Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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: | 24.24.1.115 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&2\\0&13\end{bmatrix}$, $\begin{bmatrix}17&8\\2&3\end{bmatrix}$, $\begin{bmatrix}23&4\\12&11\end{bmatrix}$, $\begin{bmatrix}23&15\\20&5\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 48.48.1-24.bc.1.1, 48.48.1-24.bc.1.2, 48.48.1-24.bc.1.3, 48.48.1-24.bc.1.4, 48.48.1-24.bc.1.5, 48.48.1-24.bc.1.6, 48.48.1-24.bc.1.7, 48.48.1-24.bc.1.8, 240.48.1-24.bc.1.1, 240.48.1-24.bc.1.2, 240.48.1-24.bc.1.3, 240.48.1-24.bc.1.4, 240.48.1-24.bc.1.5, 240.48.1-24.bc.1.6, 240.48.1-24.bc.1.7, 240.48.1-24.bc.1.8 |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 y^{2} + 4 z^{2} - w^{2} $ |
| $=$ | $12 x^{2} - z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 3 y^{2} z^{2} - 9 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{(z-w)^{3}(z+w)^{3}}{w^{2}z^{4}}$ |
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.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.12.0.m.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.cb.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.gi.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gj.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gl.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ho.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hp.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hr.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.cy.1 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.96.5.by.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 120.48.1.qe.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.bs.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.elg.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.bcm.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.qe.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.by.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.qe.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.qf.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.qg.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.qh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.qu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.qv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.qw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.qx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.by.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.qe.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.qf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.qg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.qh.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.qu.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.qv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.qw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.qx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |