Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $1800$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $7 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $10^{4}\cdot20^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.240.7.6 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&20\\58&37\end{bmatrix}$, $\begin{bmatrix}17&20\\10&57\end{bmatrix}$, $\begin{bmatrix}39&10\\10&39\end{bmatrix}$, $\begin{bmatrix}53&58\\46&27\end{bmatrix}$, $\begin{bmatrix}55&26\\36&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.120.7.b.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $9216$ |
Jacobian
| Conductor: | $2^{13}\cdot3^{8}\cdot5^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 450.2.a.c, 900.2.a.b, 1800.2.a.h, 1800.2.a.v |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w - x t + x v + y w + y t - z w - z t - z u + 2 z v $ |
| $=$ | $x t + 2 x u - 2 x v + y w + y t + y u - 2 y v + z u - 2 z v$ | |
| $=$ | $3 x w + x t - x u - x v + 2 y w - 2 y t - 3 z u$ | |
| $=$ | $2 x w + 2 x t - x u - 2 y w + y u + y v + 2 z w - z t - 2 z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 16 x^{10} + 108 x^{8} y^{2} + 180 x^{8} z^{2} + 12 x^{7} y^{3} - 108 x^{7} y z^{2} + \cdots + 36 y^{6} z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no 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 10.60.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+3y-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2x-y-3z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2x+y-2z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.120.7.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -16X^{10}+108X^{8}Y^{2}+180X^{8}Z^{2}+12X^{7}Y^{3}-108X^{7}YZ^{2}-268X^{6}Y^{4}-888X^{6}Y^{2}Z^{2}-1044X^{6}Z^{4}-52X^{5}Y^{5}+528X^{5}Y^{3}Z^{2}+1260X^{5}YZ^{4}+307X^{4}Y^{6}+645X^{4}Y^{4}Z^{2}+135X^{4}Y^{2}Z^{4}+945X^{4}Z^{6}+63X^{3}Y^{7}-129X^{3}Y^{5}Z^{2}+837X^{3}Y^{3}Z^{4}+189X^{3}YZ^{6}-167X^{2}Y^{8}-120X^{2}Y^{6}Z^{2}-216X^{2}Y^{4}Z^{4}+54X^{2}Y^{2}Z^{6}+81X^{2}Z^{8}-24XY^{9}+30XY^{7}Z^{2}-144XY^{5}Z^{4}+162XY^{3}Z^{6}+36Y^{10}-48Y^{8}Z^{2}+36Y^{6}Z^{4} $ |
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 |
| 12.24.0-12.b.1.2 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-12.b.1.2 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 20.120.3-10.a.1.4 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 60.120.3-10.a.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.480.13-60.b.1.3 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.d.1.1 | $60$ | $2$ | $2$ | $13$ | $6$ | $1^{6}$ |
| 60.480.13-60.j.1.3 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.l.1.2 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.z.1.1 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.bb.1.2 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 60.480.13-60.bh.1.1 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 60.480.13-60.bj.1.2 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 60.480.15-60.b.1.2 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.b.1.8 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.c.1.9 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.c.1.15 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.i.1.11 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.i.1.13 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.j.1.3 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-60.j.1.4 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-60.p.1.1 | $60$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 60.480.15-60.p.1.2 | $60$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 60.480.15-60.q.1.4 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.q.1.6 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.w.1.2 | $60$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 60.480.15-60.w.1.8 | $60$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
| 60.480.15-60.x.1.1 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.x.1.2 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.720.19-60.be.1.1 | $60$ | $3$ | $3$ | $19$ | $4$ | $1^{12}$ |
| 60.720.27-60.ct.1.2 | $60$ | $3$ | $3$ | $27$ | $10$ | $1^{20}$ |
| 60.960.33-60.p.1.2 | $60$ | $4$ | $4$ | $33$ | $6$ | $1^{26}$ |
| 120.480.13-120.e.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.k.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bc.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bi.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cy.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.de.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dw.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ec.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-120.d.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.d.1.30 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.g.1.26 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.g.1.32 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.u.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.u.1.30 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.x.1.26 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.x.1.32 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cd.1.9 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cd.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cg.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cg.1.13 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cw.1.9 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cw.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cz.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cz.1.13 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |