Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.234 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&14\\8&9\end{bmatrix}$, $\begin{bmatrix}11&22\\0&17\end{bmatrix}$, $\begin{bmatrix}13&22\\16&21\end{bmatrix}$, $\begin{bmatrix}15&20\\16&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.bv.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: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 9x $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{3^2}\cdot\frac{807570x^{2}y^{12}z^{2}-5785089579x^{2}y^{8}z^{6}-785817891855x^{2}y^{4}z^{10}-128505439098855x^{2}z^{14}+1548xy^{14}z+1059201279xy^{10}z^{5}+539595430704xy^{6}z^{9}+71412831316881xy^{2}z^{13}+y^{16}+146479428y^{12}z^{4}-37959767748y^{8}z^{8}-6278536444734y^{4}z^{12}+282429536481z^{16}}{zy^{4}(117x^{2}y^{8}z-3287061x^{2}y^{4}z^{5}+1219657095x^{2}z^{9}-xy^{10}+195372xy^{6}z^{4}-408678129xy^{2}z^{8}-5670y^{8}z^{3}+30823578y^{4}z^{7}-43046721z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.e.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.e.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-8.i.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.d.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1-24.d.1.9 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 24.192.1-24.ce.1.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.ce.2.2 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.cf.1.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.cf.2.2 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.cg.1.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.cg.2.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.ch.1.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.ch.2.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.288.9-24.jf.1.22 | $24$ | $3$ | $3$ | $9$ | $2$ | $1^{8}$ |
| 24.384.9-24.ez.1.18 | $24$ | $4$ | $4$ | $9$ | $3$ | $1^{8}$ |
| 48.192.3-48.bt.1.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.192.3-48.bt.2.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.192.3-48.cc.1.4 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.192.3-48.cc.2.6 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.192.3-48.cp.1.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.192.3-48.cp.2.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.192.3-48.cv.1.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.192.3-48.cv.2.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 120.192.1-120.la.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.la.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lb.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lb.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lc.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lc.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ld.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ld.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.480.17-120.ff.1.30 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.la.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.la.2.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lb.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lb.2.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lc.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lc.2.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ld.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ld.2.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-240.fr.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.fr.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ga.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ga.2.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.gh.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.gh.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.gt.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.gt.2.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.la.1.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.la.2.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.lb.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.lb.2.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.lc.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.lc.2.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ld.1.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ld.2.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.la.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.la.2.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lb.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lb.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lc.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lc.2.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ld.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ld.2.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |