Invariants
| Level: | $120$ | $\SL_2$-level: | $40$ | Newform level: | $720$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20J3 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}11&90\\99&71\end{bmatrix}$, $\begin{bmatrix}41&20\\41&93\end{bmatrix}$, $\begin{bmatrix}47&70\\119&111\end{bmatrix}$, $\begin{bmatrix}77&80\\11&57\end{bmatrix}$, $\begin{bmatrix}107&100\\92&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.3.er.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $16$ |
| Cyclic 120-torsion field degree: | $512$ |
| Full 120-torsion field degree: | $245760$ |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ z t - z u + 2 w u $ |
| $=$ | $ - x w + 2 y z$ | |
| $=$ | $x^{2} - 2 y^{2} + z^{2} + z w - 2 u^{2}$ | |
| $=$ | $x^{2} + 3 x y + y^{2} + z^{2} + z w + t u - 2 u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{6} + 9 x^{4} y^{2} + 6 x^{4} z^{2} + 6 x^{2} y^{2} z^{2} + x^{2} z^{4} - 12 y^{4} z^{2} + y^{2} z^{4} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ w^{2} $ | $=$ | $ 9 x^{4} - 9 x^{2} z^{2} + z^{4} $ |
| $0$ | $=$ | $-3 x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1741824z^{10}-5832000z^{8}u^{2}+3006720z^{6}u^{4}+9550080z^{4}u^{6}-15582720z^{2}u^{8}-1701w^{10}+136161w^{8}u^{2}-3470472w^{6}u^{4}+26061408w^{4}u^{6}+47277696w^{2}u^{8}+109575t^{10}-1116090t^{9}u+5032154t^{8}u^{2}-13563088t^{7}u^{3}+23464234t^{6}u^{4}-24530772t^{5}u^{5}+15368880t^{4}u^{6}+3220544t^{3}u^{7}-6259945t^{2}u^{8}+27335710tu^{9}-10650450u^{10}}{u^{4}(27w^{6}-4968w^{4}u^{2}+98928w^{2}u^{4}-t^{6}-6t^{5}u+489t^{4}u^{2}+1756t^{3}u^{3}-15231t^{2}u^{4}+38250tu^{5}-32425u^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.72.3.er.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{6}+9X^{4}Y^{2}+6X^{4}Z^{2}+6X^{2}Y^{2}Z^{2}-12Y^{4}Z^{2}+X^{2}Z^{4}+Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ |
| 120.24.0-12.f.1.2 | $120$ | $6$ | $6$ | $0$ | $?$ |
| 120.72.1-20.b.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.288.5-60.ec.1.11 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.ec.2.11 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.ed.1.7 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.ed.2.6 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fa.1.5 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fa.2.3 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fb.1.5 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fb.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bco.1.3 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bco.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bcv.1.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bcv.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bkg.1.5 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bkg.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bkn.1.3 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bkn.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.432.15-60.bx.1.5 | $120$ | $3$ | $3$ | $15$ |