Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
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 | $2^{4}$ | ||
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: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.217 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&0\\10&7\end{bmatrix}$, $\begin{bmatrix}13&12\\8&1\end{bmatrix}$, $\begin{bmatrix}17&7\\16&19\end{bmatrix}$, $\begin{bmatrix}23&1\\22&9\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 24-isogeny field degree: | $8$ |
Cyclic 24-torsion field degree: | $64$ |
Full 24-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{5}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x y + x z - y^{2} $ |
$=$ | $x^{2} - x y + 3 x z + y^{2} - 4 z^{2} - 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 4 x^{3} z - 3 x^{2} y^{2} - 6 x y^{2} z - 8 x z^{3} - 3 y^{2} z^{2} - 4 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot3^3\,\frac{(2z^{2}+w^{2})^{3}(2z^{2}+3w^{2})^{3}}{z^{8}(4z^{2}+3w^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.24.0.bf.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
12.24.0.n.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.cj.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.eq.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.da.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.1.dj.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.1.em.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 |
---|---|---|---|---|---|---|
24.96.1.dv.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.96.1.dv.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.144.9.edh.1 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
24.192.9.pu.1 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{8}$ |
120.96.1.tl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.96.1.tl.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.240.17.fmz.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.cgvn.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
168.96.1.tj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.96.1.tj.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.96.1.tj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.96.1.tj.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.96.1.tl.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.96.1.tl.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |