Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.75 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&21\\4&11\end{bmatrix}$, $\begin{bmatrix}5&20\\16&9\end{bmatrix}$, $\begin{bmatrix}11&20\\10&5\end{bmatrix}$, $\begin{bmatrix}13&19\\2&19\end{bmatrix}$ |
| 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: | $3072$ |
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 $ | $=$ | $ x z - y w $ |
| $=$ | $8 x^{2} - 12 y^{2} - 3 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + 3 x^{2} y^{2} - x^{2} z^{2} - 2 y^{2} z^{2} $ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{1728y^{6}+1872y^{4}w^{2}+556y^{2}w^{4}+35z^{6}-90z^{4}w^{2}+80z^{2}w^{4}+w^{6}}{1728y^{6}+144y^{4}w^{2}-36y^{2}w^{4}+27z^{6}+18z^{4}w^{2}-12z^{2}w^{4}+w^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0.u.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.12.0.n.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.bz.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 |
|---|---|---|---|---|---|---|
| 24.48.1.bm.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.cv.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ky.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lp.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.mb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.mm.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.ey.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 120.48.1.bji.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bjm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bko.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bks.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.buc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bug.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bvi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bvm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.si.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.odg.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.faw.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bjg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bjk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bkm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bkq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bua.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bue.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bvg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bvk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.mg.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.bjg.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bjk.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bkm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bkq.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bua.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bue.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bvg.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bvk.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.kk.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bji.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bjm.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bko.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bks.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.buc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bug.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bvi.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bvm.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |