Invariants
| Level: | $120$ | $\SL_2$-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 | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 96$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}17&20\\80&87\end{bmatrix}$, $\begin{bmatrix}49&56\\48&85\end{bmatrix}$, $\begin{bmatrix}87&80\\10&89\end{bmatrix}$, $\begin{bmatrix}105&68\\74&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.y.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $184320$ |
Jacobian
| Conductor: | $?$ |
| 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.y.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 |
|---|---|---|---|---|---|---|
| 40.96.1-8.k.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 120.96.0-24.i.2.3 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.i.2.15 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.j.2.3 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.j.2.9 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.v.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.v.1.11 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.w.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.w.1.11 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.1-8.k.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-24.p.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-24.p.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-24.q.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-24.q.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 240.384.5-48.b.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.h.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.q.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.s.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.cr.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ct.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cx.2.11 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.dc.1.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dd.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.di.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.fy.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ga.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.mb.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.md.1.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oy.1.15 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.pe.1.11 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.9-48.gb.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-48.gc.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-48.gh.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-48.gi.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bjh.2.22 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bji.2.23 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bjn.2.22 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bjo.2.23 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |