Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $192$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| 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: | 24J1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.3126 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&12\\16&13\end{bmatrix}$, $\begin{bmatrix}11&21\\4&5\end{bmatrix}$, $\begin{bmatrix}13&12\\0&11\end{bmatrix}$, $\begin{bmatrix}23&6\\20&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $S_3\times D_4:D_4$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.ds.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - 2 x z + w^{2} $ |
| $=$ | $2 x^{2} - 4 x y + 2 y^{2} - 2 y z + z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} - 6 x^{3} y + 2 x^{2} y^{2} + 4 x^{2} z^{2} - 4 x y z^{2} + 4 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{728xz^{23}-95488xz^{21}w^{2}+4542720xz^{19}w^{4}-93649920xz^{17}w^{6}+809693184xz^{15}w^{8}-3694362624xz^{13}w^{10}+9898164224xz^{11}w^{12}-16171663360xz^{9}w^{14}+15944122368xz^{7}w^{16}-8900313088xz^{5}w^{18}+2399141888xz^{3}w^{20}-201326592xzw^{22}+z^{24}-376z^{22}w^{2}+47616z^{20}w^{4}-2247680z^{18}w^{6}+45712128z^{16}w^{8}-382537728z^{14}w^{10}+1666793472z^{12}w^{12}-4200595456z^{10}w^{14}+6319964160z^{8}w^{16}-5546442752z^{6}w^{18}+2587885568z^{4}w^{20}-503316480z^{2}w^{22}+16777216w^{24}}{w^{2}z^{12}(z^{2}-2w^{2})(486xz^{7}-1944xz^{5}w^{2}+2336xz^{3}w^{4}-768xzw^{6}-243z^{6}w^{2}+850z^{4}w^{4}-800z^{2}w^{6}+128w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.ds.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}-6X^{3}Y+2X^{2}Y^{2}+4X^{2}Z^{2}-4XYZ^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-24.bt.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bt.1.21 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.ix.1.15 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ix.1.22 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.384.5-24.dd.3.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.do.2.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.ep.1.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.es.1.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fl.3.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fq.4.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.ft.2.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fy.3.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.9-24.bb.2.3 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.bau.4.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.baw.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bbk.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bbm.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdg.4.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdi.4.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdw.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdy.3.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bau.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.baw.1.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bbk.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bbm.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdg.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdi.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdw.1.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdy.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bau.3.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.baw.2.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bbk.1.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bbm.1.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdg.3.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdi.2.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdw.1.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdy.2.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bau.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.baw.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bbk.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bbm.2.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdg.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdi.2.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdw.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdy.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |