Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| 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: | $\begin{bmatrix}28&79\\111&110\end{bmatrix}$, $\begin{bmatrix}58&17\\59&0\end{bmatrix}$, $\begin{bmatrix}68&75\\101&22\end{bmatrix}$, $\begin{bmatrix}86&23\\35&112\end{bmatrix}$, $\begin{bmatrix}118&117\\37&44\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.36.2.dk.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 t + y t + z w $ |
| $=$ | $6 x w + z t$ | |
| $=$ | $6 x^{2} - 6 x y + z^{2}$ | |
| $=$ | $24 x^{2} + 24 x y + 24 y^{2} + 6 w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} y^{2} + 3 x^{4} z^{2} + 6 x^{2} y^{2} z^{2} + 2 x^{2} z^{4} + 4 y^{2} z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -8x^{6} - 24x^{4} - 24x^{2} - 9 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^9\cdot3^3\,\frac{576y^{2}z^{4}+720y^{2}z^{2}t^{2}-156y^{2}t^{4}-104z^{4}t^{2}-76z^{2}t^{4}-36w^{6}-75w^{2}t^{4}-2t^{6}}{4608y^{2}z^{4}+576y^{2}z^{2}t^{2}-168y^{2}t^{4}+32z^{4}t^{2}+4z^{2}t^{4}-6w^{2}t^{4}-t^{6}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.36.2.dk.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}Y^{2}+3X^{4}Z^{2}+6X^{2}Y^{2}Z^{2}+2X^{2}Z^{4}+4Y^{2}Z^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.36.2.dk.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}wt^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -6zw^{6}t^{2}-3zw^{4}t^{4}-\frac{1}{2}zw^{2}t^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w^{2}t$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.36.1-12.b.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ |
| 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 |
|---|---|---|---|---|
| 120.144.3-24.h.1.9 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.bz.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.eo.1.7 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.er.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.lg.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.lh.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.lu.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-24.lv.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.btw.1.14 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.btx.1.8 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.buk.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bul.1.6 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bwa.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bwb.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bwo.1.6 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bwp.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.360.14-120.hs.1.12 | $120$ | $5$ | $5$ | $14$ |
| 120.432.15-120.lc.1.28 | $120$ | $6$ | $6$ | $15$ |