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Properties

Label 60.48.1.n.1
Level $60$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $1800$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ Cusp orbits $2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.48.1.80

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}1&38\\57&37\end{bmatrix}$, $\begin{bmatrix}7&16\\21&7\end{bmatrix}$, $\begin{bmatrix}7&28\\45&23\end{bmatrix}$, $\begin{bmatrix}53&38\\21&53\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.96.1-60.n.1.1, 60.96.1-60.n.1.2, 60.96.1-60.n.1.3, 60.96.1-60.n.1.4, 60.96.1-60.n.1.5, 60.96.1-60.n.1.6, 60.96.1-60.n.1.7, 60.96.1-60.n.1.8, 120.96.1-60.n.1.1, 120.96.1-60.n.1.2, 120.96.1-60.n.1.3, 120.96.1-60.n.1.4, 120.96.1-60.n.1.5, 120.96.1-60.n.1.6, 120.96.1-60.n.1.7, 120.96.1-60.n.1.8, 120.96.1-60.n.1.9, 120.96.1-60.n.1.10, 120.96.1-60.n.1.11, 120.96.1-60.n.1.12, 120.96.1-60.n.1.13, 120.96.1-60.n.1.14, 120.96.1-60.n.1.15, 120.96.1-60.n.1.16
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $192$
Full 60-torsion field degree: $46080$

Jacobian

Conductor: $2^{3}\cdot3^{2}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1800.2.a.m

Models

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

$ 0 $ $=$ $ 9 x^{2} + 14 x y + 5 y^{2} + z^{2} $
$=$ $18 x^{2} - 2 x y - 5 y^{2} + 2 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 75 x^{4} + 3 x^{2} y^{2} - 50 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 10z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^2\cdot5}\cdot\frac{(15z^{2}-4w^{2})(27641250y^{2}z^{8}+1377000y^{2}z^{6}w^{2}-1231200y^{2}z^{4}w^{4}+2620800y^{2}z^{2}w^{6}-279552y^{2}w^{8}-151875z^{10}+1842750z^{8}w^{2}-7349400z^{6}w^{4}+11498400z^{4}w^{6}-2834304z^{2}w^{8}+167936w^{10})}{w^{2}z^{4}(3375y^{2}z^{4}+900y^{2}z^{2}w^{2}-480y^{2}w^{4}+13500z^{6}+1350z^{4}w^{2}+180z^{2}w^{4}+32w^{6})}$

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
12.24.0.f.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
60.12.0.f.1 $60$ $4$ $4$ $0$ $0$ full Jacobian
60.24.0.r.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.24.1.v.1 $60$ $2$ $2$ $1$ $1$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.144.5.ei.1 $60$ $3$ $3$ $5$ $2$ $1^{4}$
60.240.17.z.1 $60$ $5$ $5$ $17$ $3$ $1^{16}$
60.288.17.bh.1 $60$ $6$ $6$ $17$ $7$ $1^{16}$
60.480.33.cl.1 $60$ $10$ $10$ $33$ $7$ $1^{32}$
120.96.3.nm.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.nm.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.no.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.no.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ri.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ri.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rk.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rk.2 $120$ $2$ $2$ $3$ $?$ not computed
180.144.5.n.1 $180$ $3$ $3$ $5$ $?$ not computed
180.144.9.v.1 $180$ $3$ $3$ $9$ $?$ not computed
180.144.9.bd.1 $180$ $3$ $3$ $9$ $?$ not computed