Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| 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 | $2^{4}\cdot4^{2}$ | ||
| 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: | 24.192.1.2696 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&15\\8&17\end{bmatrix}$, $\begin{bmatrix}1&15\\20&7\end{bmatrix}$, $\begin{bmatrix}5&9\\12&23\end{bmatrix}$, $\begin{bmatrix}23&0\\8&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2\times D_4):D_{12}$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.dl.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: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} + 4 x y - 2 y^{2} + z^{2} $ |
| $=$ | $6 x^{2} - 10 x y + 2 y^{2} - z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 6 x^{2} y^{2} - 4 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3}\cdot\frac{(3z^{2}-2w^{2})^{3}(9552816y^{2}z^{16}-50948352y^{2}z^{14}w^{2}+112510944y^{2}z^{12}w^{4}-133031808y^{2}z^{10}w^{6}+87391872y^{2}z^{8}w^{8}-26735616y^{2}z^{6}w^{10}+1128960y^{2}z^{4}w^{12}-104448y^{2}z^{2}w^{14}-186368y^{2}w^{16}+6383853z^{18}-44671662z^{16}w^{2}+135576504z^{14}w^{4}-231417648z^{12}w^{6}+237536064z^{10}w^{8}-141730560z^{8}w^{10}+41014656z^{6}w^{12}-2436864z^{4}w^{14}+86784z^{2}w^{16}-10752w^{18})}{w^{8}z^{2}(3z^{2}-4w^{2})(486y^{2}z^{10}-1620y^{2}z^{8}w^{2}+1836y^{2}z^{6}w^{4}-792y^{2}z^{4}w^{6}+48y^{2}z^{2}w^{8}+32y^{2}w^{10}-243z^{12}+648z^{10}w^{2}-567z^{8}w^{4}+288z^{6}w^{6}-75z^{4}w^{8}-60z^{2}w^{10}-48w^{12})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.dl.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+6X^{2}Y^{2}-4X^{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 |
|---|---|---|---|---|---|---|
| 24.96.0-24.bs.2.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bs.2.21 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.3.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.3.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.iu.1.18 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.iu.1.25 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.cy.4.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.di.3.7 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.ei.1.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.el.1.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.ew.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.ex.2.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fs.1.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fw.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.9-24.v.2.2 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.bah.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.baj.3.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bax.3.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.baz.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bct.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bcv.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdj.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdl.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bah.3.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.baj.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bax.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.baz.2.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bct.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bcv.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdj.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdl.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bah.4.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.baj.2.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bax.1.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.baz.1.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bct.1.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bcv.2.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdj.1.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdl.1.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bah.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.baj.2.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bax.3.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.baz.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bct.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bcv.3.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdj.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdl.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |