Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| 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: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.1921 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&6\\18&19\end{bmatrix}$, $\begin{bmatrix}5&16\\18&7\end{bmatrix}$, $\begin{bmatrix}5&19\\18&19\end{bmatrix}$, $\begin{bmatrix}7&4\\18&1\end{bmatrix}$, $\begin{bmatrix}17&14\\6&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.343434 |
| Contains $-I$: | no $\quad$ (see 24.48.1.jj.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{4}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x y + z^{2} $ |
| $=$ | $6 x^{2} + 54 y^{2} - 10 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + 2 x^{2} y^{2} - 20 x^{2} z^{2} + 12 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(2z-w)(2z+w)(559104y^{2}z^{8}+26112y^{2}z^{6}w^{2}-21888y^{2}z^{4}w^{4}+43680y^{2}z^{2}w^{6}-4368y^{2}w^{8}-10240z^{10}-3072z^{8}w^{2}+5184z^{6}w^{4}-7856z^{4}w^{6}+1620z^{2}w^{8}-81w^{10})}{w^{2}z^{4}(48y^{2}z^{4}+12y^{2}z^{2}w^{2}-6y^{2}w^{4}-8z^{6}-z^{4}w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.jj.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}+2X^{2}Y^{2}-20X^{2}Z^{2}+12Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.l.1.10 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.cb.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.cb.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.cd.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.cd.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-12.l.1.24 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.192.3-24.hg.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.hh.1.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.hu.1.2 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.hv.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.hw.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.hx.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ia.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ib.1.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.5-24.bz.1.16 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.cb.1.16 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.gb.1.16 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.192.5-24.gd.1.16 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.288.5-24.iy.1.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 72.288.5-72.cd.1.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.fm.1.10 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.fp.1.6 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.3-120.um.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.un.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.uq.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ur.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.us.1.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ut.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.uw.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ux.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.5-120.he.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.hf.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.mk.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ml.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.480.17-120.fmf.1.24 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.3-168.ry.1.28 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.rz.1.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.sc.1.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.sd.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.se.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.sf.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.si.1.30 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.sj.1.28 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.5-168.io.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.ip.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.nk.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.nl.1.32 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.3-264.ry.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.rz.1.28 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.sc.1.26 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.sd.1.29 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.se.1.29 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.sf.1.26 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.si.1.28 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.sj.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.5-264.fm.1.32 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.fn.1.32 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.iw.1.32 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.ix.1.32 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.3-312.um.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.un.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.uq.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ur.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.us.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ut.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.uw.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ux.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.5-312.fu.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.fv.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.je.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.jf.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |