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Properties

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

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $600$
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: $0$
$\Q$-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: 60.48.1.123

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}9&22\\10&33\end{bmatrix}$, $\begin{bmatrix}41&12\\16&43\end{bmatrix}$, $\begin{bmatrix}49&18\\29&59\end{bmatrix}$, $\begin{bmatrix}57&4\\58&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.96.1-60.bc.1.1, 60.96.1-60.bc.1.2, 60.96.1-60.bc.1.3, 60.96.1-60.bc.1.4, 60.96.1-60.bc.1.5, 60.96.1-60.bc.1.6, 60.96.1-60.bc.1.7, 60.96.1-60.bc.1.8, 120.96.1-60.bc.1.1, 120.96.1-60.bc.1.2, 120.96.1-60.bc.1.3, 120.96.1-60.bc.1.4, 120.96.1-60.bc.1.5, 120.96.1-60.bc.1.6, 120.96.1-60.bc.1.7, 120.96.1-60.bc.1.8
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $192$
Full 60-torsion field degree: $46080$

Jacobian

Conductor: $2^{3}\cdot3\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 600.2.a.h

Models

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

$ 0 $ $=$ $ 4 x^{2} - y^{2} - z^{2} $
$=$ $x^{2} - 4 y^{2} + z^{2} + w^{2}$
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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 to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^4}{5}\cdot\frac{(5z^{2}+4w^{2})(682500xyz^{8}-102000xyz^{6}w^{2}-273600xyz^{4}w^{4}-1747200xyz^{2}w^{6}-559104xyw^{8}-228125z^{10}-102500z^{8}w^{2}-164000z^{6}w^{4}-649600z^{4}w^{6}-467200z^{2}w^{8}-74752w^{10})}{w^{2}z^{4}(750xyz^{4}-600xyz^{2}w^{2}-960xyw^{4}+250z^{6}-75z^{4}w^{2}+480z^{2}w^{4}+128w^{6})}$

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.i.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
60.12.0.u.1 $60$ $4$ $4$ $0$ $0$ full Jacobian
60.24.0.r.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.24.1.x.1 $60$ $2$ $2$ $1$ $0$ 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.lk.1 $60$ $3$ $3$ $5$ $1$ $1^{4}$
60.240.17.ks.1 $60$ $5$ $5$ $17$ $5$ $1^{16}$
60.288.17.ge.1 $60$ $6$ $6$ $17$ $6$ $1^{16}$
60.480.33.li.1 $60$ $10$ $10$ $33$ $10$ $1^{32}$
180.144.5.bc.1 $180$ $3$ $3$ $5$ $?$ not computed
180.144.9.dg.1 $180$ $3$ $3$ $9$ $?$ not computed
180.144.9.dk.1 $180$ $3$ $3$ $9$ $?$ not computed