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: | $\begin{bmatrix}30&23\\101&36\end{bmatrix}$, $\begin{bmatrix}33&92\\116&45\end{bmatrix}$, $\begin{bmatrix}68&103\\17&110\end{bmatrix}$, $\begin{bmatrix}77&108\\96&35\end{bmatrix}$, $\begin{bmatrix}91&104\\20&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.36.1.h.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $491520$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 15 x y + 16 y^{2} - 4 y z + 4 z^{2} $ |
| $=$ | $15 x^{2} - 15 x y + 15 y^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 721 x^{4} - 94 x^{3} z - 15 x^{2} y^{2} + 51 x^{2} z^{2} - 4 x z^{3} + 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^3\cdot5^2}\cdot\frac{31296131852631904800000xz^{8}+86605329040122067560000xz^{6}w^{2}-30752196775527184700400xz^{4}w^{4}-76278626344179893729340xz^{2}w^{6}+6482265681622999772224xw^{8}+295786608851929219200000y^{2}z^{7}+835705091293222060310400y^{2}z^{5}w^{2}+219615395012655648924960y^{2}z^{3}w^{4}-78353224247513506358256y^{2}zw^{6}-86347965387776712480000yz^{8}-229968291209190331881600yz^{6}w^{2}-126330691175342395212240yz^{4}w^{4}+14721035916919294659984yz^{2}w^{6}+6859406140447025656561yw^{8}+57897474238477048320000z^{9}+104268346040926272441600z^{7}w^{2}-13617931874680848677760z^{5}w^{4}-20340967025114638327824z^{3}w^{6}}{26831388762544500xz^{8}+9702532890010100xz^{6}w^{2}+467308571713200xz^{4}w^{4}-107943943968000xz^{2}w^{6}+253589342294178000y^{2}z^{7}+24385524504816209y^{2}z^{5}w^{2}-6141767144385635y^{2}z^{3}w^{4}-380052636054000y^{2}zw^{6}-74029462781015700yz^{8}-15199817237403111yz^{6}w^{2}+2629614381063565yz^{4}w^{4}+368696345520300yz^{2}w^{6}-26985985992000yw^{8}+49637752262068800z^{9}+8355515204078336z^{7}w^{2}+169232449296960z^{5}w^{4}-28785051724800z^{3}w^{6}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.36.1.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
Equation of the image curve:
| $0$ | $=$ | $ 721X^{4}-15X^{2}Y^{2}-94X^{3}Z+51X^{2}Z^{2}-4XZ^{3}+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.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.n.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ch.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ej.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.el.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.fx.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.gu.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.gx.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.hc.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.hf.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ou.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bca.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bco.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.brt.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bso.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.btx.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bus.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.360.13-60.w.1.11 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.432.13-60.bp.1.26 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |