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: | 24.72.1.38 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\12&17\end{bmatrix}$, $\begin{bmatrix}13&9\\0&7\end{bmatrix}$, $\begin{bmatrix}15&10\\22&21\end{bmatrix}$, $\begin{bmatrix}17&3\\12&5\end{bmatrix}$ |
| 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 $ | $=$ | $ 3 x^{2} - x z - 3 x w - z w $ |
| $=$ | $4 x z + 6 y^{2} - z^{2} - 2 z w + 3 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} z + 2 x^{2} y^{2} - 6 x^{2} z^{2} + 4 x y^{2} z + 4 x z^{3} + 2 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^3}\cdot\frac{1259713xz^{17}+56687085xz^{16}w+997692588xz^{15}w^{2}+8979251436xz^{14}w^{3}+45712801332xz^{13}w^{4}+137138573124xz^{12}w^{5}+240135332292xz^{11}w^{6}+219109082724xz^{10}w^{7}+41054473350xz^{9}w^{8}-102675464082xz^{8}w^{9}-67112495244xz^{7}w^{10}+63154322676xz^{6}w^{11}+162895169556xz^{5}w^{12}+193480661988xz^{4}w^{13}+164668056732xz^{3}w^{14}+101762448444xz^{2}w^{15}+40679151345xzw^{16}+8135830269xw^{17}-419904z^{18}-21415103z^{17}w-438379737z^{16}w^{2}-4716361260z^{15}w^{3}-29522588400z^{14}w^{4}-112648241916z^{13}w^{5}-264887010000z^{12}w^{6}-369635872068z^{11}w^{7}-253782541800z^{10}w^{8}+23524321158z^{9}w^{9}+185045276142z^{8}w^{10}+137381750028z^{7}w^{11}+40185442656z^{6}w^{12}+9304469028z^{5}w^{13}+21083327352z^{4}w^{14}+24507933156z^{3}w^{15}+13430576952z^{2}w^{16}+3486784401zw^{17}+129140163w^{18}}{w^{3}z^{3}(xz^{11}+18xz^{10}w+63xz^{9}w^{2}+81xz^{8}w^{3}+729xz^{6}w^{5}-9477xz^{5}w^{6}+65610xz^{4}w^{7}+505197xz^{3}w^{8}+964467xz^{2}w^{9}+708588xzw^{10}+177147xw^{11}+z^{11}w+12z^{10}w^{2}+9z^{9}w^{3}+81z^{8}w^{4}-486z^{7}w^{5}+3159z^{6}w^{6}-9477z^{5}w^{7}-297432z^{4}w^{8}-1030077z^{3}w^{9}-1397493z^{2}w^{10}-826686zw^{11}-177147w^{12})}$ |
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.br.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.f.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.cb.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.do.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.dt.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.gv.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.gw.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.hg.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.hk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.7.bo.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.ca.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.13.u.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.bbd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bbe.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bbk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bbl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bgr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bgs.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bgy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bgz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.of.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.og.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.om.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.on.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.rp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.rq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.rw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.rx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.of.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.og.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.om.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.on.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.rp.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.rq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.rw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.rx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.of.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.og.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.om.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.on.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.rp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.rq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.rw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.rx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |