Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.560 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\0&11\end{bmatrix}$, $\begin{bmatrix}11&13\\12&17\end{bmatrix}$, $\begin{bmatrix}17&3\\6&5\end{bmatrix}$, $\begin{bmatrix}17&4\\12&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.es.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 97x + 385 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(-11:0:1)$, $(5:0:1)$, $(7:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{8x^{2}y^{6}-4848x^{2}y^{4}z^{2}+784384x^{2}y^{2}z^{4}-42025984x^{2}z^{6}-136xy^{6}z+58272xy^{4}z^{3}-9451776xy^{2}z^{5}+511207424xz^{7}-y^{8}+784y^{6}z^{2}-219760y^{4}z^{4}+29704960y^{2}z^{6}-1522164736z^{8}}{z^{4}y^{2}(16x^{2}-192xz-y^{2}+560z^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-6.a.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-6.a.1.16 | $24$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.96.1-24.bx.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.cg.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.dp.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.dq.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.iy.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.iz.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.jb.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.jc.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.144.3-24.ua.1.5 | $24$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 72.144.3-72.co.1.6 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.144.5-72.bc.1.7 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.144.5-72.bg.1.3 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.1-120.bzi.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzj.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzl.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzm.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzu.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzv.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzx.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzy.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.240.9-120.xm.1.13 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.288.9-120.rvg.1.29 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.480.17-120.gii.1.29 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.96.1-168.bzg.1.11 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzh.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzj.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzk.1.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzs.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzt.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzv.1.11 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzw.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.384.13-168.pe.1.10 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.96.1-264.bzg.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzh.1.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzj.1.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzk.1.7 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzs.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzt.1.11 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzv.1.9 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzw.1.11 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzi.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzj.1.12 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzl.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzm.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzu.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzv.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzx.1.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzy.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |