Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 24.96.1.596 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&2\\20&3\end{bmatrix}$, $\begin{bmatrix}5&14\\0&19\end{bmatrix}$, $\begin{bmatrix}21&2\\4&5\end{bmatrix}$, $\begin{bmatrix}23&8\\20&15\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.bj.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 396x - 3024 $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^2}\cdot\frac{144x^{2}y^{14}+550522656x^{2}y^{12}z^{2}+283165527573504x^{2}y^{10}z^{4}+35589093251396649984x^{2}y^{8}z^{6}+1435852678252338286166016x^{2}y^{6}z^{8}+23600740106397378551611981824x^{2}y^{4}z^{10}+166825072115401190726981684035584x^{2}y^{2}z^{12}+421596930836809960564255254978232320x^{2}z^{14}+37296xy^{14}z+51202720512xy^{12}z^{3}+17060703549398784xy^{10}z^{5}+1487230108714945769472xy^{8}z^{7}+47478377603511577592856576xy^{6}z^{9}+665298645524533817351278166016xy^{4}z^{11}+4187262207135015339241996903514112xy^{2}z^{13}+9684318754352320554987748010250731520xz^{15}+y^{16}+4344192y^{14}z^{2}+4058589950208y^{12}z^{4}+792268224762138624y^{10}z^{6}+43843111579283454738432y^{8}z^{8}+944128051414930556653142016y^{6}z^{10}+9065075431123430510805612232704y^{4}z^{12}+38280102371311259261797843561611264y^{2}z^{14}+55501867011727212343338600744464941056z^{16}}{zy^{4}(10332x^{2}y^{8}z+519436800x^{2}y^{6}z^{3}+3852738452736x^{2}y^{4}z^{5}+1451188224x^{2}y^{2}z^{7}+78364164096x^{2}z^{9}+xy^{10}+516240xy^{8}z^{2}+15177570048xy^{6}z^{4}+88499423895552xy^{4}z^{6}-15237476352xy^{2}z^{8}-940369969152xz^{10}+144y^{10}z+16907616y^{8}z^{3}+217503553536y^{6}z^{5}+507198870484992y^{4}z^{7}-417942208512y^{2}z^{9}-19747769352192z^{11})}$ |
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.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-8.e.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.i.2.10 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.i.2.26 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.d.1.5 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1-24.d.1.10 | $24$ | $2$ | $2$ | $1$ | $1$ | 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.r.1.7 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.w.1.8 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.bn.1.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.bp.1.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.bw.1.6 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.by.1.8 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.cg.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.ch.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.288.9-24.hf.2.4 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.ei.2.8 | $24$ | $4$ | $4$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 120.192.1-120.hi.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.hk.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.hy.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ia.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ju.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.jw.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.kk.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.km.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.480.17-120.dr.2.22 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.hi.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.hk.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.hy.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ia.1.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ju.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.jw.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.kk.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.km.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.hi.1.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.hk.1.16 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.hy.1.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ia.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ju.1.14 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.jw.1.16 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.kk.1.14 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.km.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.hi.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.hk.1.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.hy.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ia.1.13 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ju.1.14 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.jw.1.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.kk.1.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.km.1.14 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |