Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $3600$ | ||
| Index: | $432$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $13 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $6^{6}\cdot30^{6}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $6$ | ||||||
| $\overline{\Q}$-gonality: | $6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30H13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.432.13.1645 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&4\\38&29\end{bmatrix}$, $\begin{bmatrix}5&41\\4&53\end{bmatrix}$, $\begin{bmatrix}23&12\\0&41\end{bmatrix}$, $\begin{bmatrix}27&38\\10&9\end{bmatrix}$, $\begin{bmatrix}57&38\\44&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.216.13.gn.2 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $64$ |
| Full 60-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{30}\cdot3^{26}\cdot5^{18}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2^{3}$ |
| Newforms: | 45.2.b.a$^{2}$, 90.2.a.a, 90.2.a.b, 720.2.f.d, 3600.2.a.ba, 3600.2.a.be, 3600.2.a.bj, 3600.2.a.e$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ b^{2} - c d $ |
| $=$ | $r d + s b$ | |
| $=$ | $r b + s c$ | |
| $=$ | $v^{2} + a b$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:0:1:1:1:0:0:0)$, $(0:0:0:0:0:0:0:1:-1:1:0:0:0)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 30.108.6.a.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y+z+w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w$ |
| $\displaystyle W$ | $=$ | $\displaystyle z$ |
| $\displaystyle T$ | $=$ | $\displaystyle u$ |
| $\displaystyle U$ | $=$ | $\displaystyle -x-t+u$ |
Equation of the image curve:
| $0$ | $=$ | $ ZT-WU $ |
| $=$ | $ Y^{2}-ZW $ | |
| $=$ | $ XY-YT+YU-WU $ | |
| $=$ | $ XW-YT-WT+WU $ | |
| $=$ | $ XZ-YU+ZU-WU $ | |
| $=$ | $ X^{2}-2XT+T^{2}+2XU-3TU+U^{2} $ | |
| $=$ | $ X^{3}+2Y^{3}+Y^{2}Z+2YZW-ZW^{2}-X^{2}T+X^{2}U-XTU $ | |
| $=$ | $ Y^{3}+2Y^{2}W-YW^{2}+2ZW^{2}+XT^{2}-XTU $ | |
| $=$ | $ Y^{3}-2Y^{2}Z-YZ^{2}-2Z^{2}W-XTU+XU^{2} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.1-60.cf.1.2 | $60$ | $3$ | $3$ | $1$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.216.6-30.a.2.33 | $60$ | $2$ | $2$ | $6$ | $0$ | $1^{5}\cdot2$ |
| 60.216.6-30.a.2.36 | $60$ | $2$ | $2$ | $6$ | $0$ | $1^{5}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.864.25-60.nf.2.9 | $60$ | $2$ | $2$ | $25$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.864.25-60.ng.2.9 | $60$ | $2$ | $2$ | $25$ | $3$ | $1^{6}\cdot2^{3}$ |
| 60.864.25-60.qb.2.7 | $60$ | $2$ | $2$ | $25$ | $4$ | $1^{6}\cdot2^{3}$ |
| 60.864.25-60.qc.2.7 | $60$ | $2$ | $2$ | $25$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.864.25-60.sx.2.9 | $60$ | $2$ | $2$ | $25$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.864.25-60.sy.2.9 | $60$ | $2$ | $2$ | $25$ | $4$ | $1^{6}\cdot2^{3}$ |
| 60.864.25-60.vm.2.3 | $60$ | $2$ | $2$ | $25$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.864.25-60.vn.2.8 | $60$ | $2$ | $2$ | $25$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.864.29-60.ws.2.10 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.ww.2.21 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.byu.2.5 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.bza.2.2 | $60$ | $2$ | $2$ | $29$ | $6$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.cqe.2.10 | $60$ | $2$ | $2$ | $29$ | $2$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.cqf.2.16 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.cyq.2.15 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.cyr.2.9 | $60$ | $2$ | $2$ | $29$ | $4$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.dim.2.12 | $60$ | $2$ | $2$ | $29$ | $1$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.din.2.14 | $60$ | $2$ | $2$ | $29$ | $4$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.dqu.2.13 | $60$ | $2$ | $2$ | $29$ | $4$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.dqv.2.11 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.dwi.2.9 | $60$ | $2$ | $2$ | $29$ | $2$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.dwo.2.14 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.dyw.2.6 | $60$ | $2$ | $2$ | $29$ | $2$ | $1^{8}\cdot2^{4}$ |
| 60.864.29-60.dza.2.1 | $60$ | $2$ | $2$ | $29$ | $6$ | $1^{8}\cdot2^{4}$ |
| 60.2160.73-60.biu.1.9 | $60$ | $5$ | $5$ | $73$ | $13$ | $1^{24}\cdot2^{18}$ |