Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 36$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}23&86\\106&71\end{bmatrix}$, $\begin{bmatrix}41&50\\118&47\end{bmatrix}$, $\begin{bmatrix}61&34\\22&41\end{bmatrix}$, $\begin{bmatrix}64&61\\25&38\end{bmatrix}$, $\begin{bmatrix}68&51\\57&82\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.36.1.k.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $491520$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x y - 5 y^{2} - w^{2} $ |
| $=$ | $4 x^{2} + 3 x y + 4 x z + 4 y^{2} + 4 y z - 4 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 1100 x^{4} + 40 x^{3} y - x^{2} y^{2} + 55 x^{2} z^{2} + x y z^{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 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -5\,\frac{21202042048000xz^{8}-302127480000xz^{6}w^{2}-291291092400xz^{4}w^{4}-7234155368xz^{2}w^{6}+44611127xw^{8}+35696579496000y^{2}z^{7}+2102046751200y^{2}z^{5}w^{2}-157817788040y^{2}z^{3}w^{4}-5269588720y^{2}zw^{6}+5096276864000yz^{8}-2154507184000yz^{6}w^{2}-209925794320yz^{4}w^{4}+1545892612yz^{2}w^{6}+63131992yw^{8}-13095378912000z^{9}+3002896721600z^{7}w^{2}+475070650560z^{5}w^{4}+655331160z^{3}w^{6}-520692524zw^{8}}{62246000xz^{6}w^{2}-1570580xz^{2}w^{6}-161051xw^{8}+913520000y^{2}z^{7}-80041500y^{2}z^{5}w^{2}-31677800y^{2}z^{3}w^{4}-1610510y^{2}zw^{6}-311230000yz^{8}+50718500yz^{6}w^{2}+13745600yz^{4}w^{4}+399300yz^{2}w^{6}+62246000z^{7}w^{2}-11642400z^{5}w^{4}-2582140z^{3}w^{6}-87846zw^{8}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.36.1.k.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 20z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 1100X^{4}+40X^{3}Y-X^{2}Y^{2}+55X^{2}Z^{2}+XYZ^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.36.1-12.b.1.14 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 120.36.1-12.b.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.144.3-60.u.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.cf.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ev.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ew.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ga.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.gn.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.go.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.hl.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.hm.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.og.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.beq.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bex.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bpw.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bqd.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bwi.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bwp.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.360.13-60.z.1.4 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.432.13-60.bs.1.29 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |