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Properties

Label 60.96.1.j.2
Level $60$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $72$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.96.1.480

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}34&23\\27&38\end{bmatrix}$, $\begin{bmatrix}34&33\\3&8\end{bmatrix}$, $\begin{bmatrix}38&47\\27&22\end{bmatrix}$, $\begin{bmatrix}56&39\\51&28\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.192.1-60.j.2.1, 60.192.1-60.j.2.2, 60.192.1-60.j.2.3, 60.192.1-60.j.2.4, 60.192.1-60.j.2.5, 60.192.1-60.j.2.6, 60.192.1-60.j.2.7, 60.192.1-60.j.2.8, 120.192.1-60.j.2.1, 120.192.1-60.j.2.2, 120.192.1-60.j.2.3, 120.192.1-60.j.2.4, 120.192.1-60.j.2.5, 120.192.1-60.j.2.6, 120.192.1-60.j.2.7, 120.192.1-60.j.2.8, 120.192.1-60.j.2.9, 120.192.1-60.j.2.10, 120.192.1-60.j.2.11, 120.192.1-60.j.2.12, 120.192.1-60.j.2.13, 120.192.1-60.j.2.14, 120.192.1-60.j.2.15, 120.192.1-60.j.2.16, 120.192.1-60.j.2.17, 120.192.1-60.j.2.18, 120.192.1-60.j.2.19, 120.192.1-60.j.2.20, 120.192.1-60.j.2.21, 120.192.1-60.j.2.22, 120.192.1-60.j.2.23, 120.192.1-60.j.2.24
Cyclic 60-isogeny field degree: $6$
Cyclic 60-torsion field degree: $96$
Full 60-torsion field degree: $23040$

Jacobian

Conductor: $2^{3}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 72.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 5 x^{2} - 5 y^{2} + z^{2} $
$=$ $5 x^{2} - 30 x y + 10 y^{2} + z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 25 x^{4} - 15 x^{2} y^{2} - 10 x^{2} z^{2} - 3 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{3}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3}\cdot\frac{(3z^{2}-w^{2})^{3}(17911530y^{2}z^{16}+52225560y^{2}z^{14}w^{2}+44702280y^{2}z^{12}w^{4}+230276520y^{2}z^{10}w^{6}+33791580y^{2}z^{8}w^{8}+25586280y^{2}z^{6}w^{10}+551880y^{2}z^{4}w^{12}+71640y^{2}z^{2}w^{14}+2730y^{2}w^{16}-4782969z^{18}-14348907z^{16}w^{2}-21423852z^{14}w^{4}-19522620z^{12}w^{6}-11773350z^{10}w^{8}-3445578z^{8}w^{10}-1093068z^{6}w^{12}+37044z^{4}w^{14}+1239z^{2}w^{16}+61w^{18})}{w^{2}z^{2}(3z^{2}+w^{2})^{4}(3645y^{2}z^{10}+8505y^{2}z^{8}w^{2}-10530y^{2}z^{6}w^{4}+3510y^{2}z^{4}w^{6}-315y^{2}z^{2}w^{8}-15y^{2}w^{10}+4941z^{8}w^{4}+864z^{6}w^{6}+378z^{4}w^{8}+24z^{2}w^{10}+w^{12})}$

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
12.48.1.k.1 $12$ $2$ $2$ $1$ $0$ dimension zero
60.48.0.c.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0.c.2 $60$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.288.9.bm.2 $60$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
60.480.33.bv.2 $60$ $5$ $5$ $33$ $2$ $1^{16}\cdot8^{2}$
60.576.33.ei.2 $60$ $6$ $6$ $33$ $2$ $1^{16}\cdot8^{2}$
60.960.65.hh.1 $60$ $10$ $10$ $65$ $6$ $1^{32}\cdot8^{4}$
120.192.5.vk.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.vk.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.vl.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.vl.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wc.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wc.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wd.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wd.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wp.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wp.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ww.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ww.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xf.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xf.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xi.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xi.4 $120$ $2$ $2$ $5$ $?$ not computed
180.288.9.j.1 $180$ $3$ $3$ $9$ $?$ not computed
180.288.17.s.1 $180$ $3$ $3$ $17$ $?$ not computed
180.288.17.t.1 $180$ $3$ $3$ $17$ $?$ not computed