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: | 24.72.1.80 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&10\\20&19\end{bmatrix}$, $\begin{bmatrix}11&10\\4&1\end{bmatrix}$, $\begin{bmatrix}15&13\\2&3\end{bmatrix}$, $\begin{bmatrix}23&3\\18&11\end{bmatrix}$ |
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 $ | $=$ | $ x^{2} + 2 y z $ |
$=$ | $4 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(3z^{6}-3z^{4}w^{2}-3z^{2}w^{4}-w^{6})^{3}}{z^{6}(z^{2}+w^{2})^{6}}$ |
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.m.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.36.0.ch.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.36.1.gn.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.ew.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.ue.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.wj.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.xb.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
24.144.9.dge.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.dgh.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.dgm.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.dgp.1 | $24$ | $2$ | $2$ | $9$ | $5$ | $1^{8}$ |
72.216.13.ni.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.144.9.bgcx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgcz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgdn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgdp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgfj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgfl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgfz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bggb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbzr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbzt.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcah.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcaj.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bccd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bccf.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcct.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bccv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcfr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcft.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcgh.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcgj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcid.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcif.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcit.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bciv.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbzz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcab.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcap.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcar.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bccl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bccn.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcdb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcdd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |