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: | 24.24.1.17 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&11\\10&23\end{bmatrix}$, $\begin{bmatrix}7&9\\20&5\end{bmatrix}$, $\begin{bmatrix}17&11\\0&23\end{bmatrix}$, $\begin{bmatrix}23&1\\10&13\end{bmatrix}$ |
| 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} - 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 |
|---|
| $(0:1:0)$, $(-6:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{36x^{2}y^{6}+225747x^{2}y^{4}z^{2}+309889152x^{2}y^{2}z^{4}+121321662057x^{2}z^{6}+882xy^{6}z+3475872xy^{4}z^{3}+3968099361xy^{2}z^{5}+1393412438538xz^{7}+y^{8}+14256y^{6}z^{2}+29116260y^{4}z^{4}+19590962292y^{2}z^{6}+3992895328617z^{8}}{z^{2}(x^{2}y^{4}+12744x^{2}y^{2}z^{2}+13868496x^{2}z^{4}+36xy^{4}z+193104xy^{2}z^{3}+159283584xz^{5}+756y^{4}z^{2}+1492992y^{2}z^{4}+456435648z^{6})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.br.1 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.o.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.da.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ec.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.el.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.kd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.kn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.kz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ln.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.kt.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.el.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 120.48.1.bhp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bht.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bif.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bij.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bsj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bsn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bsz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.btd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.qp.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.oah.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.ewr.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bhn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bhr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bid.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bih.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bsh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bsl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bsx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.btb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.kn.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.bhn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bhr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bid.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bih.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bsh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bsl.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bsx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.btb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.ir.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bhp.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bht.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bif.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bij.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bsj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bsn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bsz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.btd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |