Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| 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: | 24.48.1.57 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&11\\0&11\end{bmatrix}$, $\begin{bmatrix}17&12\\0&13\end{bmatrix}$, $\begin{bmatrix}19&0\\0&13\end{bmatrix}$, $\begin{bmatrix}23&17\\6&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.er.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: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 876x + 9520 $ |
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 |
|---|
| $(0:1:0)$, $(-34:0:1)$, $(14:0:1)$, $(20:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot3^6}\cdot\frac{24x^{2}y^{6}-392688x^{2}y^{4}z^{2}+1715447808x^{2}y^{2}z^{4}-2481592329216x^{2}z^{6}-1176xy^{6}z+13374720xy^{4}z^{3}-58582206720xy^{2}z^{5}+85595676880896xz^{7}-y^{8}+19968y^{6}z^{2}-146437632y^{4}z^{4}+524385073152y^{2}z^{6}-720863480254464z^{8}}{z^{4}y^{2}(48x^{2}-1632xz-y^{2}+13440z^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-6.a.1.10 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.by.1.20 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ch.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.eo.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ep.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ie.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.if.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.in.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.io.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.144.3-24.qm.1.4 | $24$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 72.144.3-72.cn.1.7 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.144.5-72.bb.1.10 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.144.5-72.bf.1.12 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.1-120.byq.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.byr.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.byt.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.byu.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzc.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzd.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzf.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzg.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.240.9-120.xl.1.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.288.9-120.rvf.1.17 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.480.17-120.gih.1.32 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.96.1-168.byo.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byp.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byr.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bys.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bza.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzb.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzd.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bze.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.384.13-168.pd.1.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.96.1-264.byo.1.11 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.byp.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.byr.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bys.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bza.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzb.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bzd.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.bze.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.byq.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.byr.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.byt.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.byu.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzc.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzd.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzf.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzg.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |