Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $192$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot6^{2}\cdot8\cdot24$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 24F2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.2.132 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&15\\20&5\end{bmatrix}$, $\begin{bmatrix}11&15\\20&19\end{bmatrix}$, $\begin{bmatrix}13&12\\22&11\end{bmatrix}$, $\begin{bmatrix}17&9\\18&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.135402 |
| Contains $-I$: | no $\quad$ (see 24.48.2.e.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $2$ |
| Newforms: | 192.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y w - z w^{2} $ |
| $=$ | $x y z - z^{2} w$ | |
| $=$ | $x y^{2} - y z w$ | |
| $=$ | $x^{2} y - x z w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + 9 x^{3} z^{2} - x^{2} y^{2} z - y^{2} z^{3} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{5} + 10x^{3} + 9x $ |
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.
| Embedded model |
|---|
| $(0:0:1:0)$, $(0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{729x^{2}z^{8}-270x^{2}z^{4}w^{4}+9x^{2}w^{8}+756xz^{6}w^{3}-36xz^{2}w^{7}-756y^{2}z^{6}w^{2}+324y^{2}z^{2}w^{6}-729yz^{9}+2430yz^{5}w^{4}-81yzw^{8}+8w^{10}}{w^{2}z^{6}(xw-y^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.2.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}-X^{2}Y^{2}Z+9X^{3}Z^{2}-Y^{2}Z^{3} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.48.2.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y^{4}zw+y^{2}z^{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle yz$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.f.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.f.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.3-24.bj.2.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.192.3-24.cs.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.192.3-24.dj.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.192.3-24.do.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.192.3-24.fh.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.192.3-24.fj.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.192.3-24.fl.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.192.3-24.fn.1.7 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.288.7-24.xo.1.15 | $24$ | $3$ | $3$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 72.288.7-72.c.1.10 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.288.10-72.f.1.9 | $72$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 72.288.10-72.j.1.15 | $72$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 120.192.3-120.qq.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qr.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qu.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qv.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.rg.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.rh.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.rk.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.rl.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.18-120.n.2.3 | $120$ | $5$ | $5$ | $18$ | $?$ | not computed |
| 168.192.3-168.oc.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.od.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.og.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.oh.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.os.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ot.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ow.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ox.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.oc.1.16 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.od.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.og.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.oh.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.os.1.14 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ot.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ow.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ox.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qq.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qr.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qu.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qv.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.rg.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.rh.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.rk.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.rl.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |