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: | 24.48.1.245 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&13\\10&5\end{bmatrix}$, $\begin{bmatrix}19&6\\20&5\end{bmatrix}$, $\begin{bmatrix}23&15\\22&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| 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 $ | $=$ | $ 2 x^{2} - y^{2} + y z - z^{2} + w^{2} $ |
| $=$ | $4 x w + y^{2} + 2 y z - 2 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 4 x^{3} y - 2 x^{2} z^{2} + 8 x y^{3} + 4 x y z^{2} + 4 y^{4} - 8 y^{2} z^{2} + 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{4492800xyz^{9}w+32732160xyz^{7}w^{3}+39648000xyz^{5}w^{5}+10053888xyz^{3}w^{7}+396384xyzw^{9}-4845312xz^{10}w-67176960xz^{8}w^{3}-148233344xz^{6}w^{5}-73155072xz^{4}w^{7}-7313136xz^{2}w^{9}-67168xw^{11}-1347840yz^{11}-36823680yz^{9}w^{2}-98553216yz^{7}w^{4}-50404416yz^{5}w^{6}-4027024yz^{3}w^{8}+44856yzw^{10}+986688z^{12}+28833408z^{10}w^{2}+80379984z^{8}w^{4}+32714048z^{6}w^{6}-6896788z^{4}w^{8}-1640760z^{2}w^{10}-19673w^{12}}{w^{8}(24xyzw-60xz^{2}w-24xw^{3}-36yz^{3}-18yzw^{2}+27z^{4}+18z^{2}w^{2}-7w^{4})}$ |
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.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.em.2 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.1 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.192.9.qm.2 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.96.5.hs.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.hu.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.jo.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.jq.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.sa.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.sc.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.tw.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.ty.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 120.240.17.fnx.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.288.17.cgwr.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 240.96.5.bzc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bze.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bzk.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bzm.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.cqq.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.cqs.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.cqy.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.cra.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |