Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $450$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $13 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $5^{4}\cdot10^{4}\cdot15^{4}\cdot30^{4}$ | Cusp orbits | $4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12$) |
Other labels
| Cummins and Pauli (CP) label: | 30N13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.480.13.1096 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&12\\30&29\end{bmatrix}$, $\begin{bmatrix}17&18\\6&43\end{bmatrix}$, $\begin{bmatrix}19&54\\24&41\end{bmatrix}$, $\begin{bmatrix}56&25\\45&26\end{bmatrix}$, $\begin{bmatrix}59&12\\42&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.240.13.b.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{7}\cdot3^{17}\cdot5^{26}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}$ |
| Newforms: | 50.2.a.b$^{2}$, 75.2.a.a$^{2}$, 75.2.a.b$^{2}$, 150.2.a.b, 225.2.a.a$^{2}$, 450.2.a.b, 450.2.a.f, 450.2.a.g$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ x^{2} - x z - x w - x t + x u + x r - x a + x c - y u - y b - z r - t r + u r + u s $ |
| $=$ | $x y + x w - 2 y^{2} - y z - y w + y t - y v + z^{2} - w v$ | |
| $=$ | $x s - y^{2} - y z + y w + y u - y v - 2 y r - 2 y s + y b - 2 y c + y d + z s - v s$ | |
| $=$ | $x w + x s + 4 y^{2} + y z - y w - 6 y u - 6 y v + y r - 2 y a + y b - y c + z^{2} + 2 z s - w v + \cdots - s d$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has 2 rational CM points but no rational cusps or other known rational points.
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 30.120.7.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle w$ |
| $\displaystyle T$ | $=$ | $\displaystyle t$ |
| $\displaystyle U$ | $=$ | $\displaystyle u$ |
| $\displaystyle V$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ XY-2Y^{2}+YZ+Z^{2}-XW+YW-YT+YV-WV $ |
| $=$ | $ 2X^{2}+XY-2XZ-2Z^{2}+2XW-YW+ZW+W^{2}+XT-YT+WT-XU-YU+ZU+WU-TU-U^{2}-ZV $ | |
| $=$ | $ X^{2}-3XZ+YZ+Z^{2}+2XW-2ZW-XT-WT-T^{2}-XU+YU-3ZU-2TU+XV+YV-WV-UV $ | |
| $=$ | $ X^{2}-XY-3YZ+YW-2ZW-YT-ZT-WT-T^{2}-XU-3ZU-2TU+XV-YV+TV-UV $ | |
| $=$ | $ 2XZ+YZ+2XW-2YW+ZW+2XT-2YT+ZT+WT+T^{2}+YU+2ZU+TU+YV+ZV-WV $ | |
| $=$ | $ 2XY+2XZ+Z^{2}+2XW-ZW+XT+2ZT-WT+YU-2ZU-TU-WV $ | |
| $=$ | $ X^{2}-XZ+YZ+XW-2YW+ZW+XT-2ZT-WT-XU+2YU+TU+XV-YV+2ZV+WV-TV-UV $ | |
| $=$ | $ 2XY-3XZ-2YZ-2Z^{2}-XW-ZW-2XT-2YT-2ZT-T^{2}+YU-ZU-TU-ZV-WV $ | |
| $=$ | $ 2X^{2}-2XZ+YZ-Z^{2}-XW-XT+2ZT+T^{2}-2XU+2YU-2XV+YV-3ZV-WV-2TV+2UV $ | |
| $=$ | $ X^{2}-XZ+YZ-2YW-W^{2}+3XT-2YT-ZT+T^{2}+3XU+YU+ZU-WU+2TU+U^{2}-4XV+YV-WV-TV-UV-V^{2} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.240.7-30.h.1.14 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{6}$ |
| 60.240.7-30.h.1.35 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.960.29-30.b.1.13 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-30.o.1.9 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{16}$ |
| 60.960.29-30.x.1.9 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-30.z.1.9 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-60.eb.1.23 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-60.fk.1.5 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-60.fm.1.6 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-60.lb.1.15 | $60$ | $2$ | $2$ | $29$ | $16$ | $1^{16}$ |
| 60.960.29-60.le.1.1 | $60$ | $2$ | $2$ | $29$ | $16$ | $1^{16}$ |
| 60.960.29-60.lf.1.2 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{16}$ |
| 60.960.29-60.na.1.15 | $60$ | $2$ | $2$ | $29$ | $10$ | $1^{16}$ |
| 60.960.29-60.nc.1.4 | $60$ | $2$ | $2$ | $29$ | $10$ | $1^{16}$ |
| 60.960.29-60.ne.1.3 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-60.no.1.15 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{16}$ |
| 60.960.29-60.nq.1.8 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{16}$ |
| 60.960.29-60.ns.1.7 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.33-60.pd.1.10 | $60$ | $2$ | $2$ | $33$ | $10$ | $1^{20}$ |
| 60.960.33-60.pf.1.9 | $60$ | $2$ | $2$ | $33$ | $11$ | $1^{20}$ |
| 60.960.33-60.pt.1.14 | $60$ | $2$ | $2$ | $33$ | $8$ | $1^{20}$ |
| 60.960.33-60.pv.1.13 | $60$ | $2$ | $2$ | $33$ | $11$ | $1^{20}$ |
| 60.960.33-60.qg.1.15 | $60$ | $2$ | $2$ | $33$ | $9$ | $1^{20}$ |
| 60.960.33-60.qh.1.16 | $60$ | $2$ | $2$ | $33$ | $4$ | $1^{20}$ |
| 60.960.33-60.qn.1.11 | $60$ | $2$ | $2$ | $33$ | $9$ | $1^{20}$ |
| 60.960.33-60.qp.1.20 | $60$ | $2$ | $2$ | $33$ | $18$ | $1^{20}$ |
| 60.1440.37-30.d.1.11 | $60$ | $3$ | $3$ | $37$ | $4$ | $1^{24}$ |
| 60.1440.45-30.d.1.11 | $60$ | $3$ | $3$ | $45$ | $9$ | $1^{28}\cdot2^{2}$ |
| 60.1440.45-30.e.1.24 | $60$ | $3$ | $3$ | $45$ | $7$ | $1^{28}\cdot2^{2}$ |