Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $64$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.64 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&11\\22&17\end{bmatrix}$, $\begin{bmatrix}17&0\\0&19\end{bmatrix}$, $\begin{bmatrix}17&17\\22&23\end{bmatrix}$, $\begin{bmatrix}17&19\\14&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 4x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(-2:0:1)$, $(0:0:1)$, $(2:0:1)$ |
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.n.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ch.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gn.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 |
|---|---|---|---|---|---|---|
| 24.144.9.cg.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.uf.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.xt.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.yd.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.dfw.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.dfz.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.dgu.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.dgx.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
| 72.216.13.ne.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.9.bgdf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgdh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgdv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgdx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgfr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgft.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcab.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcap.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcar.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bccl.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bccn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcgb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcgp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcgr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcil.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcin.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcah.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcaj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcax.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcaz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcct.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bccv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |