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 | $2^{2}\cdot4^{2}$ | ||
| 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: | 24.72.1.40 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\6&7\end{bmatrix}$, $\begin{bmatrix}1&18\\18&17\end{bmatrix}$, $\begin{bmatrix}7&15\\12&19\end{bmatrix}$, $\begin{bmatrix}21&20\\14&3\end{bmatrix}$ |
| 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 $ | $=$ | $ 3 x^{2} + y w - w^{2} $ |
| $=$ | $3 y^{2} - 6 z^{2} - 4 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 6 x^{2} z^{2} - 2 y^{2} z^{2} - 3 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{354294yz^{16}w+4802652yz^{14}w^{3}+21218274yz^{12}w^{5}+45139680yz^{10}w^{7}+53274024yz^{8}w^{9}+36716760yz^{6}w^{11}+14680872yz^{4}w^{13}+3152064yz^{2}w^{15}+280904yw^{17}-59049z^{18}-2243862z^{16}w^{2}-16612452z^{14}w^{4}-52562358z^{12}w^{6}-89258760z^{10}w^{8}-89249040z^{8}w^{10}-54155628z^{6}w^{12}-19590552z^{4}w^{14}-3882960z^{2}w^{16}-324360w^{18}}{w^{6}(3z^{2}+2w^{2})(29160yz^{8}w+244512yz^{6}w^{3}+563058yz^{4}w^{5}+485370yz^{2}w^{7}+140452yw^{9}-5832z^{10}-148716z^{8}w^{2}-646542z^{6}w^{4}-1024893z^{4}w^{6}-682092z^{2}w^{8}-162180w^{10})}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.0.c.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.b.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.l.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ca.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.eb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.eh.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.hi.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.hj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.hp.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.hr.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 72.216.7.br.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.cd.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.13.x.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.bbr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bbs.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bby.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bbz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bhf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bhg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bhm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.bhn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ot.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ou.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.pa.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.pb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.sd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.se.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.sk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.sl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ot.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ou.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.pa.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.pb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.sd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.se.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.sk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.sl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ot.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ou.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.pa.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.pb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.sd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.se.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.sk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.sl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |