Invariants
Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
Cusps: | $2$ (none of which are rational) | Cusp widths | $12^{2}$ | Cusp orbits | $2$ | ||
Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12G1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.122 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&20\\20&11\end{bmatrix}$, $\begin{bmatrix}13&3\\0&23\end{bmatrix}$, $\begin{bmatrix}15&17\\4&21\end{bmatrix}$, $\begin{bmatrix}20&9\\9&19\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 24-isogeny field degree: | $48$ |
Cyclic 24-torsion field degree: | $384$ |
Full 24-torsion field degree: | $3072$ |
Jacobian
Conductor: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x y - z^{2} $ |
$=$ | $216 x^{2} + 2 x y - 72 x z + 6 y^{2} - 12 y z + 5 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 6 x^{3} z + 6 x^{2} y^{2} + x^{2} z^{2} - 2 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^7\cdot3^5\,\frac{2160xz^{5}-4656xz^{3}w^{2}-240y^{2}z^{4}+540y^{2}z^{2}w^{2}-9y^{2}w^{4}+120yz^{5}-36yzw^{4}+156z^{6}+554z^{4}w^{2}}{38880xz^{5}-6912xz^{3}w^{2}+216xzw^{4}-4320y^{2}z^{4}+360y^{2}z^{2}w^{2}-6y^{2}w^{4}+2160yz^{5}-288yz^{3}w^{2}+12yzw^{4}+2808z^{6}-900z^{4}w^{2}+54z^{2}w^{4}-w^{6}}$ |
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 |
---|---|---|---|---|---|---|
12.12.0.q.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.6.0.f.1 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.48.3.j.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.48.3.l.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
24.48.3.p.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.48.3.r.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
24.48.3.bh.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.48.3.bj.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
24.48.3.bn.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.48.3.bp.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
24.72.3.ye.1 | $24$ | $3$ | $3$ | $3$ | $2$ | $1^{2}$ |
24.96.5.ix.1 | $24$ | $4$ | $4$ | $5$ | $3$ | $1^{4}$ |
72.72.3.cq.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
72.216.15.zl.1 | $72$ | $9$ | $9$ | $15$ | $?$ | not computed |
120.48.3.dl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.48.3.dn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.48.3.dr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.48.3.dt.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.48.3.dx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.48.3.dz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.48.3.ed.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.48.3.ef.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.120.9.bxd.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
120.144.9.bgzn.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
120.240.17.lqp.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
168.48.3.cr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.48.3.ct.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.48.3.cx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.48.3.cz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.48.3.dd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.48.3.df.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.48.3.dj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.48.3.dl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.15.ef.1 | $168$ | $8$ | $8$ | $15$ | $?$ | not computed |
264.48.3.cr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.48.3.ct.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.48.3.cx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.48.3.cz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.48.3.dd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.48.3.df.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.48.3.dj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.48.3.dl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.288.23.ef.1 | $264$ | $12$ | $12$ | $23$ | $?$ | not computed |
312.48.3.cr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.48.3.ct.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.48.3.cx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.48.3.cz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.48.3.dd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.48.3.df.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.48.3.dj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.48.3.dl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |