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: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.112 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&21\\2&13\end{bmatrix}$, $\begin{bmatrix}15&16\\20&19\end{bmatrix}$, $\begin{bmatrix}17&14\\12&1\end{bmatrix}$, $\begin{bmatrix}23&19\\8&9\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 48.48.1-24.bq.1.1, 48.48.1-24.bq.1.2, 48.48.1-24.bq.1.3, 48.48.1-24.bq.1.4, 48.48.1-24.bq.1.5, 48.48.1-24.bq.1.6, 48.48.1-24.bq.1.7, 48.48.1-24.bq.1.8, 240.48.1-24.bq.1.1, 240.48.1-24.bq.1.2, 240.48.1-24.bq.1.3, 240.48.1-24.bq.1.4, 240.48.1-24.bq.1.5, 240.48.1-24.bq.1.6, 240.48.1-24.bq.1.7, 240.48.1-24.bq.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 $ | $=$ | $ 6 x y - z w $ |
| $=$ | $48 x^{2} - 3 y^{2} - 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 12 x^{4} + 18 x^{2} y^{2} - x^{2} z^{2} - 3 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{420y^{2}z^{4}+1620y^{2}z^{2}w^{2}+1455y^{2}w^{4}+216z^{6}+1124z^{4}w^{2}+1562z^{2}w^{4}+27w^{6}}{12y^{2}z^{4}-12y^{2}z^{2}w^{2}-3y^{2}w^{4}+8z^{6}-4z^{4}w^{2}+6z^{2}w^{4}+w^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.1.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.12.0.n.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.bv.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.g.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.cd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fa.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fc.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fs.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hr.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ht.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.gg.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.cw.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 120.48.1.su.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.to.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.xs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.xw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.dm.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.gfa.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.boe.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.ss.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.sw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ti.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.tm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.wk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.wo.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.xq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.xu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.du.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.ss.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.sw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ti.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.tm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.wk.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.wo.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.xq.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.xu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.du.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.su.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.sy.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.tk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.to.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.wm.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.wq.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.xs.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.xw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |