Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.102 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&15\\0&11\end{bmatrix}$, $\begin{bmatrix}5&22\\18&23\end{bmatrix}$, $\begin{bmatrix}9&10\\2&15\end{bmatrix}$, $\begin{bmatrix}11&4\\18&17\end{bmatrix}$ |
| 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: | $3072$ |
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 $ | $=$ | $ y^{2} + y z + z^{2} - w^{2} $ |
| $=$ | $6 x^{2} + y w - z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 3 x^{2} y z + 3 y^{2} z^{2} - 36 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3^3\,\frac{z^{3}(yw^{2}+z^{3}-zw^{2})}{w^{4}(3yz-w^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.12.0.l.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.ca.1 | $24$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.48.1.ga.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gc.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hh.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hi.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hj.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.cw.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.bw.1 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 120.48.1.pg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ph.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.pi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.pj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.pw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.px.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.py.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.pz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.bq.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.ele.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.bck.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.pg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ph.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.pi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.pj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.pw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.px.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.py.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.pz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.bw.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.pg.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ph.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.pi.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.pj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.pw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.px.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.py.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.pz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.bw.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.pg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.ph.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.pi.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.pj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.pw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.px.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.py.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.pz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |