Invariants
Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $2\cdot4$ | ||
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: | 12K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.17 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&22\\20&19\end{bmatrix}$, $\begin{bmatrix}3&20\\16&9\end{bmatrix}$, $\begin{bmatrix}13&19\\8&7\end{bmatrix}$, $\begin{bmatrix}23&5\\22&5\end{bmatrix}$ |
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: | $2048$ |
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 z - 3 z^{2} + 2 w^{2} $ |
$=$ | $4 x^{2} + 4 x y + x z + 4 y^{2} - 4 y z + 4 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 2 x^{2} y^{2} - 3 x^{2} z^{2} - 2 x y z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^4\cdot3^2\,\frac{1152xy^{6}w^{2}-5184xy^{4}w^{4}+576xy^{2}w^{6}+2064xw^{8}-768y^{9}+3456y^{7}w^{2}-4608y^{5}w^{4}-2368y^{3}w^{6}+19683yz^{8}-13122yz^{6}w^{2}-37908yz^{4}w^{4}+23832yz^{2}w^{6}-2112yw^{8}-324z^{9}+68202z^{7}w^{2}-78624z^{5}w^{4}+28704z^{3}w^{6}-4016zw^{8}}{z^{3}(3z^{2}-2w^{2})^{3}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
$X_{\mathrm{ns}}(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
8.6.0.d.1 | $8$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
6.18.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.0.i.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.d.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 |
---|---|---|---|---|---|---|
24.72.3.d.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.do.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.hj.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.ho.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.kh.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.kj.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.kv.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.kx.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.bl.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.108.7.bn.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.bh.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.bid.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bif.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bir.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bit.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bkh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bkj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bkv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bkx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.cf.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.ex.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.bfv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bfx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bgj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bgl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bhz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bib.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bin.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bip.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.sw.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.bfv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bfx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bgj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bgl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bhz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bib.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bin.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bip.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bfv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bfx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bgj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bgl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bhz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bib.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bin.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bip.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |