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: | 24.72.1.344 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&4\\20&21\end{bmatrix}$, $\begin{bmatrix}11&1\\2&23\end{bmatrix}$, $\begin{bmatrix}11&20\\16&11\end{bmatrix}$, $\begin{bmatrix}23&2\\8&19\end{bmatrix}$ |
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 no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
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.bs.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.fb.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.rr.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.vx.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.wf.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
24.144.9.eig.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.eii.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.eiw.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.eiy.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
72.216.13.nz.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.144.9.bghu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bghw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgik.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgim.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgkg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgki.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgkw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgky.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bceo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bceq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcfe.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcfg.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcha.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bchc.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bchq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bchs.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcko.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bckq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcle.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bclg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcna.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcnc.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcnq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcns.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcew.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcey.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcfm.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcfo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bchi.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bchk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bchy.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcia.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |