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: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | 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: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.50 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&9\\8&23\end{bmatrix}$, $\begin{bmatrix}15&14\\2&17\end{bmatrix}$, $\begin{bmatrix}17&1\\0&19\end{bmatrix}$, $\begin{bmatrix}17&7\\16&19\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| 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} - 99x - 378 $ |
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 |
|---|
| $(-6:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{3^2}\cdot\frac{36x^{2}y^{6}-197073x^{2}y^{4}z^{2}+1330255872x^{2}y^{2}z^{4}-7764582533463x^{2}z^{6}-738xy^{6}z+11261592xy^{4}z^{3}-10883905119xy^{2}z^{5}-89178419095542xz^{7}+y^{8}-44064y^{6}z^{2}+235787760y^{4}z^{4}-557256356748y^{2}z^{6}-255545542837143z^{8}}{z(873x^{2}y^{4}z+2923776x^{2}y^{2}z^{3}+1997063424x^{2}z^{5}+xy^{6}+17226xy^{4}z^{2}+40310784xy^{2}z^{4}+22936836096xz^{6}+36y^{6}z+217161y^{4}z^{3}+250822656y^{2}z^{5}+65726733312z^{7})}$ |
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.r.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.12.0.o.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.bq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.dc.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.dj.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.do.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.jn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.jz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.kc.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ko.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.kl.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.ed.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 120.48.1.bgj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bgn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bgz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bhd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.brd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.brh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.brt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.brx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.qh.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.nzz.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.ewj.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bgh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bgl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bgx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bhb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.brb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.brf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.brr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.brv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.kf.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.bgh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bgl.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bgx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bhb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.brb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.brf.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.brr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.brv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.ij.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bgj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bgn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bgz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bhd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.brd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.brh.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.brt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.brx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |