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: | 24.72.1.3 |
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} - 8 $ |
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 |
|---|
| $(2:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3}\cdot\frac{(y^{2}-24z^{2})^{3}(y^{6}-1800y^{4}z^{2}-25920y^{2}z^{4}-124416z^{6})^{3}}{z^{2}y^{6}(y^{2}+8z^{2})^{2}(y^{2}+72z^{2})^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 8.6.0.a.1 | $8$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
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.p.1 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.24.1.bx.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.36.0.f.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.g.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.y.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ba.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bg.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bh.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bk.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.bl.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bs.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bu.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.7.e.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.f.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.h.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.10.a.1 | $72$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 72.216.13.c.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.ge.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.gf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.gm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.gn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.hc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.hd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.hk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.hl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.dc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.dd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.dk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.dl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ds.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.dt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ea.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.eb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.dc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.dd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.dk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.dl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ds.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.dt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ea.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.eb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.dc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.dd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.dk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.dl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ds.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.dt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ea.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.eb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |