Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
Index: | $48$ | $\PSL_2$-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 | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
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: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.6 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&6\\2&19\end{bmatrix}$, $\begin{bmatrix}1&10\\18&23\end{bmatrix}$, $\begin{bmatrix}13&2\\22&7\end{bmatrix}$, $\begin{bmatrix}17&6\\14&19\end{bmatrix}$, $\begin{bmatrix}19&14\\6&5\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 24.96.1-24.a.1.1, 24.96.1-24.a.1.2, 24.96.1-24.a.1.3, 24.96.1-24.a.1.4, 24.96.1-24.a.1.5, 24.96.1-24.a.1.6, 120.96.1-24.a.1.1, 120.96.1-24.a.1.2, 120.96.1-24.a.1.3, 120.96.1-24.a.1.4, 120.96.1-24.a.1.5, 120.96.1-24.a.1.6, 168.96.1-24.a.1.1, 168.96.1-24.a.1.2, 168.96.1-24.a.1.3, 168.96.1-24.a.1.4, 168.96.1-24.a.1.5, 168.96.1-24.a.1.6, 264.96.1-24.a.1.1, 264.96.1-24.a.1.2, 264.96.1-24.a.1.3, 264.96.1-24.a.1.4, 264.96.1-24.a.1.5, 264.96.1-24.a.1.6, 312.96.1-24.a.1.1, 312.96.1-24.a.1.2, 312.96.1-24.a.1.3, 312.96.1-24.a.1.4, 312.96.1-24.a.1.5, 312.96.1-24.a.1.6 |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + z w $ |
$=$ | $4 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} + y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{2}-zw+w^{2})^{3}(z^{2}+zw+w^{2})^{3}}{w^{4}z^{4}(z^{2}+w^{2})^{2}}$ |
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 |
---|---|---|---|---|---|---|
4.24.0.a.1 | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.j.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.ds.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.ec.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.b.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.24.1.dc.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.24.1.dm.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
24.96.3.a.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
24.96.3.c.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.144.9.c.1 | $24$ | $3$ | $3$ | $9$ | $2$ | $1^{8}$ |
24.192.9.c.1 | $24$ | $4$ | $4$ | $9$ | $3$ | $1^{8}$ |
120.96.3.d.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.96.3.e.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.240.17.a.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.a.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
168.96.3.d.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.96.3.e.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.96.3.d.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.96.3.e.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.96.3.d.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.96.3.e.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |