Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $24$ | $\PSL_2$-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: | 24.24.1.1 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\6&19\end{bmatrix}$, $\begin{bmatrix}7&15\\9&8\end{bmatrix}$, $\begin{bmatrix}8&9\\3&22\end{bmatrix}$, $\begin{bmatrix}23&21\\15&10\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 24.48.1-24.bx.1.1, 24.48.1-24.bx.1.2, 24.48.1-24.bx.1.3, 24.48.1-24.bx.1.4, 120.48.1-24.bx.1.1, 120.48.1-24.bx.1.2, 120.48.1-24.bx.1.3, 120.48.1-24.bx.1.4, 168.48.1-24.bx.1.1, 168.48.1-24.bx.1.2, 168.48.1-24.bx.1.3, 168.48.1-24.bx.1.4, 264.48.1-24.bx.1.1, 264.48.1-24.bx.1.2, 264.48.1-24.bx.1.3, 264.48.1-24.bx.1.4, 312.48.1-24.bx.1.1, 312.48.1-24.bx.1.2, 312.48.1-24.bx.1.3, 312.48.1-24.bx.1.4 |
| Cyclic 24-isogeny field degree: | $12$ |
| Cyclic 24-torsion field degree: | $96$ |
| Full 24-torsion field degree: | $3072$ |
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} + 216 $ |
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 |
|---|
| $(0:1:0)$, $(-6:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3}\cdot\frac{(y^{2}-1944z^{2})^{3}(y^{2}-216z^{2})}{z^{2}y^{6}}$ |
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 | $8$ | $12$ | $12$ | $0$ | $0$ | 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 |
| 24.8.0.a.1 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.12.0.bx.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.n.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 |
|---|---|---|---|---|---|---|
| 24.72.1.c.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.96.5.dl.1 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 72.72.3.c.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.72.3.f.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.72.3.r.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.72.4.i.1 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 72.72.5.b.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.120.9.dt.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.ipd.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.brr.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.192.13.eb.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.288.21.eb.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |