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 | $4^{2}$ | ||
| 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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.469 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&10\\4&9\end{bmatrix}$, $\begin{bmatrix}11&5\\20&1\end{bmatrix}$, $\begin{bmatrix}23&14\\10&17\end{bmatrix}$ |
| 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 $ | $=$ | $ 6 x^{2} - y^{2} + y z - z^{2} $ |
| $=$ | $6 x^{2} + 6 y^{2} - 9 y z + 6 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 20 x^{2} y^{2} + 6 x^{2} z^{2} + 49 y^{4} - 42 y^{2} z^{2} + 9 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^2}\cdot\frac{471801510yz^{11}-1981474488yz^{9}w^{2}+1768761036yz^{7}w^{4}+48231288yz^{5}w^{6}-160641306yz^{3}w^{8}+60505200yzw^{10}-316758519z^{12}+377270622z^{10}w^{2}+1960700175z^{8}w^{4}-3223754100z^{6}w^{6}+1369211067z^{4}w^{8}-175681170z^{2}w^{10}-2100875w^{12}}{7766280yz^{11}+3882060yz^{9}w^{2}-12674340yz^{7}w^{4}+2765952yz^{5}w^{6}+3764768yz^{3}w^{8}-537824yzw^{10}-5214132z^{12}+8432856z^{10}w^{2}+7286517z^{8}w^{4}-8598324z^{6}w^{6}-48020z^{4}w^{8}+499408z^{2}w^{10}-268912w^{12}}$ |
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.h.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.0.q.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.s.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.et.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.fb.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.bl.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.bt.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.tj.1 | $24$ | $3$ | $3$ | $9$ | $2$ | $1^{8}$ |
| 24.192.9.hl.1 | $24$ | $4$ | $4$ | $9$ | $3$ | $1^{8}$ |
| 48.96.5.cn.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.da.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.dj.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.dp.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 120.240.17.mt.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.288.17.oim.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 240.96.5.im.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.iw.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.jn.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.kb.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |