Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $24$ | ||
| 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 | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | 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: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.2124 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&15\\6&1\end{bmatrix}$, $\begin{bmatrix}11&9\\6&1\end{bmatrix}$, $\begin{bmatrix}13&14\\0&23\end{bmatrix}$, $\begin{bmatrix}13&21\\6&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.168745 |
| Contains $-I$: | no $\quad$ (see 24.48.1.ct.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: | $2^{3}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 24.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 2 y z $ |
| $=$ | $8 x^{2} + 2 y^{2} + 4 y z + 18 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} + 20 x^{2} z^{2} + 2 y^{2} z^{2} + 4 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3}\cdot\frac{23855104yz^{11}-16056320yz^{9}w^{2}+3760128yz^{7}w^{4}-342016yz^{5}w^{6}+8384yz^{3}w^{8}-48yzw^{10}+23592960z^{12}-14614528z^{10}w^{2}+2764800z^{8}w^{4}-114688z^{6}w^{6}-11264z^{4}w^{8}+336z^{2}w^{10}-w^{12}}{w^{2}z^{6}(2592yz^{3}-36yzw^{2}+2592z^{4}+126z^{2}w^{2}-w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.ct.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}+20X^{2}Z^{2}+2Y^{2}Z^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.k.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.p.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.p.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bx.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bx.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-12.k.1.16 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.192.5-24.m.1.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.o.1.4 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.192.5-24.q.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.s.1.6 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.288.5-24.bs.1.4 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 72.288.5-72.k.1.7 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.s.1.4 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.w.1.3 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.5-120.m.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.o.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.q.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.s.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.480.17-120.jy.1.3 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.5-168.s.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.u.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.w.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.y.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.m.1.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.o.1.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.q.1.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.s.1.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.m.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.o.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.q.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.s.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |