Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
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: | $0$ | ||||||
$\Q$-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: | 24.48.1.418 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&3\\4&1\end{bmatrix}$, $\begin{bmatrix}21&20\\14&7\end{bmatrix}$, $\begin{bmatrix}23&4\\16&3\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + z^{2} + w^{2} $ |
$=$ | $3 x^{2} - 6 y^{2} - 2 z^{2} - 2 z w - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 12 x^{2} y^{2} + 36 y^{4} + 12 y^{2} z^{2} + 9 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(3z^{2}-2zw+3w^{2})^{3}(5z^{2}+2zw+5w^{2})^{3}}{(z+w)^{8}(z^{2}+w^{2})^{2}}$ |
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 |
---|---|---|---|---|---|---|
8.24.1.bd.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.0.da.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.de.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.ed.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.ei.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.bf.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.1.bu.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 |
---|---|---|---|---|---|---|
24.144.9.bsf.1 | $24$ | $3$ | $3$ | $9$ | $4$ | $1^{8}$ |
24.192.9.ov.1 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{8}$ |
120.240.17.bbx.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.zfh.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |