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Properties

Label 60.48.1.ba.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: 60.48.1.127

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}17&46\\45&43\end{bmatrix}$, $\begin{bmatrix}23&6\\54&59\end{bmatrix}$, $\begin{bmatrix}29&0\\48&41\end{bmatrix}$, $\begin{bmatrix}53&56\\48&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.96.1-60.ba.1.1, 60.96.1-60.ba.1.2, 60.96.1-60.ba.1.3, 60.96.1-60.ba.1.4, 60.96.1-60.ba.1.5, 60.96.1-60.ba.1.6, 60.96.1-60.ba.1.7, 60.96.1-60.ba.1.8, 120.96.1-60.ba.1.1, 120.96.1-60.ba.1.2, 120.96.1-60.ba.1.3, 120.96.1-60.ba.1.4, 120.96.1-60.ba.1.5, 120.96.1-60.ba.1.6, 120.96.1-60.ba.1.7, 120.96.1-60.ba.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 $ $=$ $ 6 x y - 3 y^{2} - z^{2} $
$=$ $135 x^{2} - 18 x y - 21 y^{2} - 32 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{4} - 5 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{5}w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^4}{5}\cdot\frac{(20z^{2}+w^{2})(873600000y^{2}z^{8}+109200000y^{2}z^{6}w^{2}+684000y^{2}z^{4}w^{4}+10200y^{2}z^{2}w^{6}-2730y^{2}w^{8}-2624000000z^{10}-590480000z^{8}w^{2}-31940000z^{6}w^{4}-272200z^{4}w^{6}-910z^{2}w^{8}-w^{10})}{w^{4}z^{2}(12000y^{2}z^{4}+300y^{2}z^{2}w^{2}-15y^{2}w^{4}-4000z^{6}+300z^{4}w^{2}-30z^{2}w^{4}+4w^{6})}$

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.h.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
60.12.0.s.1 $60$ $4$ $4$ $0$ $0$ full Jacobian
60.24.0.s.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.lq.1 $60$ $3$ $3$ $5$ $1$ $1^{4}$
60.240.17.kq.1 $60$ $5$ $5$ $17$ $5$ $1^{16}$
60.288.17.gc.1 $60$ $6$ $6$ $17$ $6$ $1^{16}$
60.480.33.lg.1 $60$ $10$ $10$ $33$ $10$ $1^{32}$
180.144.5.ba.1 $180$ $3$ $3$ $5$ $?$ not computed
180.144.9.dc.1 $180$ $3$ $3$ $9$ $?$ not computed
180.144.9.de.1 $180$ $3$ $3$ $9$ $?$ not computed