Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6^{2}$ | Cusp orbits | $1^{2}$ | ||
| 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: | 6B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.12.1.4 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}0&7\\23&12\end{bmatrix}$, $\begin{bmatrix}0&13\\11&3\end{bmatrix}$, $\begin{bmatrix}12&23\\5&0\end{bmatrix}$, $\begin{bmatrix}14&21\\3&2\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $24$ |
| Cyclic 24-torsion field degree: | $192$ |
| Full 24-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 540x + 4752 $ |
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 |
|---|
| $(12:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3}\cdot\frac{36x^{2}y^{2}-509328x^{2}z^{2}-1188xy^{2}z+13296960xz^{3}-y^{4}+22896y^{2}z^{2}-84960576z^{4}}{z^{2}(36x^{2}-972xz-y^{2}+6480z^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.2.0.a.1 | $8$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.6.0.l.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.6.1.a.1 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.24.1.bx.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.by.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.ca.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.cb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.g.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.48.3.t.1 | $24$ | $4$ | $4$ | $3$ | $1$ | $1^{2}$ |
| 72.36.2.a.1 | $72$ | $3$ | $3$ | $2$ | $?$ | not computed |
| 72.36.3.v.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 120.24.1.bx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.by.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.ca.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.cb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.60.5.n.1 | $120$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 120.72.5.kb.1 | $120$ | $6$ | $6$ | $5$ | $?$ | not computed |
| 120.120.9.gz.1 | $120$ | $10$ | $10$ | $9$ | $?$ | not computed |
| 168.24.1.bx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.by.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.ca.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.cb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.7.n.1 | $168$ | $8$ | $8$ | $7$ | $?$ | not computed |
| 168.252.19.cj.1 | $168$ | $21$ | $21$ | $19$ | $?$ | not computed |
| 264.24.1.bx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.by.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.ca.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.cb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.144.11.n.1 | $264$ | $12$ | $12$ | $11$ | $?$ | not computed |
| 312.24.1.bx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.by.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.ca.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.cb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.168.13.bl.1 | $312$ | $14$ | $14$ | $13$ | $?$ | not computed |