Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12D2 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}0&49\\67&82\end{bmatrix}$, $\begin{bmatrix}17&86\\90&97\end{bmatrix}$, $\begin{bmatrix}26&49\\19&52\end{bmatrix}$, $\begin{bmatrix}31&54\\10&11\end{bmatrix}$, $\begin{bmatrix}75&118\\46&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.36.2.u.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $491520$ |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x w + y t $ |
| $=$ | $ - x t + y w + z t$ | |
| $=$ | $x^{2} - x z + y^{2}$ | |
| $=$ | $4 x^{2} + 4 x z + 4 z^{2} + w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + x^{4} y^{2} + 3 x^{4} z^{2} + 3 x^{2} y^{2} z^{2} + 3 x^{2} z^{4} + 3 y^{2} z^{4} + z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{6} - 4x^{4} - 6x^{2} - 3 $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{432xz^{3}t^{2}-324xzt^{4}-432z^{6}+864z^{4}t^{2}-432z^{2}t^{4}-7w^{6}-54w^{4}t^{2}-135w^{2}t^{4}-72t^{6}}{16xz^{3}t^{2}+4xzt^{4}-64z^{6}+32z^{4}t^{2}-w^{6}-2w^{4}t^{2}-w^{2}t^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.36.2.u.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}+X^{4}Y^{2}+3X^{4}Z^{2}+3X^{2}Y^{2}Z^{2}+3X^{2}Z^{4}+3Y^{2}Z^{4}+Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 12.36.2.u.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w^{3}+wt^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2zw^{8}-10zw^{6}t^{2}-20zw^{4}t^{4}-18zw^{2}t^{6}-6zt^{8}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w^{2}t+t^{3}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.36.1-12.b.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ |
| 120.36.1-12.b.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.144.3-12.j.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-12.t.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-12.bl.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-12.bm.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.cg.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.ez.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.gm.1.8 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.go.1.1 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.gu.1.1 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.gw.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.jl.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.js.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bpq.1.7 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bqe.1.6 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bru.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bsi.1.1 | $120$ | $2$ | $2$ | $3$ |
| 120.360.14-60.ca.1.4 | $120$ | $5$ | $5$ | $14$ |
| 120.432.15-60.cy.1.14 | $120$ | $6$ | $6$ | $15$ |