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: | 24.72.1.11 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&7\\2&9\end{bmatrix}$, $\begin{bmatrix}3&13\\8&21\end{bmatrix}$, $\begin{bmatrix}11&9\\18&7\end{bmatrix}$, $\begin{bmatrix}23&18\\0&5\end{bmatrix}$ |
| 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 $ | $=$ | $ y^{2} - y w + z^{2} + 2 z w + w^{2} $ |
| $=$ | $6 x^{2} + 3 y^{2} - 4 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 49 x^{4} - 28 x^{3} z + 84 x^{2} y^{2} + 6 x^{2} z^{2} - 24 x y^{2} z - 4 x z^{3} + 36 y^{4} - 12 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{797508yz^{17}+12694347yz^{16}w+84267648yz^{15}w^{2}+323165808yz^{14}w^{3}+821155320yz^{13}w^{4}+1487441718yz^{12}w^{5}+2003069088yz^{11}w^{6}+2053665900yz^{10}w^{7}+1621701324yz^{9}w^{8}+988205913yz^{8}w^{9}+461311200yz^{7}w^{10}+162047952yz^{6}w^{11}+41456772yz^{5}w^{12}+7289271yz^{4}w^{13}+787320yz^{3}w^{14}+39366yz^{2}w^{15}+3243581z^{18}+43864506z^{17}w+275592177z^{16}w^{2}+1079295264z^{15}w^{3}+2974252878z^{14}w^{4}+6151206852z^{13}w^{5}+9912817692z^{12}w^{6}+12731211288z^{11}w^{7}+13202693415z^{10}w^{8}+11127149550z^{9}w^{9}+7630503507z^{8}w^{10}+4242161808z^{7}w^{11}+1895235975z^{6}w^{12}+670232394z^{5}w^{13}+183209364z^{4}w^{14}+37292724z^{3}w^{15}+5314410z^{2}w^{16}+472392zw^{17}+19683w^{18}}{z^{6}(z+w)^{6}(2z+w)^{3}(117yz^{2}+54yzw-44z^{3}-243z^{2}w-162zw^{2}-27w^{3})}$ |
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.cn.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.36.0.a.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.ga.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.gi.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.hc.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.hk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ig.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.im.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ji.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.jo.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.7.dl.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.en.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.13.ds.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.ctv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ctz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cux.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cvb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dlz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dmd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dnb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dnf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bjv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bjz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bkx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.blb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bup.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.but.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bvr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bvv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bjv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bjz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bkx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.blb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bup.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.but.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bvr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bvv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bjv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bjz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bkx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.blb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bup.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.but.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bvr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bvv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |