Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| 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: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.58 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&23\\22&1\end{bmatrix}$, $\begin{bmatrix}17&4\\20&17\end{bmatrix}$, $\begin{bmatrix}19&10\\10&21\end{bmatrix}$, $\begin{bmatrix}21&19\\22&3\end{bmatrix}$ |
| 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: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 396x - 3024 $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{312x^{2}y^{6}+180556272x^{2}y^{4}z^{2}+6653452409856x^{2}y^{2}z^{4}+44673552633996288x^{2}z^{6}+42168xy^{6}z+7912594944xy^{4}z^{3}+190470929469696xy^{2}z^{5}+1026176675378909184xz^{7}+y^{8}+3287168y^{6}z^{2}+228433858560y^{4}z^{4}+2605015062042624y^{2}z^{6}+5881128527428227072z^{8}}{24x^{2}y^{6}-22032x^{2}y^{4}z^{2}-2239488x^{2}y^{2}z^{4}+60466176x^{2}z^{6}-72xy^{6}z+248832xy^{4}z^{3}+28553472xy^{2}z^{5}-725594112xz^{7}+y^{8}-10368y^{6}z^{2}+5971968y^{4}z^{4}+524040192y^{2}z^{6}-15237476352z^{8}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.s.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.br.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.by.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
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.d.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.cn.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ef.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ej.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ke.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.kk.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.le.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.lg.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.72.5.ko.1 | $24$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 24.96.5.eg.1 | $24$ | $4$ | $4$ | $5$ | $2$ | $1^{4}$ |
| 48.48.2.cu.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.cv.1 | $48$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 48.48.2.cw.1 | $48$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 48.48.2.cx.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.cy.1 | $48$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 48.48.2.cz.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.da.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.db.1 | $48$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 120.48.1.bhg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bhk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bhw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bia.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bsa.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bse.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bsq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bsu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.qk.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.oac.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.ewm.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bhe.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bhi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bhu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bhy.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bry.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bsc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bso.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bss.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.ki.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 240.48.2.cw.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.cx.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.cy.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.cz.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.da.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.db.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.dc.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.dd.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.48.1.bhe.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bhi.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bhu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bhy.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bry.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bsc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bso.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bss.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.im.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bhg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bhk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bhw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bia.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bsa.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bse.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bsq.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bsu.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |