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: | 24.36.1.141 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&4\\2&1\end{bmatrix}$, $\begin{bmatrix}3&10\\2&15\end{bmatrix}$, $\begin{bmatrix}13&23\\16&19\end{bmatrix}$, $\begin{bmatrix}17&19\\10&19\end{bmatrix}$ |
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: | $2048$ |
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} + 6 x y + 2 x z - 2 z^{2} $ |
$=$ | $2 x^{2} - 3 x y + x z + 24 y^{2} - z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 19 x^{4} + 14 x^{3} z + 6 x^{2} y^{2} - 3 x^{2} z^{2} - 4 x z^{3} + 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^3}{3}\cdot\frac{5160653174952xz^{8}-1659617480052xz^{6}w^{2}-3369163899402xz^{4}w^{4}+488697525999xz^{2}w^{6}-7205619565xw^{8}+40686470429184y^{2}z^{7}+47658493319616y^{2}z^{5}w^{2}-27563700682944y^{2}z^{3}w^{4}+1183914650448y^{2}zw^{6}-3158712093600yz^{8}-10836831810240yz^{6}w^{2}+2915017745400yz^{4}w^{4}+13864782600yz^{2}w^{6}-8938717390yw^{8}-2650692148992z^{9}+3726871563888z^{7}w^{2}+3032150148936z^{5}w^{4}-1251768676860z^{3}w^{6}+49329777102zw^{8}}{7963970949xz^{8}+2633354991xz^{6}w^{2}+169614900xz^{4}w^{4}+6775248xz^{2}w^{6}-411540xw^{8}+62787763008y^{2}z^{7}-12205099008y^{2}z^{5}w^{2}-1237219200y^{2}z^{3}w^{4}-76381824y^{2}zw^{6}-4874555700yz^{8}+5623538490yz^{6}w^{2}+102928320yz^{4}w^{4}-3292320yz^{2}w^{6}-4090574304z^{9}+27126846z^{7}w^{2}-413415072z^{5}w^{4}-44088208z^{3}w^{6}-3182576zw^{8}}$ |
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 |
---|---|---|---|---|---|---|
12.18.0.h.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.1.f.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
24.18.0.b.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.a.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.ck.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.cm.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.dc.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.de.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.ec.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.ee.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ek.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.em.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
72.108.7.h.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.l.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.ji.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.jk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.jq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.js.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.mu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.mw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.nc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ne.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.w.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.bq.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.iy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ja.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.jg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ji.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.mc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.me.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.mk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.mm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.qr.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.iy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ja.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.jg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ji.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.mc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.me.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.mk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.mm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.iy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ja.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.jg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ji.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.mc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.me.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.mk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.mm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |