Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $4^{2}$ | ||
| 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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.288 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&10\\2&13\end{bmatrix}$, $\begin{bmatrix}19&21\\16&5\end{bmatrix}$, $\begin{bmatrix}21&22\\20&9\end{bmatrix}$ |
| 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: | $1536$ |
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} - 9x $ |
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 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{120x^{2}y^{14}-1379970x^{2}y^{12}z^{2}+424900080x^{2}y^{10}z^{4}-26017638939x^{2}y^{8}z^{6}+643362474600x^{2}y^{6}z^{8}-7702995489345x^{2}y^{4}z^{10}+47559739229640x^{2}y^{2}z^{12}-128505439098855x^{2}z^{14}-5412xy^{14}z+12583440xy^{12}z^{3}-2085866559xy^{10}z^{5}+108913124880xy^{8}z^{7}-2304804347136xy^{6}z^{9}+21163489912440xy^{4}z^{11}-71412831316881xy^{2}z^{13}-y^{16}+115840y^{14}z^{2}-74009052y^{12}z^{4}+6457336200y^{10}z^{6}-174985098012y^{8}z^{8}+1822459992480y^{6}z^{10}-6578141622894y^{4}z^{12}+418414128120y^{2}z^{14}-282429536481z^{16}}{24x^{2}y^{14}-23166x^{2}y^{12}z^{2}+874800x^{2}y^{10}z^{4}-118865637x^{2}y^{8}z^{6}+29628898632x^{2}y^{6}z^{8}+1146118946625x^{2}y^{4}z^{10}+9511947845928x^{2}y^{2}z^{12}+128505439098855x^{2}z^{14}+36xy^{14}z+120528xy^{12}z^{3}-30449601xy^{10}z^{5}-751103280xy^{8}z^{7}+107750725632xy^{6}z^{9}+4232697982488xy^{4}z^{11}+71412831316881xy^{2}z^{13}+y^{16}-3456y^{14}z^{2}+1723356y^{12}z^{4}-105973272y^{10}z^{6}+4198383900y^{8}z^{8}+350342913312y^{6}z^{10}+6346722450798y^{4}z^{12}+83682825624y^{2}z^{14}+282429536481z^{16}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.0.bo.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.cm.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.ct.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.eo.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.dg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.ei.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 |
|---|---|---|---|---|---|---|
| 24.144.9.efs.1 | $24$ | $3$ | $3$ | $9$ | $4$ | $1^{8}$ |
| 24.192.9.qr.1 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
| 48.96.3.qn.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.qp.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.th.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.ti.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.tj.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.tk.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.ub.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.ud.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.5.ox.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 48.96.5.oy.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 120.240.17.fpk.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.288.17.cgzm.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 240.96.3.dyt.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dyv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dzj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dzk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dzl.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dzm.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.ead.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.eaf.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.clb.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.clc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |