Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.8 |
Level structure
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 216 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-6:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3\cdot3^3}\cdot\frac{(y^{2}+648z^{2})^{3}(y^{6}+48600y^{4}z^{2}-18895680y^{2}z^{4}+2448880128z^{6})^{3}}{z^{2}y^{6}(y^{2}-1944z^{2})^{6}(y^{2}-216z^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.36.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.ca.1 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.24.1.cl.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.36.0.f.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.cy.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.fv.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.gb.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.gx.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.hd.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.hz.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ih.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.jb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.jj.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.7.cp.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.dj.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.ek.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.10.bl.1 | $72$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 72.216.13.dn.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.csy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ctc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cua.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cue.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dlc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dlg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dme.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dmi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.biy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bjc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bka.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bke.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bts.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.btw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.buu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.buy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.biy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bjc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bka.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bke.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bts.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.btw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.buu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.buy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.biy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bjc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bka.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bke.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bts.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.btw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.buu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.buy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |