Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-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 | $2^{4}$ | ||
| 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: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.782 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&0\\16&13\end{bmatrix}$, $\begin{bmatrix}11&2\\12&19\end{bmatrix}$, $\begin{bmatrix}15&4\\16&5\end{bmatrix}$, $\begin{bmatrix}19&14\\4&15\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.q.1 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x y + z^{2} $ |
| $=$ | $6 x^{2} + 12 y^{2} + 3 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 3 x^{2} y^{2} + 9 x^{2} z^{2} + 18 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{378y^{2}z^{10}-1134y^{2}z^{8}w^{2}+108y^{2}z^{6}w^{4}+108y^{2}z^{4}w^{6}-1134y^{2}z^{2}w^{8}+378y^{2}w^{10}+31z^{12}-60z^{10}w^{2}-48z^{8}w^{4}+64z^{6}w^{6}-255z^{4}w^{8}+192z^{2}w^{10}-32w^{12}}{w^{4}z^{4}(6y^{2}z^{2}+6y^{2}w^{2}+z^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.q.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-3X^{2}Y^{2}+9X^{2}Z^{2}+18Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.c.1.7 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.h.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.h.1.32 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.i.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.i.1.32 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-8.c.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.c.2.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.d.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.f.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.k.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.q.2.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.v.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.y.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.z.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.de.2.31 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.br.1.17 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.192.5-48.bd.2.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.192.5-48.be.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.192.5-48.bi.2.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.192.5-48.bl.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.192.5-48.bo.2.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.5-48.br.1.6 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.5-48.bu.2.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.5-48.bv.1.6 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 120.192.1-120.r.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.s.2.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.bd.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.bg.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.cc.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.cf.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.cn.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.co.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.480.17-120.bh.2.13 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.r.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.s.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.bd.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.bg.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.cc.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.cf.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.cn.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.co.2.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.5-240.dr.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ds.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ea.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ed.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.en.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.eq.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ey.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ez.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.1-264.r.2.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.s.1.14 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.bd.1.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.bg.1.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.cc.2.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.cf.1.11 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.cn.1.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.co.1.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.r.2.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.s.1.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.bd.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.bg.2.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.cc.2.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.cf.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.cn.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.co.2.16 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |