Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.63 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&17\\4&15\end{bmatrix}$, $\begin{bmatrix}3&8\\16&23\end{bmatrix}$, $\begin{bmatrix}11&15\\18&1\end{bmatrix}$, $\begin{bmatrix}21&23\\2&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 36x $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{120x^{2}y^{6}+945648x^{2}y^{4}z^{2}-83980800x^{2}y^{2}z^{4}+3930301440x^{2}z^{6}+5304xy^{6}z+2643840xy^{4}z^{3}-213311232xy^{2}z^{5}+y^{8}+107200y^{6}z^{2}+9953280y^{4}z^{4}+201553920y^{2}z^{6}+2176782336z^{8}}{24x^{2}y^{6}+22032x^{2}y^{4}z^{2}-16796160x^{2}y^{2}z^{4}-3930301440x^{2}z^{6}+72xy^{6}z+528768xy^{4}z^{3}+213311232xy^{2}z^{5}-y^{8}-5184y^{6}z^{2}-2985984y^{4}z^{4}+40310784y^{2}z^{6}-2176782336z^{8}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.y.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.12.0.m.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.bz.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 |
|---|---|---|---|---|---|---|
| 24.48.1.ka.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.kb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.kc.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.kd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lo.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lp.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lr.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.oy.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.ga.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 120.48.1.bxe.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.xc.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.rik.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.gfu.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bxc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxe.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxs.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxt.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.oc.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.bxc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxe.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxf.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxs.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxt.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.mg.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bxe.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxh.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxu.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |