Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $180$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $2^{3}\cdot6^{3}\cdot10^{3}\cdot30^{3}$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30S7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.7.2834 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&15\\10&13\end{bmatrix}$, $\begin{bmatrix}23&30\\42&11\end{bmatrix}$, $\begin{bmatrix}47&15\\42&23\end{bmatrix}$, $\begin{bmatrix}53&45\\40&17\end{bmatrix}$, $\begin{bmatrix}59&15\\48&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.144.7.e.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $2$ |
| Cyclic 60-torsion field degree: | $32$ |
| Full 60-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{7}\cdot3^{11}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 15.2.a.a$^{2}$, 30.2.a.a, 90.2.c.a, 180.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} - y z $ |
| $=$ | $x^{2} + x z + x w - x t + z w$ | |
| $=$ | $x^{2} + y t - y v - w u$ | |
| $=$ | $x^{2} + x y + x w + y w - y t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{10} - 5 x^{9} y - 4 x^{8} y^{2} - 7 x^{8} y z - 3 x^{7} y^{3} - 18 x^{7} y^{2} z + \cdots + 5 y^{6} z^{4} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1:0:0:0)$, $(0:0:0:0:0:1:1)$, $(0:0:0:0:0:1:0)$, $(0:0:0:1/3:2/3:0:1)$ |
Maps to other modular curves
Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 30.48.3.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x+w-t+v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y-z-2w+3t+u-v$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{4}-10X^{3}Y-9X^{2}Y^{2}-2XY^{3}-2X^{3}Z+12X^{2}YZ-2Y^{3}Z+3X^{2}Z^{2}+12XYZ^{2}+3Y^{2}Z^{2}+2XZ^{3}+2YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 30.144.7.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{10}-5X^{9}Y-4X^{8}Y^{2}-7X^{8}YZ-3X^{7}Y^{3}-18X^{7}Y^{2}Z-2X^{6}Y^{4}-11X^{6}Y^{3}Z-12X^{6}Y^{2}Z^{2}+2X^{5}Y^{5}-X^{5}Y^{4}Z-21X^{5}Y^{3}Z^{2}+2X^{4}Y^{6}+6X^{4}Y^{5}Z-10X^{4}Y^{3}Z^{3}+7X^{3}Y^{6}Z+9X^{3}Y^{5}Z^{2}-10X^{3}Y^{4}Z^{3}+12X^{2}Y^{6}Z^{2}+10X^{2}Y^{5}Z^{3}-5X^{2}Y^{4}Z^{4}+10XY^{6}Z^{3}+5Y^{6}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.3-30.a.1.9 | $60$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 60.144.3-30.a.1.26 | $60$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.576.13-30.b.2.7 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-30.c.1.9 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-30.f.1.7 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-30.g.1.13 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2$ |
| 60.576.13-60.ge.1.16 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.jw.1.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.ng.1.12 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.ns.1.16 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{4}\cdot2$ |
| 60.576.17-60.bl.1.1 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.bn.1.11 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.ed.1.9 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.eg.1.11 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.gv.1.2 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.gx.1.4 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.hg.1.10 | $60$ | $2$ | $2$ | $17$ | $4$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.hj.1.12 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.js.1.3 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.jv.1.6 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kb.1.11 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kd.1.14 | $60$ | $2$ | $2$ | $17$ | $4$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.ki.1.6 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kl.1.8 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kr.1.14 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kt.1.24 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.864.25-30.q.1.5 | $60$ | $3$ | $3$ | $25$ | $0$ | $1^{8}\cdot2^{5}$ |
| 60.1440.43-30.q.1.6 | $60$ | $5$ | $5$ | $43$ | $0$ | $1^{16}\cdot2^{10}$ |