Invariants
Level: | $120$ | $\SL_2$-level: | $60$ | Newform level: | $450$ | ||
Index: | $360$ | $\PSL_2$-index: | $180$ | ||||
Genus: | $12 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $15^{4}\cdot30^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 12$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 12$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 30A12 |
Level structure
$\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}7&31\\0&101\end{bmatrix}$, $\begin{bmatrix}41&116\\12&55\end{bmatrix}$, $\begin{bmatrix}47&72\\84&107\end{bmatrix}$, $\begin{bmatrix}47&102\\24&23\end{bmatrix}$, $\begin{bmatrix}101&2\\36&109\end{bmatrix}$, $\begin{bmatrix}113&41\\78&97\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 30.180.12.a.1 for the level structure with $-I$) |
Cyclic 120-isogeny field degree: | $24$ |
Cyclic 120-torsion field degree: | $768$ |
Full 120-torsion field degree: | $98304$ |
Models
Canonical model in $\mathbb{P}^{ 11 }$ defined by 45 equations
$ 0 $ | $=$ | $ x z - x w + x t + x s + x a + y w - y u + y r - y b + w u + r a - s a - s c $ |
$=$ | $2 x^{2} - x a - x b + 3 s b - a^{2} + a b - b^{2}$ | |
$=$ | $2 x y - x w - x s + x c - y w + y u - y r - y s + y b - w u - r s - r a + s b - s c$ | |
$=$ | $x y - x w - x t - x u + y^{2} - y w + y t + y v - y r + y b - z w - w u + w s + v s$ | |
$=$ | $\cdots$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Canonical model |
---|
$(0:0:0:0:0:0:0:0:1:-1:1:1)$, $(0:0:1:0:0:1:1:1:0:0:0:0)$, $(0:0:0:0:0:0:0:0:1:0:0:0)$, $(0:0:0:0:-1:0:1:-1:0:0:0:1)$ |
Maps to other modular curves
Map of degree 4 from the canonical model of this modular curve to the canonical model of the modular curve 30.45.3.a.1 :
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle v$ |
$\displaystyle Z$ | $=$ | $\displaystyle -2y+2z-2t+v-2r+s-a+b-4c$ |
Equation of the image curve:
$0$ | $=$ | $ 7X^{4}-6X^{3}Y+5X^{2}Y^{2}-5XY^{3}-Y^{4}-3X^{3}Z-3X^{2}YZ-5XY^{2}Z+2Y^{3}Z-3X^{2}Z^{2}+3XYZ^{2}-Y^{2}Z^{2}-XZ^{3} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
120.72.0-6.a.1.3 | $120$ | $5$ | $5$ | $0$ | $?$ |
120.120.4-30.b.1.13 | $120$ | $3$ | $3$ | $4$ | $?$ |
120.120.4-30.b.1.14 | $120$ | $3$ | $3$ | $4$ | $?$ |