Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}7&50\\12&47\end{bmatrix}$, $\begin{bmatrix}41&18\\28&65\end{bmatrix}$, $\begin{bmatrix}69&16\\92&9\end{bmatrix}$, $\begin{bmatrix}83&96\\76&53\end{bmatrix}$, $\begin{bmatrix}99&94\\4&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.bh.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $368640$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x y + 2 y^{2} + z^{2} $ |
| $=$ | $5 x^{2} - 3 x y + 2 y^{2} + z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 10 x^{2} y^{2} + 3 x^{2} z^{2} + 2 z^{4} $ |
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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{5^3}\cdot\frac{1968750y^{2}z^{10}-2362500y^{2}z^{8}w^{2}+90000y^{2}z^{6}w^{4}+36000y^{2}z^{4}w^{6}-151200y^{2}z^{2}w^{8}+20160y^{2}w^{10}+484375z^{12}-375000z^{10}w^{2}-120000z^{8}w^{4}+64000z^{6}w^{6}-102000z^{4}w^{8}+30720z^{2}w^{10}-2048w^{12}}{w^{4}z^{4}(10y^{2}z^{2}+4y^{2}w^{2}+5z^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.48.1.bh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{10}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-10X^{2}Y^{2}+3X^{2}Z^{2}+2Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.0-8.d.2.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 120.48.0-8.d.2.2 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.48.0-40.i.1.2 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.48.0-40.i.1.21 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.48.1-40.d.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1-40.d.1.20 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.192.1-40.a.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.r.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.bj.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.bn.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.bs.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.bw.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.ce.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.cg.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.gi.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.go.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.hp.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.hv.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.mq.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.mw.2.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.nw.2.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.oc.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-120.si.1.57 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-120.jo.1.2 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 120.480.17-40.cb.2.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |