Invariants
| Level: | $60$ | $\SL_2$-level: | $60$ | Newform level: | $1200$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $5^{2}\cdot15^{2}\cdot20\cdot60$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 60C8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.240.8.55 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}17&3\\6&41\end{bmatrix}$, $\begin{bmatrix}19&36\\54&49\end{bmatrix}$, $\begin{bmatrix}35&1\\42&35\end{bmatrix}$, $\begin{bmatrix}37&55\\30&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.120.8.n.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $9216$ |
Jacobian
| Conductor: | $2^{18}\cdot3^{4}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 50.2.a.b$^{2}$, 75.2.a.a$^{2}$, 400.2.a.f$^{2}$, 1200.2.a.c, 1200.2.a.n |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x v + y z $ |
| $=$ | $x v + z^{2} - w v + v^{2}$ | |
| $=$ | $x y + w u + t v + t r$ | |
| $=$ | $x y - x z - y w + y v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} y^{2} + 12 x^{6} y^{4} + 5 x^{5} y^{4} z - 10 x^{5} y^{2} z^{3} + x^{5} z^{5} + 30 x^{4} y^{6} + \cdots + 9 y^{10} $ |
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 |
|---|
| $(1:0:0:0:0:0:0:0)$, $(0:0:0:-1:0:0:-1:1)$ |
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 30.60.4.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t$ |
| $\displaystyle W$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}-4XY+XZ+3YZ-XW+2W^{2} $ |
| $=$ | $ X^{3}-X^{2}Y+X^{2}Z-2XYZ-Y^{2}Z+YZ^{2}-2X^{2}W-XYW-XZW+XW^{2}+ZW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.120.8.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{8}Y^{2}+12X^{6}Y^{4}+5X^{5}Y^{4}Z-10X^{5}Y^{2}Z^{3}+X^{5}Z^{5}+30X^{4}Y^{6}+30X^{3}Y^{6}Z-60X^{3}Y^{4}Z^{3}+6X^{3}Y^{2}Z^{5}+36X^{2}Y^{8}-80X^{2}Y^{6}Z^{2}+40X^{2}Y^{4}Z^{4}-15XY^{8}Z+30XY^{6}Z^{3}-3XY^{4}Z^{5}+9Y^{10} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.f.1.3 | $12$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.f.1.3 | $12$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 30.120.4-30.b.1.1 | $30$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 60.120.4-30.b.1.2 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.480.17-60.a.1.24 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{9}$ |
| 60.480.17-60.i.1.8 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{9}$ |
| 60.480.17-60.q.1.6 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{9}$ |
| 60.480.17-60.r.1.5 | $60$ | $2$ | $2$ | $17$ | $6$ | $1^{9}$ |
| 60.480.17-60.u.1.6 | $60$ | $2$ | $2$ | $17$ | $2$ | $1^{9}$ |
| 60.480.17-60.v.1.3 | $60$ | $2$ | $2$ | $17$ | $5$ | $1^{9}$ |
| 60.480.17-60.y.1.1 | $60$ | $2$ | $2$ | $17$ | $6$ | $1^{9}$ |
| 60.480.17-60.z.1.3 | $60$ | $2$ | $2$ | $17$ | $3$ | $1^{9}$ |
| 60.720.22-60.bp.1.9 | $60$ | $3$ | $3$ | $22$ | $2$ | $1^{14}$ |
| 60.720.25-60.br.1.2 | $60$ | $3$ | $3$ | $25$ | $7$ | $1^{17}$ |
| 60.960.29-60.fo.1.11 | $60$ | $4$ | $4$ | $29$ | $5$ | $1^{21}$ |
| 120.480.16-120.fe.1.28 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.fe.2.27 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.ff.1.20 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.ff.2.19 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.fi.1.21 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.fi.2.17 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.fj.1.19 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.fj.2.17 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.17-120.is.1.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.nz.1.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bpq.1.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bpt.1.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bqc.1.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bqf.1.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bqo.1.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bqr.1.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.18-120.i.1.2 | $120$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 120.480.18-120.i.2.6 | $120$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 120.480.18-120.j.1.2 | $120$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 120.480.18-120.j.2.4 | $120$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 120.480.18-120.m.1.5 | $120$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 120.480.18-120.m.2.7 | $120$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 120.480.18-120.n.1.1 | $120$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 120.480.18-120.n.2.3 | $120$ | $2$ | $2$ | $18$ | $?$ | not computed |