Invariants
Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
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: | 24.72.1.343 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&1\\14&1\end{bmatrix}$, $\begin{bmatrix}13&5\\10&19\end{bmatrix}$, $\begin{bmatrix}13&9\\18&17\end{bmatrix}$, $\begin{bmatrix}21&20\\20&9\end{bmatrix}$ |
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: | $2^{5}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} + z w $ |
$=$ | $6 y^{2} - 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{(16z^{6}+w^{6})^{3}}{w^{6}z^{12}}$ |
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 |
---|---|---|---|---|---|---|
24.36.0.bv.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.36.0.ch.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.36.1.gq.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 |
---|---|---|---|---|---|---|
24.144.9.bv.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.rq.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.yf.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.yr.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
24.144.9.eio.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.eiq.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.eje.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.ejg.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
72.216.13.nv.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.144.9.bgic.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgie.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgis.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgiu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgko.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgkq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgle.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bglg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcew.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcey.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcfm.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcfo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bchi.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bchk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bchy.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcia.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bckw.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcky.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bclm.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bclo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcni.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcnk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcny.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcoa.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcfe.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcfg.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcfu.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcfw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bchq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bchs.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcig.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcii.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |