Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $150$ | ||
| 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: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30N13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.480.13.1072 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&6\\46&19\end{bmatrix}$, $\begin{bmatrix}5&33\\38&29\end{bmatrix}$, $\begin{bmatrix}11&27\\52&7\end{bmatrix}$, $\begin{bmatrix}13&9\\24&35\end{bmatrix}$, $\begin{bmatrix}35&9\\44&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.240.13.a.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: | $2^{7}\cdot3^{9}\cdot5^{26}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}$ |
| Newforms: | 50.2.a.a$^{2}$, 50.2.a.b$^{2}$, 75.2.a.a$^{2}$, 75.2.a.b$^{2}$, 75.2.a.c$^{2}$, 150.2.a.a, 150.2.a.b, 150.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ 3 x^{2} + x y - 2 x z + x u + x v - 3 x r + x s - x a + x b - 2 y^{2} - y z - y w + y t + 2 y u + \cdots + c^{2} $ |
| $=$ | $2 x^{2} - x y - x z + 4 x w + x t + x u + 2 x v + x s - 3 x a - x b + x c - 2 x d - 3 y^{2} + y z + \cdots - c^{2}$ | |
| $=$ | $x^{2} + 5 x y - x z - x w + x t - 2 x u - 3 x v + 2 x r + x a - 3 x b - 2 x c + 4 y^{2} - y z + 3 y w + \cdots + c d$ | |
| $=$ | $4 x^{2} - 5 x y + 3 x z + 3 x w + 4 x t - x u + 3 x v + x r - 2 x s + 3 x a - x b + 2 x c - x d + \cdots + c d$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=29$, and therefore no 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 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z$ |
| $\displaystyle W$ | $=$ | $\displaystyle -w$ |
| $\displaystyle T$ | $=$ | $\displaystyle u$ |
| $\displaystyle U$ | $=$ | $\displaystyle -v$ |
| $\displaystyle V$ | $=$ | $\displaystyle -x+y+z+w-u-v+a-b$ |
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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(5)$ | $5$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 12.24.0-6.a.1.6 | $12$ | $20$ | $20$ | $0$ | $0$ | full Jacobian |
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.1.44 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{6}$ |
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.a.1.13 | $60$ | $2$ | $2$ | $29$ | $1$ | $1^{16}$ |
| 60.960.29-30.k.1.5 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-30.w.1.9 | $60$ | $2$ | $2$ | $29$ | $1$ | $1^{16}$ |
| 60.960.29-30.y.1.9 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{16}$ |
| 60.960.29-60.ea.1.20 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{16}$ |
| 60.960.29-60.fg.1.1 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{16}$ |
| 60.960.29-60.fi.1.6 | $60$ | $2$ | $2$ | $29$ | $1$ | $1^{16}$ |
| 60.960.29-60.id.1.10 | $60$ | $2$ | $2$ | $29$ | $8$ | $1^{16}$ |
| 60.960.29-60.if.1.7 | $60$ | $2$ | $2$ | $29$ | $8$ | $1^{16}$ |
| 60.960.29-60.ih.1.4 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-60.mt.1.10 | $60$ | $2$ | $2$ | $29$ | $6$ | $1^{16}$ |
| 60.960.29-60.mv.1.8 | $60$ | $2$ | $2$ | $29$ | $6$ | $1^{16}$ |
| 60.960.29-60.mx.1.3 | $60$ | $2$ | $2$ | $29$ | $1$ | $1^{16}$ |
| 60.960.29-60.nh.1.10 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-60.nj.1.2 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{16}$ |
| 60.960.29-60.nl.1.5 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{16}$ |
| 60.960.33-60.ov.1.12 | $60$ | $2$ | $2$ | $33$ | $8$ | $1^{20}$ |
| 60.960.33-60.ox.1.15 | $60$ | $2$ | $2$ | $33$ | $11$ | $1^{20}$ |
| 60.960.33-60.pl.1.14 | $60$ | $2$ | $2$ | $33$ | $4$ | $1^{20}$ |
| 60.960.33-60.pn.1.9 | $60$ | $2$ | $2$ | $33$ | $5$ | $1^{20}$ |
| 60.960.33-60.qb.1.13 | $60$ | $2$ | $2$ | $33$ | $7$ | $1^{20}$ |
| 60.960.33-60.qd.1.10 | $60$ | $2$ | $2$ | $33$ | $10$ | $1^{20}$ |
| 60.960.33-60.qj.1.11 | $60$ | $2$ | $2$ | $33$ | $5$ | $1^{20}$ |
| 60.960.33-60.ql.1.24 | $60$ | $2$ | $2$ | $33$ | $6$ | $1^{20}$ |
| 60.1440.37-30.c.1.13 | $60$ | $3$ | $3$ | $37$ | $0$ | $1^{24}$ |
| 60.1440.45-30.a.1.12 | $60$ | $3$ | $3$ | $45$ | $6$ | $1^{28}\cdot2^{2}$ |