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: | 24.48.1.540 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&6\\18&23\end{bmatrix}$, $\begin{bmatrix}13&13\\6&17\end{bmatrix}$, $\begin{bmatrix}17&10\\12&1\end{bmatrix}$, $\begin{bmatrix}23&9\\12&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.et.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.d |
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.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-6.a.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.bw.1.9 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.cf.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.er.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.es.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.je.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.jf.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.jh.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ji.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.144.3-24.ug.1.3 | $24$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 72.144.3-72.cp.1.14 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.144.5-72.bd.1.3 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.144.5-72.bh.1.11 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.1-120.bzo.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzp.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzr.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzs.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.caa.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.cab.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.cad.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.cae.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.240.9-120.xn.1.5 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.288.9-120.rvh.1.37 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.480.17-120.gij.1.25 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.96.1-168.bzm.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzn.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzp.1.11 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzq.1.11 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzy.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzz.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cab.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cac.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.384.13-168.pf.1.2 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.96.1-264.bzm.1.7 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzn.1.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzp.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzq.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzy.1.9 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzz.1.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.cab.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.cac.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzo.1.12 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzp.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzr.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzs.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.caa.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cab.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cad.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cae.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |