Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $1 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $6$ | Cusp orbits | $1$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 6A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.6.1.3 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}4&1\\17&13\end{bmatrix}$, $\begin{bmatrix}5&22\\23&5\end{bmatrix}$, $\begin{bmatrix}7&18\\21&5\end{bmatrix}$, $\begin{bmatrix}13&6\\15&1\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $48$ |
| Cyclic 24-torsion field degree: | $384$ |
| Full 24-torsion field degree: | $12288$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.e |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 8 $ |
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^3\cdot3^3\,\frac{y^{2}-8z^{2}}{z^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.2.0.b.1 | $24$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.12.1.bf.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.12.1.bg.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.12.1.bi.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.12.1.bj.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.12.1.br.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.12.1.bs.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.12.1.bu.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.12.1.bv.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.18.1.f.1 | $24$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
| 24.24.2.d.1 | $24$ | $4$ | $4$ | $2$ | $1$ | $1$ |
| 72.18.2.d.1 | $72$ | $3$ | $3$ | $2$ | $?$ | not computed |
| 72.54.4.l.1 | $72$ | $9$ | $9$ | $4$ | $?$ | not computed |
| 120.12.1.bf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.30.3.d.1 | $120$ | $5$ | $5$ | $3$ | $?$ | not computed |
| 120.36.3.d.1 | $120$ | $6$ | $6$ | $3$ | $?$ | not computed |
| 120.60.5.cx.1 | $120$ | $10$ | $10$ | $5$ | $?$ | not computed |
| 168.12.1.bf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.br.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bs.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.4.d.1 | $168$ | $8$ | $8$ | $4$ | $?$ | not computed |
| 168.126.10.d.1 | $168$ | $21$ | $21$ | $10$ | $?$ | not computed |
| 168.168.13.d.1 | $168$ | $28$ | $28$ | $13$ | $?$ | not computed |
| 264.12.1.bf.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bg.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bi.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.br.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bs.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.6.d.1 | $264$ | $12$ | $12$ | $6$ | $?$ | not computed |
| 312.12.1.bf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bi.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.br.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bs.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bu.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.84.7.d.1 | $312$ | $14$ | $14$ | $7$ | $?$ | not computed |