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: | 24.24.1.118 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&13\\22&11\end{bmatrix}$, $\begin{bmatrix}15&11\\16&9\end{bmatrix}$, $\begin{bmatrix}17&2\\6&11\end{bmatrix}$, $\begin{bmatrix}21&11\\2&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 48.48.1-24.bj.1.1, 48.48.1-24.bj.1.2, 48.48.1-24.bj.1.3, 48.48.1-24.bj.1.4, 48.48.1-24.bj.1.5, 48.48.1-24.bj.1.6, 48.48.1-24.bj.1.7, 48.48.1-24.bj.1.8, 240.48.1-24.bj.1.1, 240.48.1-24.bj.1.2, 240.48.1-24.bj.1.3, 240.48.1-24.bj.1.4, 240.48.1-24.bj.1.5, 240.48.1-24.bj.1.6, 240.48.1-24.bj.1.7, 240.48.1-24.bj.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 $ |
| $=$ | $96 x^{2} - 6 y^{2} - 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6 x^{4} + 18 x^{2} y^{2} - x^{2} z^{2} - 6 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{13440y^{2}z^{4}+12960y^{2}z^{2}w^{2}+2910y^{2}w^{4}+13824z^{6}+17984z^{4}w^{2}+6248z^{2}w^{4}+27w^{6}}{384y^{2}z^{4}-96y^{2}z^{2}w^{2}-6y^{2}w^{4}+512z^{6}-64z^{4}w^{2}+24z^{2}w^{4}+w^{6}}$ |
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 |
| 24.12.0.br.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.bt.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.l.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ce.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ee.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.eg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gm.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hf.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.et.1 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.96.5.cp.1 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 120.48.1.rp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.vh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.vl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.vx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.cp.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.gdn.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.bmb.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.rn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.rr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.sd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.sh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.vf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.vj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.vv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.vz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.cx.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.rn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.rr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.sd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.sh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.vf.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.vj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.vv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.vz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.cx.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.rp.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.rt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.sf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.sj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.vh.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.vl.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.vx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.wb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |