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: | 60.288.7.2810 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&0\\54&17\end{bmatrix}$, $\begin{bmatrix}17&0\\0&13\end{bmatrix}$, $\begin{bmatrix}23&25\\24&23\end{bmatrix}$, $\begin{bmatrix}29&10\\18&53\end{bmatrix}$, $\begin{bmatrix}49&10\\18&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.144.7.e.2 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 $ |
| $=$ | $x^{2} + x u + z w$ | |
| $=$ | $x y - x w + y u$ | |
| $=$ | $x v + z w + w^{2} - 2 w t + w u + w v + u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 4 x^{10} + 4 x^{9} y + 4 x^{8} y^{2} - 5 x^{8} y z + 3 x^{7} y^{3} + 12 x^{7} y^{2} z + \cdots + 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 |
|---|
| $(0:0:0:0:1:0:1)$, $(0:0:0:0:0:0:1)$, $(0:0:0:-1:0:1:0)$, $(0:0:0:1:1:1:0)$ |
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.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 5x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -8x-y-2z+w+3t-4u-2v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -6x-2y+z+2w+t+2u+v$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{4}-2X^{3}Y+3X^{2}Y^{2}+2XY^{3}-10X^{3}Z+12X^{2}YZ+12XY^{2}Z+2Y^{3}Z-9X^{2}Z^{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.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -4X^{10}+4X^{9}Y+4X^{8}Y^{2}-5X^{8}YZ+3X^{7}Y^{3}+12X^{7}Y^{2}Z-10X^{6}Y^{4}-7X^{6}Y^{3}Z-6X^{6}Y^{2}Z^{2}-X^{5}Y^{5}+5X^{5}Y^{4}Z+9X^{5}Y^{3}Z^{2}+4X^{4}Y^{6}-2X^{4}Y^{3}Z^{3}-5X^{3}Y^{6}Z+3X^{3}Y^{5}Z^{2}+2X^{3}Y^{4}Z^{3}+6X^{2}Y^{6}Z^{2}+2X^{2}Y^{5}Z^{3}-X^{2}Y^{4}Z^{4}-2XY^{6}Z^{3}+Y^{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.1.29 | $60$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
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.1.7 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-30.c.2.6 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-30.f.2.6 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-30.g.2.18 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2$ |
| 60.576.13-60.ge.2.16 | $60$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2$ |
| 60.576.13-60.jw.2.16 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.ng.2.12 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2$ |
| 60.576.13-60.ns.2.16 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{4}\cdot2$ |
| 60.576.17-60.bl.2.1 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.bn.2.11 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.ed.2.9 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.eg.2.11 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.gv.2.2 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.gx.2.4 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.hg.2.10 | $60$ | $2$ | $2$ | $17$ | $4$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.hj.2.12 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 60.576.17-60.js.2.5 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.jv.2.7 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kb.2.13 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kd.2.15 | $60$ | $2$ | $2$ | $17$ | $4$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.ki.2.6 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kl.2.8 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kr.2.14 | $60$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 60.576.17-60.kt.2.24 | $60$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.864.25-30.q.2.5 | $60$ | $3$ | $3$ | $25$ | $0$ | $1^{8}\cdot2^{5}$ |
| 60.1440.43-30.q.1.6 | $60$ | $5$ | $5$ | $43$ | $0$ | $1^{16}\cdot2^{10}$ |