Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
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 | $4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.446 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&11\\20&3\end{bmatrix}$, $\begin{bmatrix}11&16\\18&17\end{bmatrix}$, $\begin{bmatrix}21&22\\14&23\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 48.96.1-24.gz.1.1, 48.96.1-24.gz.1.2, 48.96.1-24.gz.1.3, 48.96.1-24.gz.1.4, 240.96.1-24.gz.1.1, 240.96.1-24.gz.1.2, 240.96.1-24.gz.1.3, 240.96.1-24.gz.1.4 |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{6}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y^{2} + 2 y z - 2 z^{2} - 2 w^{2} $ |
$=$ | $24 x^{2} + y^{2} - y z + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 8 x^{2} y^{2} + 6 x^{2} z^{2} + y^{4} + 6 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 4x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{z^{3}(3z^{2}+2w^{2})(7020yz^{6}+7020yz^{4}w^{2}+1752yz^{2}w^{4}+64yw^{6}-5139z^{7}-9192z^{5}w^{2}-4660z^{3}w^{4}-608zw^{6})}{w^{4}(4536yz^{7}+4536yz^{5}w^{2}+1296yz^{3}w^{4}+96yzw^{6}-3321z^{8}-5940z^{6}w^{2}-3132z^{4}w^{4}-528z^{2}w^{6}-16w^{8})}$ |
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.z.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.0.cn.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.cs.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.dv.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.ej.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.y.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.1.bj.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.bpx.1 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{8}$ |
24.192.9.nt.1 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
120.240.17.zp.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.zbt.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |