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Properties

Label 60.120.8.m.1
Level $60$
Index $120$
Genus $8$
Analytic rank $2$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $3600$
Index: $120$ $\PSL_2$-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 $20^{6}$ Cusp orbits $1^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-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: 20A8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.120.8.41

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}3&44\\13&41\end{bmatrix}$, $\begin{bmatrix}27&8\\8&43\end{bmatrix}$, $\begin{bmatrix}37&56\\44&33\end{bmatrix}$, $\begin{bmatrix}47&32\\31&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.240.8-60.m.1.1, 60.240.8-60.m.1.2, 60.240.8-60.m.1.3, 60.240.8-60.m.1.4, 120.240.8-60.m.1.1, 120.240.8-60.m.1.2, 120.240.8-60.m.1.3, 120.240.8-60.m.1.4, 120.240.8-60.m.1.5, 120.240.8-60.m.1.6, 120.240.8-60.m.1.7, 120.240.8-60.m.1.8, 120.240.8-60.m.1.9, 120.240.8-60.m.1.10, 120.240.8-60.m.1.11, 120.240.8-60.m.1.12
Cyclic 60-isogeny field degree: $24$
Cyclic 60-torsion field degree: $384$
Full 60-torsion field degree: $18432$

Jacobian

Conductor: $2^{22}\cdot3^{8}\cdot5^{16}$
Simple: no
Squarefree: no
Decomposition: $1^{8}$
Newforms: 50.2.a.b$^{2}$, 400.2.a.a, 400.2.a.f, 450.2.a.g, 1800.2.a.r, 3600.2.a.l, 3600.2.a.m

Models

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

$ 0 $ $=$ $ x y - x z + x t - x v + x r - y u - y v - y r + z t + z u + w t $
$=$ $x y - 2 x t + x v - x r - 2 y u - y v - y r + z^{2} + z w + 2 z u$
$=$ $3 x^{2} + x u + x v + x r + y z - y t + y v - y r - z t - z u - w u$
$=$ $x^{2} - 2 x u - x v - x r + 2 y z - 2 y t + y v - y r + z^{2} - 2 z t$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 264196 x^{12} - 83760 x^{10} y^{2} + 120764 x^{10} y z - 35984 x^{10} z^{2} + 19109 x^{8} y^{4} + \cdots + 16 y^{4} z^{8} $
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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
$(0:0:0:0:0:0:1:0)$, $(0:0:0:0:0:0:0:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

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

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 20.60.4.h.1 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle -y$
$\displaystyle Z$ $=$ $\displaystyle -z$
$\displaystyle W$ $=$ $\displaystyle -w$

Equation of the image curve:

$0$ $=$ $ 7X^{2}+Y^{2}+2ZW+W^{2} $
$=$ $ X^{3}-XY^{2}+XZ^{2}+YZ^{2}+YZW $

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
20.60.4.h.1 $20$ $2$ $2$ $4$ $1$ $1^{4}$
60.24.0.h.1 $60$ $5$ $5$ $0$ $0$ full Jacobian
60.60.4.ci.1 $60$ $2$ $2$ $4$ $2$ $1^{4}$
60.60.4.cj.1 $60$ $2$ $2$ $4$ $1$ $1^{4}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.360.22.bm.1 $60$ $3$ $3$ $22$ $6$ $1^{14}$
60.360.28.cc.1 $60$ $3$ $3$ $28$ $11$ $1^{20}$
60.480.29.ep.1 $60$ $4$ $4$ $29$ $9$ $1^{21}$
60.480.35.ce.1 $60$ $4$ $4$ $35$ $10$ $1^{27}$
120.240.17.nk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.no.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.pg.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.pn.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.ra.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.rd.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.rq.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.rx.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.tl.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.tu.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.ur.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.uw.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.vl.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.vu.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.wo.1 $120$ $2$ $2$ $17$ $?$ not computed
120.240.17.ws.1 $120$ $2$ $2$ $17$ $?$ not computed