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: | 12B2 |
Level structure
$\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}7&103\\100&63\end{bmatrix}$, $\begin{bmatrix}7&114\\114&85\end{bmatrix}$, $\begin{bmatrix}21&100\\20&33\end{bmatrix}$, $\begin{bmatrix}21&113\\86&25\end{bmatrix}$, $\begin{bmatrix}97&7\\82&69\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.36.2.q.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 $ | $=$ | $ 3 z^{2} + w^{2} + w t + t^{2} $ |
$=$ | $ - x w - 2 x t + 3 y^{2}$ | |
$=$ | $4 x y + z w + 2 z t$ | |
$=$ | $4 x^{2} + 3 y z$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 108 x^{6} - 18 x^{3} y z^{2} + y^{2} z^{4} + z^{6} $ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ -x^{6} - 27 $ |
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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3^3\,\frac{t^{3}(w+t)^{3}}{(w+2t)^{2}(w^{2}+wt+t^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.36.2.q.1 :
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{3}{2}t$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{3}{2}z$ |
Equation of the image curve:
$0$ | $=$ | $ 108X^{6}-18X^{3}YZ^{2}+Y^{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.q.1 :
$\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}z$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}x^{3}-\frac{1}{8}z^{2}t$ |
$\displaystyle Z$ | $=$ | $\displaystyle -\frac{1}{3}x$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
120.24.0-12.e.1.1 | $120$ | $3$ | $3$ | $0$ | $?$ |
120.36.1-12.b.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ |
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.bk.1.5 | $120$ | $2$ | $2$ | $3$ |
120.144.3-12.bl.1.4 | $120$ | $2$ | $2$ | $3$ |
120.144.3-12.bs.1.2 | $120$ | $2$ | $2$ | $3$ |
120.144.3-12.bt.1.1 | $120$ | $2$ | $2$ | $3$ |
120.144.3-60.ei.1.2 | $120$ | $2$ | $2$ | $3$ |
120.144.3-60.ej.1.2 | $120$ | $2$ | $2$ | $3$ |
120.144.3-60.eu.1.4 | $120$ | $2$ | $2$ | $3$ |
120.144.3-60.ev.1.3 | $120$ | $2$ | $2$ | $3$ |
120.144.3-24.jc.1.2 | $120$ | $2$ | $2$ | $3$ |
120.144.3-24.jj.1.2 | $120$ | $2$ | $2$ | $3$ |
120.144.3-24.lg.1.3 | $120$ | $2$ | $2$ | $3$ |
120.144.3-24.ln.1.4 | $120$ | $2$ | $2$ | $3$ |
120.144.3-120.bbs.1.4 | $120$ | $2$ | $2$ | $3$ |
120.144.3-120.bbz.1.4 | $120$ | $2$ | $2$ | $3$ |
120.144.3-120.bei.1.1 | $120$ | $2$ | $2$ | $3$ |
120.144.3-120.bep.1.5 | $120$ | $2$ | $2$ | $3$ |
120.360.14-60.bg.1.2 | $120$ | $5$ | $5$ | $14$ |
120.432.15-60.bw.1.5 | $120$ | $6$ | $6$ | $15$ |