Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $6^{2}$ | Cusp orbits | $2$ | ||
| 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: | 6B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.12.1.44 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&9\\0&5\end{bmatrix}$, $\begin{bmatrix}22&1\\7&11\end{bmatrix}$, $\begin{bmatrix}23&6\\6&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $48$ |
| Cyclic 24-torsion field degree: | $384$ |
| Full 24-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 36 x^{2} + 3 x z + y^{2} $ |
| $=$ | $48 x z + 8 y^{2} - 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} - 2 x^{2} y^{2} + 3 x^{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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{24}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}y$ |
Maps to other modular curves
$j$-invariant map of degree 12 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^7\cdot3^3\,\frac{z^{3}}{12xw^{2}-2z^{3}+zw^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.6.0.i.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.6.0.h.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.6.1.c.1 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.36.1.cw.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.48.3.by.1 | $24$ | $4$ | $4$ | $3$ | $1$ | $1^{2}$ |
| 72.36.3.bt.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.108.7.fu.1 | $72$ | $9$ | $9$ | $7$ | $?$ | not computed |
| 120.60.5.dm.1 | $120$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 120.72.5.vi.1 | $120$ | $6$ | $6$ | $5$ | $?$ | not computed |
| 120.120.9.uw.1 | $120$ | $10$ | $10$ | $9$ | $?$ | not computed |
| 168.96.7.di.1 | $168$ | $8$ | $8$ | $7$ | $?$ | not computed |
| 168.252.19.la.1 | $168$ | $21$ | $21$ | $19$ | $?$ | not computed |
| 264.144.11.di.1 | $264$ | $12$ | $12$ | $11$ | $?$ | not computed |
| 312.168.13.ha.1 | $312$ | $14$ | $14$ | $13$ | $?$ | not computed |