Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| 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: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.180 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&10\\20&11\end{bmatrix}$, $\begin{bmatrix}13&0\\12&5\end{bmatrix}$, $\begin{bmatrix}17&20\\12&13\end{bmatrix}$, $\begin{bmatrix}19&2\\12&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.be.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
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 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^2}\cdot\frac{72x^{2}y^{14}-34407666x^{2}y^{12}z^{2}+2212230684168x^{2}y^{10}z^{4}-34754973878317041x^{2}y^{8}z^{6}+175274985138224888448x^{2}y^{6}z^{8}-360118715002401406122009x^{2}y^{4}z^{10}+318193573218157178358043068x^{2}y^{2}z^{12}-100516541203691949978889287705x^{2}z^{14}-9324xy^{14}z+1600085016xy^{12}z^{3}-66643373239839xy^{10}z^{5}+726186576520969614xy^{8}z^{7}-2897850195526829687064xy^{6}z^{9}+5075825847812910593805528xy^{4}z^{11}-3993284423003211344949719337xy^{2}z^{13}+1154460758489646977780788899690xz^{15}-y^{16}+543024y^{14}z^{2}-63415467972y^{12}z^{4}+1547398876488552y^{10}z^{6}-10703884662910999692y^{8}z^{8}+28812501569059160054112y^{6}z^{10}-34580518459790918391439866y^{4}z^{12}+18253375230460767393969461232y^{2}z^{14}-3308169067604971667727148577241z^{16}}{zy^{4}(2583x^{2}y^{8}z-16232400x^{2}y^{6}z^{3}+15049759581x^{2}y^{4}z^{5}-708588x^{2}y^{2}z^{7}+4782969x^{2}z^{9}+xy^{10}-64530xy^{8}z^{2}+237149532xy^{6}z^{4}-172850437296xy^{4}z^{6}-3720087xy^{2}z^{8}+28697814xz^{10}-72y^{10}z+1056726y^{8}z^{3}-1699246512y^{6}z^{5}+495311396958y^{4}z^{7}+51018336y^{2}z^{9}-301327047z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.e.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-8.e.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.h.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.h.1.32 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.c.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1-24.c.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.192.1-24.h.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.x.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bd.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bh.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bv.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bz.2.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.cb.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.cd.1.8 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.gv.2.32 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.dz.2.18 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.192.1-120.fx.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.gb.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.gn.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.gr.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ij.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.in.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.iz.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.jd.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.480.17-120.dh.2.9 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.fx.1.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.gb.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.gn.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.gr.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ij.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.in.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.iz.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.jd.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.fx.1.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.gb.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.gn.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.gr.1.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ij.1.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.in.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.iz.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.jd.1.14 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.fx.1.13 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.gb.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.gn.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.gr.1.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ij.1.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.in.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.iz.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.jd.1.14 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |