Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $64$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $16$ 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: | 24H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.94 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\12&13\end{bmatrix}$, $\begin{bmatrix}3&13\\8&21\end{bmatrix}$, $\begin{bmatrix}5&15\\6&7\end{bmatrix}$, $\begin{bmatrix}15&23\\4&21\end{bmatrix}$, $\begin{bmatrix}17&7\\10&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + z w $ |
| $=$ | $3 y^{2} + 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 3 y^{2} z^{2} - 9 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(4z^{3}-w^{3})^{3}(4z^{3}+w^{3})^{3}}{w^{6}z^{12}}$ |
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.s.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.cf.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gp.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.rm.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.rn.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ro.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.rp.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ta.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tc.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.td.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.uk.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.ul.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.um.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.un.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.va.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.vc.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vd.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.fy.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.uc.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bah.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bas.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.btp.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.btv.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.buf.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.bul.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 72.216.13.na.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.ksg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kus.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kut.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfzf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgch.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hsf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hur.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hus.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hut.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.huu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbvz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyl.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hsg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hus.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hut.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcbz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcel.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcen.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hsg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hus.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hut.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbwh.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyt.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |