Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $32$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 24H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.67 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&6\\12&5\end{bmatrix}$, $\begin{bmatrix}7&0\\18&17\end{bmatrix}$, $\begin{bmatrix}15&22\\22&21\end{bmatrix}$, $\begin{bmatrix}17&19\\2&11\end{bmatrix}$, $\begin{bmatrix}19&5\\20&17\end{bmatrix}$ |
| 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: | $1024$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 4x $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}}\cdot\frac{336x^{2}y^{20}z^{2}-18432x^{2}y^{16}z^{6}+57999360x^{2}y^{12}z^{10}-257370882048x^{2}y^{8}z^{14}+23093770715136x^{2}y^{4}z^{18}-70351564308480x^{2}z^{22}+24xy^{22}z-6656xy^{18}z^{5}-23986176xy^{14}z^{9}+25839009792xy^{10}z^{13}-9619787218944xy^{6}z^{17}+140741783322624xy^{2}z^{21}+y^{24}+512y^{20}z^{4}+3219456y^{16}z^{8}-1028653056y^{12}z^{12}+1375916261376y^{8}z^{16}-30769145708544y^{4}z^{20}+68719476736z^{24}}{z^{8}y^{12}(4xz-y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.0.p.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ci.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.go.1 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.144.5.xi.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.xj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.xk.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.xl.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.yo.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.yp.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.yq.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.yr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bba.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bbb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bbc.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.bbd.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.bbq.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bbr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bbs.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.bbt.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.bc.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.nv.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bfo.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bft.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dgy.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dhb.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dhw.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dhz.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 72.216.13.nk.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kys.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kzi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kzj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kzk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kzl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lbe.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lbf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lbg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lbh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lbu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lbv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lbw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lbx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bgeh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgej.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgex.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgez.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bghj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bghl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hyr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hys.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hzh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hzi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hzj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hzk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ibd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ibe.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ibf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ibg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ibt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ibu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ibv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ibw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bcbb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbt.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bced.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcef.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hys.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hzi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hzj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hzk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hzl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ibe.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ibf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ibg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ibh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ibu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ibv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ibw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ibx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bchb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bchd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bchr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcht.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjn.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bckd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bckf.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hys.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hzi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hzj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hzk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hzl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ibe.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ibf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ibg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ibh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ibu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ibv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ibw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ibx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bcbj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bccb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcel.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcen.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |