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: | 24.96.1.1422 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&13\\0&7\end{bmatrix}$, $\begin{bmatrix}13&20\\18&5\end{bmatrix}$, $\begin{bmatrix}19&22\\6&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.69761 |
| Contains $-I$: | no $\quad$ (see 24.48.1.ci.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 $ | $=$ | $ 3 x y - z^{2} $ |
| $=$ | $2 x^{2} + 2 x y + 18 y^{2} + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + 6 x^{2} y^{2} + 10 x^{2} z^{2} + 3 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3\cdot3}\cdot\frac{(8z^{2}+3w^{2})(8945664y^{2}z^{8}-626688y^{2}z^{6}w^{2}-787968y^{2}z^{4}w^{4}-2358720y^{2}z^{2}w^{6}-353808y^{2}w^{8}+327680z^{10}-147456z^{8}w^{2}-373248z^{6}w^{4}-848448z^{4}w^{6}-262440z^{2}w^{8}-19683w^{10})}{w^{2}z^{4}(96y^{2}z^{4}-36y^{2}z^{2}w^{2}-27y^{2}w^{4}+32z^{6}-6z^{4}w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.ci.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}+6X^{2}Y^{2}+10X^{2}Z^{2}+3Z^{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.j.1.8 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.j.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.p.1.10 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.p.1.15 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.eq.1.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1-24.eq.1.16 | $24$ | $2$ | $2$ | $1$ | $1$ | 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.bk.1.1 | $24$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 72.288.5-72.h.1.5 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.n.1.8 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.p.1.8 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.480.17-120.iv.1.8 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |