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: | 24.96.1.1714 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&20\\6&11\end{bmatrix}$, $\begin{bmatrix}7&22\\6&23\end{bmatrix}$, $\begin{bmatrix}17&6\\0&1\end{bmatrix}$, $\begin{bmatrix}17&19\\0&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.136648 |
| Contains $-I$: | no $\quad$ (see 24.48.1.er.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.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 2 y z $ |
| $=$ | $4 x^{2} + y^{2} - 2 y z + 9 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{4} + 20 x^{2} z^{2} + y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{372736yz^{11}-501760yz^{9}w^{2}+235008yz^{7}w^{4}-42752yz^{5}w^{6}+2096yz^{3}w^{8}-24yzw^{10}-368640z^{12}+456704z^{10}w^{2}-172800z^{8}w^{4}+14336z^{6}w^{6}+2816z^{4}w^{8}-168z^{2}w^{10}+w^{12}}{w^{2}z^{6}(648yz^{3}-18yzw^{2}-648z^{4}-63z^{2}w^{2}+w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.er.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 6z$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{4}+20X^{2}Z^{2}+Y^{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.d.1.10 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.d.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bx.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bx.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.et.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1-24.et.1.9 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.dz.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.dz.2.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.ea.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.ea.2.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.288.5-24.dn.1.4 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 72.288.5-72.v.1.7 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.bn.1.10 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.bt.1.3 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.3-120.kd.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kd.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ke.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ke.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.17-120.qv.1.5 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.3-168.id.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.id.2.11 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ie.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ie.2.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.id.1.11 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.id.2.4 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ie.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ie.2.4 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kd.1.3 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kd.2.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ke.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ke.2.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |