Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| 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: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.1077 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&22\\16&17\end{bmatrix}$, $\begin{bmatrix}5&14\\4&5\end{bmatrix}$, $\begin{bmatrix}7&10\\20&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3:\GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.i.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $384$ |
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 $ | $=$ | $ y^{2} - y z + z^{2} - w^{2} $ |
| $=$ | $6 x^{2} + 2 y^{2} + y z - z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} - 12 x^{2} z^{2} + 4 y^{4} + 12 y^{2} z^{2} + 9 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{275562yz^{15}w^{8}-1285956yz^{13}w^{10}+2388204yz^{11}w^{12}-2245320yz^{9}w^{14}+1074789yz^{7}w^{16}-190026yz^{5}w^{18}-21438yz^{3}w^{20}+6084yzw^{22}-531441z^{24}+4251528z^{22}w^{2}-14880348z^{20}w^{4}+29918160z^{18}w^{6}-38145654z^{16}w^{8}+32030802z^{14}w^{10}-17646174z^{12}w^{12}+5998212z^{10}w^{14}-954504z^{8}w^{16}-133623z^{6}w^{18}+111969z^{4}w^{20}-20826z^{2}w^{22}-2197w^{24}}{w^{8}(729yz^{15}-3402yz^{13}w^{2}+6318yz^{11}w^{4}-5940yz^{9}w^{6}+3006yz^{7}w^{8}-828yz^{5}w^{10}+124yz^{3}w^{12}-8yzw^{14}-243z^{14}w^{2}+1053z^{12}w^{4}-1782z^{10}w^{6}+1449z^{8}w^{8}-546z^{6}w^{10}+66z^{4}w^{12}+4z^{2}w^{14}-w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.i.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-2X^{2}Y^{2}+4Y^{4}-12X^{2}Z^{2}+12Y^{2}Z^{2}+9Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.0-24.g.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.g.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.i.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.i.2.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.t.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.t.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.v.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.v.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-8.i.2.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.m.2.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.m.2.11 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.p.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.p.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.576.17-24.lb.2.1 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.dv.1.10 | $24$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 48.384.9-48.dr.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.ds.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.dw.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.dx.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 240.384.9-240.st.1.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.su.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.sz.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ta.1.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |