Invariants
Level: | $24$ | $\SL_2$-level: | $6$ | 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 | $6^{6}$ | 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: | 6E1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.32 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&15\\12&1\end{bmatrix}$, $\begin{bmatrix}9&14\\10&15\end{bmatrix}$, $\begin{bmatrix}23&1\\16&19\end{bmatrix}$, $\begin{bmatrix}23&11\\2&13\end{bmatrix}$ |
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 $ | $=$ | $ x^{2} + x y - x z + y^{2} + y z $ |
$=$ | $3 x^{2} + 2 x z + 3 y^{2} - 2 y z + 16 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 7 x^{4} + 8 x^{3} z + 6 x^{2} y^{2} + 18 x^{2} z^{2} - 12 x y^{2} z + 8 x z^{3} + 6 y^{2} z^{2} + 7 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle y$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^3\,\frac{864xz^{6}w^{2}+432xz^{4}w^{4}+48xz^{2}w^{6}-xw^{8}-864yz^{6}w^{2}-432yz^{4}w^{4}-48yz^{2}w^{6}+yw^{8}-6912z^{9}-3456z^{7}w^{2}+168z^{3}w^{6}+16zw^{8}}{z^{6}(xw^{2}-yw^{2}-32z^{3}-4zw^{2})}$ |
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 |
---|---|---|---|---|---|---|
6.18.0.d.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.1.bp.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
24.18.0.c.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.e.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.lp.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.lt.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.mr.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.mv.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.uw.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.uz.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.vy.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.wb.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.ew.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.fb.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.ddy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dea.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.def.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.deh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.duu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.duw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dvb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dvd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.pl.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.ob.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.czu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.czw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dab.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dad.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dni.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dnk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dnp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dnr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.bgb.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.czu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.czw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dab.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dad.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dni.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dnk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dnp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dnr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.czu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.czw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dab.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dad.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dni.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dnk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dnp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dnr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |