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: | $12$ (none of which are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| 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: | 6F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.12 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&9\\12&7\end{bmatrix}$, $\begin{bmatrix}7&3\\6&23\end{bmatrix}$, $\begin{bmatrix}15&10\\22&3\end{bmatrix}$, $\begin{bmatrix}15&19\\2&21\end{bmatrix}$ |
| 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: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x y + y^{2} - 2 y z + z^{2} $ |
| $=$ | $3 x^{2} - 4 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 49 x^{4} - 84 x^{3} y + 66 x^{2} y^{2} + 28 x^{2} z^{2} - 36 x y^{3} - 24 x y z^{2} + 9 y^{4} + \cdots + 4 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{280904xz^{17}+1050688xz^{15}w^{2}+1631208xz^{13}w^{4}+1359880xz^{11}w^{6}+657704xz^{9}w^{8}+185760xz^{7}w^{10}+29106xz^{5}w^{12}+2196xz^{3}w^{14}+54xzw^{16}-324360z^{18}-1294320z^{16}w^{2}-2176728z^{14}w^{4}-2005764z^{12}w^{6}-1101840z^{10}w^{8}-367320z^{8}w^{10}-72102z^{6}w^{12}-7596z^{4}w^{14}-342z^{2}w^{16}-3w^{18}}{z^{6}(2z^{2}+w^{2})(140452xz^{9}+161790xz^{7}w^{2}+62562xz^{5}w^{4}+9056xz^{3}w^{6}+360xzw^{8}-162180z^{10}-227364z^{8}w^{2}-113877z^{6}w^{4}-23946z^{4}w^{6}-1836z^{2}w^{8}-24w^{10})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.by.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.36.0.a.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.bd.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bf.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bq.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.br.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.by.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bz.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.cb.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.cd.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.7.m.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.o.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.13.f.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.jq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.jr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.jy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.jz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.mk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ml.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ms.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.mt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fa.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.gg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.gh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fa.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.gg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.gh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fa.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.gg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.gh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |