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Properties

Label 60.120.8.o.1
Level $60$
Index $120$
Genus $8$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $6$

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Invariants

Level: $60$ $\SL_2$-level: $60$ Newform level: $300$
Index: $120$ $\PSL_2$-index:$120$
Genus: $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $5^{2}\cdot15^{2}\cdot20\cdot60$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3 \le \gamma \le 5$
$\overline{\Q}$-gonality: $3 \le \gamma \le 5$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 60C8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.120.8.1

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}13&18\\48&47\end{bmatrix}$, $\begin{bmatrix}25&26\\24&55\end{bmatrix}$, $\begin{bmatrix}25&31\\24&55\end{bmatrix}$, $\begin{bmatrix}29&23\\24&47\end{bmatrix}$, $\begin{bmatrix}43&47\\12&17\end{bmatrix}$, $\begin{bmatrix}59&27\\24&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.240.8-60.o.1.1, 60.240.8-60.o.1.2, 60.240.8-60.o.1.3, 60.240.8-60.o.1.4, 60.240.8-60.o.1.5, 60.240.8-60.o.1.6, 60.240.8-60.o.1.7, 60.240.8-60.o.1.8, 60.240.8-60.o.1.9, 60.240.8-60.o.1.10, 60.240.8-60.o.1.11, 60.240.8-60.o.1.12, 60.240.8-60.o.1.13, 60.240.8-60.o.1.14, 60.240.8-60.o.1.15, 60.240.8-60.o.1.16, 60.240.8-60.o.1.17, 60.240.8-60.o.1.18, 60.240.8-60.o.1.19, 60.240.8-60.o.1.20, 60.240.8-60.o.1.21, 60.240.8-60.o.1.22, 60.240.8-60.o.1.23, 60.240.8-60.o.1.24, 120.240.8-60.o.1.1, 120.240.8-60.o.1.2, 120.240.8-60.o.1.3, 120.240.8-60.o.1.4, 120.240.8-60.o.1.5, 120.240.8-60.o.1.6, 120.240.8-60.o.1.7, 120.240.8-60.o.1.8, 120.240.8-60.o.1.9, 120.240.8-60.o.1.10, 120.240.8-60.o.1.11, 120.240.8-60.o.1.12, 120.240.8-60.o.1.13, 120.240.8-60.o.1.14, 120.240.8-60.o.1.15, 120.240.8-60.o.1.16, 120.240.8-60.o.1.17, 120.240.8-60.o.1.18, 120.240.8-60.o.1.19, 120.240.8-60.o.1.20, 120.240.8-60.o.1.21, 120.240.8-60.o.1.22, 120.240.8-60.o.1.23, 120.240.8-60.o.1.24, 120.240.8-60.o.1.25, 120.240.8-60.o.1.26, 120.240.8-60.o.1.27, 120.240.8-60.o.1.28, 120.240.8-60.o.1.29, 120.240.8-60.o.1.30, 120.240.8-60.o.1.31, 120.240.8-60.o.1.32, 120.240.8-60.o.1.33, 120.240.8-60.o.1.34, 120.240.8-60.o.1.35, 120.240.8-60.o.1.36, 120.240.8-60.o.1.37, 120.240.8-60.o.1.38, 120.240.8-60.o.1.39, 120.240.8-60.o.1.40, 120.240.8-60.o.1.41, 120.240.8-60.o.1.42, 120.240.8-60.o.1.43, 120.240.8-60.o.1.44, 120.240.8-60.o.1.45, 120.240.8-60.o.1.46, 120.240.8-60.o.1.47, 120.240.8-60.o.1.48, 120.240.8-60.o.1.49, 120.240.8-60.o.1.50, 120.240.8-60.o.1.51, 120.240.8-60.o.1.52, 120.240.8-60.o.1.53, 120.240.8-60.o.1.54, 120.240.8-60.o.1.55, 120.240.8-60.o.1.56
Cyclic 60-isogeny field degree: $6$
Cyclic 60-torsion field degree: $96$
Full 60-torsion field degree: $18432$

Jacobian

Conductor: $2^{6}\cdot3^{4}\cdot5^{16}$
Simple: no
Squarefree: no
Decomposition: $1^{8}$
Newforms: 50.2.a.b$^{4}$, 75.2.a.a$^{3}$, 300.2.a.b

Models

Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations

$ 0 $ $=$ $ x u + y z $
$=$ $x u + z^{2} - w u$
$=$ $x y - x z - y w$
$=$ $y u - z r - t u - u v$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{8} y^{5} - 5 x^{8} y^{4} z + 10 x^{8} y^{3} z^{2} - 10 x^{8} y^{2} z^{3} + 5 x^{8} y z^{4} + \cdots + 3 y^{8} z^{5} $
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Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(1:0:0:0:0:0:0:0)$, $(-3:-1:-3/2:3/2:2:1/2:0:1)$, $(0:0:-1:1:0:1:0:0)$, $(0:0:0:0:0:-1:0:1)$, $(0:0:1:1:0:1:0:0)$, $(-3:1:3/2:3/2:-2:1/2:0:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle z$

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 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle -z$
$\displaystyle Z$ $=$ $\displaystyle -t$
$\displaystyle W$ $=$ $\displaystyle v$

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} $

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(3)$ $3$ $30$ $30$ $0$ $0$ full Jacobian
$X_0(4)$ $4$ $20$ $20$ $0$ $0$ full Jacobian
$X_{S_4}(5)$ $5$ $24$ $24$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_0(12)$ $12$ $5$ $5$ $0$ $0$ full Jacobian
20.30.2.c.1 $20$ $4$ $4$ $2$ $0$ $1^{6}$
30.60.4.b.1 $30$ $2$ $2$ $4$ $0$ $1^{4}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.240.16.e.1 $60$ $2$ $2$ $16$ $0$ $8$
60.240.16.e.2 $60$ $2$ $2$ $16$ $0$ $8$
60.240.16.e.3 $60$ $2$ $2$ $16$ $0$ $8$
60.240.16.e.4 $60$ $2$ $2$ $16$ $0$ $8$
60.240.16.f.1 $60$ $2$ $2$ $16$ $0$ $8$
60.240.16.f.2 $60$ $2$ $2$ $16$ $0$ $8$
60.240.16.f.3 $60$ $2$ $2$ $16$ $0$ $8$
60.240.16.f.4 $60$ $2$ $2$ $16$ $0$ $8$
60.240.17.b.1 $60$ $2$ $2$ $17$ $1$ $1^{9}$
60.240.17.p.1 $60$ $2$ $2$ $17$ $3$ $1^{9}$
60.240.17.s.1 $60$ $2$ $2$ $17$ $2$ $1^{9}$
60.240.17.t.1 $60$ $2$ $2$ $17$ $4$ $1^{9}$
60.240.17.w.1 $60$ $2$ $2$ $17$ $1$ $1^{9}$
60.240.17.x.1 $60$ $2$ $2$ $17$ $4$ $1^{9}$
60.240.17.ba.1 $60$ $2$ $2$ $17$ $3$ $1^{9}$
60.240.17.bb.1 $60$ $2$ $2$ $17$ $4$ $1^{9}$
60.360.22.bq.1 $60$ $3$ $3$ $22$ $0$ $1^{14}$
60.360.25.bt.1 $60$ $3$ $3$ $25$ $4$ $1^{17}$
60.480.29.fq.1 $60$ $4$ $4$ $29$ $1$ $1^{21}$
120.240.16.fg.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fg.2 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fh.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fh.2 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fk.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fk.2 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fl.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fl.2 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fm.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fm.2 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fm.3 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fm.4 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fn.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fn.2 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fn.3 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fn.4 $120$ $2$ $2$ $16$ $?$ not computed
120.240.17.it.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.qw.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bpw.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bpz.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bqi.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bql.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bqu.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bqx.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bra.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brb.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brc.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brd.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bre.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brf.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brg.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brh.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bri.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brj.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brl.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brm.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brn.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.bro.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.brp.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.18.k.1 $120$ $2$ $2$ $18$ $?$ not computed
120.240.18.k.2 $120$ $2$ $2$ $18$ $?$ not computed
120.240.18.l.1 $120$ $2$ $2$ $18$ $?$ not computed
120.240.18.l.2 $120$ $2$ $2$ $18$ $?$ not computed
120.240.18.o.1 $120$ $2$ $2$ $18$ $?$ not computed
120.240.18.o.2 $120$ $2$ $2$ $18$ $?$ not computed
120.240.18.p.1 $120$ $2$ $2$ $18$ $?$ not computed
120.240.18.p.2 $120$ $2$ $2$ $18$ $?$ not computed