Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | 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 | $3^{8}\cdot12^{4}$ | 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: | 12S1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.37 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&4\\2&15\end{bmatrix}$, $\begin{bmatrix}13&21\\6&11\end{bmatrix}$, $\begin{bmatrix}15&20\\20&9\end{bmatrix}$, $\begin{bmatrix}21&11\\10&15\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| 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 z - x w - z w + w^{2} $ |
| $=$ | $x^{2} + 4 x z + 2 x w + 6 y^{2} - z^{2} + 2 z w - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 8 x^{3} z - 6 x^{2} y^{2} + 12 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{16777215xz^{17}-16777197xz^{16}w-100663452xz^{15}w^{2}+184550832xz^{14}w^{3}+16763463xz^{13}w^{4}-251571501xz^{12}w^{5}+213366474xz^{11}w^{6}-97219602xz^{10}w^{7}+67607460xz^{9}w^{8}-43216092xz^{8}w^{9}+11340324xz^{7}w^{10}-2169504xz^{6}w^{11}+1493721xz^{5}w^{12}-124659xz^{4}w^{13}-196830xz^{3}w^{14}+39366xz^{2}w^{15}-z^{18}-16777197z^{17}w+33554277z^{16}w^{2}+67110300z^{15}w^{3}-234894600z^{14}w^{4}+167857245z^{13}w^{5}+100135872z^{12}w^{6}-210570030z^{11}w^{7}+146915613z^{10}w^{8}-101974734z^{9}w^{9}+78535899z^{8}w^{10}-40774428z^{7}w^{11}+16202754z^{6}w^{12}-8089713z^{5}w^{13}+3503574z^{4}w^{14}-931662z^{3}w^{15}+295245z^{2}w^{16}-118098zw^{17}+19683w^{18}}{w^{3}z^{3}(z-w)(xz^{10}-15xz^{9}w+99xz^{8}w^{2}-372xz^{7}w^{3}+846xz^{6}w^{4}-1071xz^{5}w^{5}+270xz^{4}w^{6}+1458xz^{3}w^{7}-1944xz^{2}w^{8}+729xzw^{9}+z^{11}-15z^{10}w+98z^{9}w^{2}-356z^{8}w^{3}+730z^{7}w^{4}-566z^{6}w^{5}-1215z^{5}w^{6}+4671z^{4}w^{7}-7722z^{3}w^{8}+7290z^{2}w^{9}-3645zw^{10}+729w^{11})}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.0.b.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.a.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.dn.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.o.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.by.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ct.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.cy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ic.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.im.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.iw.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ja.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.7.cw.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.du.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.13.da.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.cpx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cpy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cqe.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cqf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dib.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dic.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dii.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dij.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bfx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bfy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bge.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bgf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bqr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bqs.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bqy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bqz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bfx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bfy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bge.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bgf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bqr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bqs.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bqy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bqz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bfx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bfy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bge.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bgf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bqr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bqs.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bqy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bqz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |