Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| 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 | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| 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: | 12V1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.1662 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&8\\6&17\end{bmatrix}$, $\begin{bmatrix}7&10\\12&13\end{bmatrix}$, $\begin{bmatrix}7&18\\18&7\end{bmatrix}$, $\begin{bmatrix}11&16\\18&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $S_3\times C_2^3:D_4$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.cl.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + 2 x y + 2 x z + y^{2} - 2 y z $ |
| $=$ | $x^{2} + 8 x z - y^{2} + 4 y z + 6 z^{2} - 3 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} + 28 x^{3} z - 6 x^{2} y^{2} + 42 x^{2} z^{2} + 12 x y^{2} z + 28 x z^{3} - 6 y^{2} z^{2} + 5 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{28545045561344xz^{23}+41645860143104xz^{21}w^{2}+15147512119296xz^{19}w^{4}-3814343970816xz^{17}w^{6}-2873455271936xz^{15}w^{8}-18759088128xz^{13}w^{10}+170438625280xz^{11}w^{12}+7655824896xz^{9}w^{14}-4152119040xz^{7}w^{16}-288096192xz^{5}w^{18}+13736768xz^{3}w^{20}+928560xzw^{22}-3989691170816y^{2}z^{22}-5879689928704y^{2}z^{20}w^{2}-2221593632768y^{2}z^{18}w^{4}+475987388416y^{2}z^{16}w^{6}+400207814656y^{2}z^{14}w^{8}+10282898432y^{2}z^{12}w^{10}-22403136512y^{2}z^{10}w^{12}-1372647168y^{2}z^{8}w^{14}+526256640y^{2}z^{6}w^{16}+49868576y^{2}z^{4}w^{18}-2964640y^{2}z^{2}w^{20}-334136y^{2}w^{22}+22251905122304yz^{23}+33214650630144yz^{21}w^{2}+12906477936640yz^{19}w^{4}-2556036100096yz^{17}w^{6}-2319927734272yz^{15}w^{8}-84745824256yz^{13}w^{10}+130533690368yz^{11}w^{12}+9517279744yz^{9}w^{14}-3023409024yz^{7}w^{16}-266296640yz^{5}w^{18}+18587872yz^{3}w^{20}+1135120yzw^{22}+22869112492032z^{24}+23193936150528z^{22}w^{2}-3215571843072z^{20}w^{4}-9231187791872z^{18}w^{6}-1271713837312z^{16}w^{8}+1043744901120z^{14}w^{10}+192160997120z^{12}w^{12}-50454067712z^{10}w^{14}-8244004752z^{8}w^{16}+998683712z^{6}w^{18}+163449352z^{4}w^{20}-1147920z^{2}w^{22}-893295w^{24}}{w^{4}(2z^{2}+w^{2})(4096xz^{17}+56832xz^{15}w^{2}+243968xz^{13}w^{4}+511360xz^{11}w^{6}-19997376xz^{9}w^{8}-2681440xz^{7}w^{10}+2656944xz^{5}w^{12}+86088xz^{3}w^{14}-13428xzw^{16}-1024y^{2}z^{16}-12288y^{2}z^{14}w^{2}-48128y^{2}z^{12}w^{4}-92672y^{2}z^{10}w^{6}+2777984y^{2}z^{8}w^{8}+409600y^{2}z^{6}w^{10}-354112y^{2}z^{4}w^{12}-17056y^{2}z^{2}w^{14}+396y^{2}w^{16}+4096yz^{17}+48640yz^{15}w^{2}+188672yz^{13}w^{4}+360832yz^{11}w^{6}-15673024yz^{9}w^{8}-2724448yz^{7}w^{10}+1926960yz^{5}w^{12}+99208yz^{3}w^{14}-13812yzw^{16}+3072z^{18}+44032z^{16}w^{2}+178560z^{14}w^{4}+324928z^{12}w^{6}-16218528z^{10}w^{8}+4944528z^{8}w^{10}+3425608z^{6}w^{12}-811012z^{4}w^{14}-74922z^{2}w^{16}+3159w^{18})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.cl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}-6X^{2}Y^{2}+28X^{3}Z+12XY^{2}Z+42X^{2}Z^{2}-6Y^{2}Z^{2}+28XZ^{3}+5Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-12.a.2.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.o.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.o.2.17 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.bx.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bx.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 |
|---|---|---|---|---|---|---|
| 24.384.5-24.e.2.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.f.2.12 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.g.2.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.h.4.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bw.4.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bx.2.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bz.2.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.ca.4.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.9-24.d.1.6 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.iz.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ja.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jc.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jd.4.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jq.3.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jr.3.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jt.3.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ju.4.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.iz.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ja.1.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jc.3.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jd.3.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jq.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jr.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jt.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ju.3.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.iz.3.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ja.2.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jc.4.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jd.4.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jq.3.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jr.1.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jt.1.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ju.3.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.iz.4.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ja.4.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jc.4.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jd.4.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jq.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jr.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jt.4.12 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ju.2.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |