5\\6&5\end{bmatrix}$, $\begin{bmatrix}19&7\\6&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.69821 |
| Contains $-I$: | no $\quad$ (see 24.48.1.dx.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: | $2^{3}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x y + z^{2} $ |
| $=$ | $2 x^{2} - 2 x y + 18 y^{2} + 3 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} - 6 x^{2} y^{2} + 20 x^{2} z^{2} + 12 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{(4z^{2}-3w^{2})(559104y^{2}z^{8}+78336y^{2}z^{6}w^{2}-196992y^{2}z^{4}w^{4}+1179360y^{2}z^{2}w^{6}-353808y^{2}w^{8}+10240z^{10}+9216z^{8}w^{2}-46656z^{6}w^{4}+212112z^{4}w^{6}-131220z^{2}w^{8}+19683w^{10})}{w^{2}z^{4}(48y^{2}z^{4}+36y^{2}z^{2}w^{2}-54y^{2}w^{4}+8z^{6}+3z^{4}w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.dx.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}-6X^{2}Y^{2}+20X^{2}Z^{2}+12Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.i.1.4 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.y.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.y.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.cc.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.cc.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-12.i.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.288.5-24.dc.1.4 | $24$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 72.288.5-72.r.1.6 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.bg.1.6 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.bj.1.5 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.480.17-120.oz.1.3 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |