Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $6^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.143 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&6\\6&1\end{bmatrix}$, $\begin{bmatrix}13&17\\8&17\end{bmatrix}$, $\begin{bmatrix}19&2\\14&5\end{bmatrix}$, $\begin{bmatrix}23&3\\12&5\end{bmatrix}$ |
| 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: | $2048$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 4 x y - 2 x z + y^{2} + 2 y z $ |
| $=$ | $3 x y + x z - y z - 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 8 x^{3} z - 6 x^{2} y^{2} + 30 x^{2} z^{2} + 12 x y^{2} z + 8 x z^{3} - 6 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^3\,\frac{864xz^{6}w^{2}-432xz^{4}w^{4}+48xz^{2}w^{6}+xw^{8}-864yz^{6}w^{2}+432yz^{4}w^{4}-48yz^{2}w^{6}-yw^{8}-6912z^{9}+3456z^{7}w^{2}-168z^{3}w^{6}+16zw^{8}}{z^{6}(xw^{2}-yw^{2}-32z^{3}+4zw^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.0.h.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.bm.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.18.0.d.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.e.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 |
|---|---|---|---|---|---|---|
| 24.72.3.li.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.lm.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.mk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.mo.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.up.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.us.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.vr.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.vu.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.7.et.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.ey.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.3.ddk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ddm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ddr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ddt.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dug.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dui.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dun.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dup.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.pi.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.ny.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.czg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.czi.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.czn.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.czp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dmu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dmw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dnb.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dnd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.bfy.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.3.czg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.czi.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.czn.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.czp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dmu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dmw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dnb.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dnd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.czg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.czi.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.czn.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.czp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dmu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dmw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dnb.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dnd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |