Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $144$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{4}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.574 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&21\\21&14\end{bmatrix}$, $\begin{bmatrix}11&6\\3&19\end{bmatrix}$, $\begin{bmatrix}13&9\\18&17\end{bmatrix}$, $\begin{bmatrix}17&3\\3&10\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.1.b.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $12$ |
| Cyclic 24-torsion field degree: | $96$ |
| Full 24-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 27 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-3:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(y^{2}-243z^{2})^{3}(y^{2}-27z^{2})}{z^{2}y^{6}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.16.0-12.a.1.2 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.16.0-12.a.1.4 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.24.0-3.a.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-3.a.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.96.3-12.b.1.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.96.3-12.b.1.4 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.96.3-24.f.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.96.3-24.f.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.144.1-12.b.1.2 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.192.5-12.c.1.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 72.144.3-36.b.1.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.144.3-36.b.1.2 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.144.3-36.d.1.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.144.3-36.g.1.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.144.4-36.h.1.1 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 72.144.4-36.h.1.3 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 72.144.5-36.b.1.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.3-60.b.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3-60.b.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3-120.b.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3-120.b.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.240.9-60.b.1.7 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.288.9-60.b.1.4 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.480.17-60.bd.1.7 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.96.3-84.b.1.4 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-84.b.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.b.1.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.b.1.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.384.13-84.b.1.12 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.96.3-132.b.1.2 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3-132.b.1.7 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3-264.b.1.2 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3-264.b.1.16 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3-156.b.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3-156.b.1.7 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3-312.b.1.2 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3-312.b.1.16 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |