Invariants
| Level: | $120$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 72$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20H1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}14&109\\105&88\end{bmatrix}$, $\begin{bmatrix}28&103\\105&106\end{bmatrix}$, $\begin{bmatrix}39&28\\52&65\end{bmatrix}$, $\begin{bmatrix}60&101\\31&90\end{bmatrix}$, $\begin{bmatrix}108&91\\59&70\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.1.k.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $16$ |
| Cyclic 120-torsion field degree: | $512$ |
| Full 120-torsion field degree: | $245760$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} - y^{2} + y z + y w $ |
| $=$ | $y^{2} - 2 y z - 2 y w + 2 z^{2} + z w + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + 3 x^{2} y z - 6 x^{2} z^{2} + 3 y^{2} z^{2} - 9 y z^{3} + 9 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 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -3^3\,\frac{236yz^{17}-2420yz^{16}w+11024yz^{15}w^{2}-32880yz^{14}w^{3}+70800yz^{13}w^{4}-109488yz^{12}w^{5}+133232yz^{11}w^{6}-109328yz^{10}w^{7}+42920yz^{9}w^{8}+42920yz^{8}w^{9}-109328yz^{7}w^{10}+133232yz^{6}w^{11}-109488yz^{5}w^{12}+70800yz^{4}w^{13}-32880yz^{3}w^{14}+11024yz^{2}w^{15}-2420yzw^{16}+236yw^{17}-121z^{18}+990z^{17}w-5265z^{16}w^{2}+23296z^{15}w^{3}-64980z^{14}w^{4}+133416z^{13}w^{5}-246116z^{12}w^{6}+342144z^{11}w^{7}-445374z^{10}w^{8}+462580z^{9}w^{9}-445374z^{8}w^{10}+342144z^{7}w^{11}-246116z^{6}w^{12}+133416z^{5}w^{13}-64980z^{4}w^{14}+23296z^{3}w^{15}-5265z^{2}w^{16}+990zw^{17}-121w^{18}}{(z+w)^{5}(z^{2}-zw+w^{2})^{5}(2yz^{2}-8yzw+2yw^{2}+6z^{3}+9z^{2}w+9zw^{2}+6w^{3})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.72.1.k.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}y$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}+3X^{2}YZ-6X^{2}Z^{2}+3Y^{2}Z^{2}-9YZ^{3}+9Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 120.72.1-20.b.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.288.5-60.bc.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bx.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ez.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fa.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fg.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fi.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fs.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fx.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fy.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ni.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bka.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bkh.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bly.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bml.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqm.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqt.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-60.bc.1.29 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |