Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $8^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $4$ 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: | 8I1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.233 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\22&11\end{bmatrix}$, $\begin{bmatrix}1&4\\4&21\end{bmatrix}$, $\begin{bmatrix}7&1\\20&9\end{bmatrix}$, $\begin{bmatrix}13&15\\20&11\end{bmatrix}$ |
| 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 $ | $=$ | $ y^{2} + y z - y w + z w $ |
| $=$ | $3 x^{2} + 2 y z + 2 y w + z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 3 x^{2} y^{2} - 6 x^{2} z^{2} - 6 x y^{2} z - 3 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(z-w)^{3}(3376yz^{8}-68832yz^{7}w+512896yz^{6}w^{2}-1697056yz^{5}w^{3}+2532000yz^{4}w^{4}-1697056yz^{3}w^{5}+512896yz^{2}w^{6}-68832yzw^{7}+3376yw^{8}+z^{9}+3391z^{8}w-61972z^{7}w^{2}+395988z^{6}w^{3}-1008346z^{5}w^{4}+1008346z^{4}w^{5}-395988z^{3}w^{6}+61972z^{2}w^{7}-3391zw^{8}-w^{9})}{(z+w)^{4}(yz^{7}-13yz^{6}w+61yz^{5}w^{2}-129yz^{4}w^{3}+129yz^{3}w^{4}-61yz^{2}w^{5}+13yzw^{6}-yw^{7}+z^{7}w-11z^{6}w^{2}+41z^{5}w^{3}-63z^{4}w^{4}+41z^{3}w^{5}-11z^{2}w^{6}+zw^{7})}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.0.bt.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.3.q.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.96.3.dd.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.3.ev.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.96.3.ex.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.3.if.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.3.ii.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.96.3.in.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.3.iq.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.144.7.cox.1 | $24$ | $3$ | $3$ | $7$ | $2$ | $1^{6}$ |
| 24.192.11.p.1 | $24$ | $4$ | $4$ | $11$ | $2$ | $1^{10}$ |
| 48.96.3.we.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.wg.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.yg.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.ym.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.zg.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.zi.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.zu.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.baa.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.5.lk.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.5.lq.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.mc.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.me.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.5.pg.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.96.5.pm.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.rm.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 48.96.5.ro.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 120.96.3.zz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bad.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bah.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bal.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bat.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bbb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bbj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bbr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.240.17.lql.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.288.17.dmsf.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 168.96.3.xl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.xp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.xt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.xx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.yf.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.yn.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.yv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.zd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fpa.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fpc.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.frc.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fri.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.ftq.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fts.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fum.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fus.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.dia.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dig.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dja.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.djc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dma.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dmg.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dog.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.doi.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.96.3.xl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.xp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.xt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.xx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.yf.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.yn.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.yv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.zd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.zz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bad.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bah.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bal.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bat.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bbb.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bbj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bbr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |