Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 24.96.1.1723 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&22\\0&23\end{bmatrix}$, $\begin{bmatrix}11&14\\18&11\end{bmatrix}$, $\begin{bmatrix}19&3\\12&23\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.69763 |
| Contains $-I$: | no $\quad$ (see 24.48.1.ii.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 2 y z $ |
| $=$ | $12 x^{2} - 3 y^{2} - 6 y z - 27 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{4} - 20 x^{2} z^{2} - 3 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^3}\cdot\frac{271724544yz^{11}+121927680yz^{9}w^{2}+19035648yz^{7}w^{4}+1154304yz^{5}w^{6}+18864yz^{3}w^{8}+72yzw^{10}+268738560z^{12}+110979072z^{10}w^{2}+13996800z^{8}w^{4}+387072z^{6}w^{6}-25344z^{4}w^{8}-504z^{2}w^{10}-w^{12}}{w^{2}z^{6}(5832yz^{3}+54yzw^{2}+5832z^{4}-189z^{2}w^{2}-w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.ii.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 6z$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{4}-20X^{2}Z^{2}-3Y^{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.h.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.h.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bw.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bw.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.eq.1.6 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1-24.eq.1.7 | $24$ | $2$ | $2$ | $1$ | $1$ | 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.fs.1.4 | $24$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
| 72.288.5-72.bc.1.7 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.ch.1.6 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.ck.1.5 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.480.17-120.bpu.1.5 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |