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: | 24.36.1.16 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&13\\2&3\end{bmatrix}$, $\begin{bmatrix}17&0\\18&17\end{bmatrix}$, $\begin{bmatrix}17&5\\4&23\end{bmatrix}$, $\begin{bmatrix}19&4\\20&13\end{bmatrix}$ |
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} + 4 x y + 2 w^{2} $ |
$=$ | $x^{2} - x y + 3 x z + 4 y^{2} + 3 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} + 6 x^{3} y + 6 x^{2} y^{2} + 3 x^{2} 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{17640xz^{8}+15276xz^{6}w^{2}-20658xz^{4}w^{4}+7745xz^{2}w^{6}-70xw^{8}+171648y^{2}z^{5}w^{2}+5376y^{2}z^{3}w^{4}-1440y^{2}zw^{6}+69120yz^{8}+34320yz^{6}w^{2}-85560yz^{4}w^{4}+4700yz^{2}w^{6}-50yw^{8}-9504z^{7}w^{2}-26472z^{5}w^{4}+28272z^{3}w^{6}-1210zw^{8}}{360xz^{8}-1356xz^{6}w^{2}-4674xz^{4}w^{4}+3155xz^{2}w^{6}-130xw^{8}-4608y^{2}z^{5}w^{2}-4992y^{2}z^{3}w^{4}+8640y^{2}zw^{6}-240yz^{6}w^{2}-1320yz^{4}w^{4}+9740yz^{2}w^{6}-1190yw^{8}-3456z^{7}w^{2}-4440z^{5}w^{4}+5736z^{3}w^{6}-2110zw^{8}}$ |
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.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.1.z.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
24.18.0.h.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.jd.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.jh.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.kf.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.kj.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.ra.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.rd.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.sc.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.sf.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.dg.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.108.7.eo.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.dt.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.chf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.chh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.chm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.cho.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.cpv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.cpx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.cqc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.cqe.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.jl.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.hl.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.cdz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ceb.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ceg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.cei.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.cmp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.cmr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.cmw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.cmy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.bab.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.cdz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ceb.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ceg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.cei.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.cmp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.cmr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.cmw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.cmy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.cdz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ceb.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ceg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.cei.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.cmp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.cmr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.cmw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.cmy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |