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: | 24.192.1.770 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&4\\16&1\end{bmatrix}$, $\begin{bmatrix}11&10\\8&9\end{bmatrix}$, $\begin{bmatrix}17&0\\20&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3:\GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.t.2 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 $ | $=$ | $ 3 x^{2} + y^{2} - y z + z^{2} $ |
| $=$ | $y^{2} + 2 y z - 2 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{2} y^{2} - 6 x^{2} z^{2} + y^{4} - 12 y^{2} z^{2} + 36 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^2}\cdot\frac{1120043793960yz^{23}-4106827244520yz^{21}w^{2}+6533588798100yz^{19}w^{4}-5911342245900yz^{17}w^{6}+3349947712608yz^{15}w^{8}-1234558910592yz^{13}w^{10}+297508921164yz^{11}w^{12}-45962823060yz^{9}w^{14}+4338801936yz^{7}w^{16}-228578112yz^{5}w^{18}+5637492yz^{3}w^{20}-42588yzw^{22}-819928963881z^{24}+3653063786844z^{22}w^{2}-7046220506787z^{20}w^{4}+7740317901870z^{18}w^{6}-5353309709730z^{16}w^{8}+2430698284800z^{14}w^{10}-732624100299z^{12}w^{12}+144836980794z^{10}w^{14}-18123718086z^{8}w^{16}+1341005760z^{6}w^{18}-51761943z^{4}w^{20}+802854z^{2}w^{22}-2197w^{24}}{w^{8}(7919856yz^{15}-18479664yz^{13}w^{2}+17159688yz^{11}w^{4}-8066520yz^{9}w^{6}+2016504yz^{7}w^{8}-256536yz^{5}w^{10}+14224yz^{3}w^{12}-224yzw^{14}-5797737z^{16}+18100584z^{14}w^{2}-22468914z^{12}w^{4}+14288076z^{10}w^{6}-4969089z^{8}w^{8}+928644z^{6}w^{10}-84444z^{4}w^{12}+2896z^{2}w^{14}-16w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.t.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+4X^{2}Y^{2}+Y^{4}-6X^{2}Z^{2}-12Y^{2}Z^{2}+36Z^{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.2.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.0-24.h.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.h.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.j.2.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.j.2.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.u.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.u.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.w.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.w.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-8.i.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.p.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.p.1.13 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.s.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.s.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 |
|---|---|---|---|---|---|---|
| 24.576.17-24.os.1.3 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.fm.2.5 | $24$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 48.384.9-48.fk.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.fl.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.fp.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.fq.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 240.384.9-240.bhm.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bhn.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bhs.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bht.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |