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 | $2^{2}\cdot4$ | ||
| 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: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.255 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&23\\2&21\end{bmatrix}$, $\begin{bmatrix}17&0\\8&7\end{bmatrix}$, $\begin{bmatrix}19&9\\10&19\end{bmatrix}$ |
| 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: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 8 x^{2} + y^{2} - z^{2} - z w - w^{2} $ |
| $=$ | $8 x y + z^{2} - 2 z w - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 8 x^{3} z - 8 x^{2} y^{2} - 4 x y^{2} z + 4 x z^{3} + 2 y^{4} - 2 y^{2} z^{2} + 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{664y^{2}z^{10}+6104y^{2}z^{9}w+22104y^{2}z^{8}w^{2}+42816y^{2}z^{7}w^{3}+48576y^{2}z^{6}w^{4}+32832y^{2}z^{5}w^{5}+10944y^{2}z^{4}w^{6}-768y^{2}z^{3}w^{7}-1152y^{2}z^{2}w^{8}-640y^{2}zw^{9}-128y^{2}w^{10}-87z^{12}-204z^{11}w+1344z^{10}w^{2}+5368z^{9}w^{3}+4572z^{8}w^{4}-9312z^{7}w^{5}-25216z^{6}w^{6}-24384z^{5}w^{7}-12432z^{4}w^{8}-3520z^{3}w^{9}+384zw^{11}+64w^{12}}{(z^{2}-2zw-2w^{2})^{4}(8y^{2}z^{2}+8y^{2}zw+8y^{2}w^{2}-7z^{4}-20z^{3}w-24z^{2}w^{2}-8zw^{3}-4w^{4})}$ |
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.bl.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.em.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.ee.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.eef.2 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.192.9.qm.1 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.96.5.hs.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.hu.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.jo.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.jq.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.sa.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.sc.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.tw.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.ty.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 120.240.17.fnx.2 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.288.17.cgwr.2 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 240.96.5.bzc.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bze.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bzk.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bzm.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.cqq.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.cqs.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.cqy.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.cra.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |