Properties

Label 1.37.6t1.a.a
Dimension $1$
Group $C_6$
Conductor $37$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(37\)
Artin field: Galois closure of 6.6.69343957.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{37}(27,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 15x^{4} + 28x^{3} + 15x^{2} - 38x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 22 a + 17 + \left(16 a + 5\right)\cdot 23 + \left(21 a + 19\right)\cdot 23^{2} + \left(10 a + 4\right)\cdot 23^{3} + \left(14 a + 12\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 18 + \left(2 a + 13\right)\cdot 23 + \left(13 a + 8\right)\cdot 23^{2} + 19 a\cdot 23^{3} + \left(15 a + 3\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 a + 19 + \left(20 a + 6\right)\cdot 23 + \left(9 a + 9\right)\cdot 23^{2} + \left(3 a + 3\right)\cdot 23^{3} + \left(7 a + 15\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 15 a + 20 + \left(18 a + 12\right)\cdot 23 + \left(18 a + 6\right)\cdot 23^{2} + \left(12 a + 1\right)\cdot 23^{3} + \left(12 a + 21\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 8 a + 4 + \left(4 a + 12\right)\cdot 23 + \left(4 a + 2\right)\cdot 23^{2} + \left(10 a + 8\right)\cdot 23^{3} + \left(10 a + 10\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( a + 15 + \left(6 a + 17\right)\cdot 23 + \left(a + 22\right)\cdot 23^{2} + \left(12 a + 4\right)\cdot 23^{3} + \left(8 a + 7\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2,5,6,3,4)$
$(1,6)(2,3)(4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,6)(2,3)(4,5)$$-1$
$1$$3$$(1,5,3)(2,6,4)$$\zeta_{3}$
$1$$3$$(1,3,5)(2,4,6)$$-\zeta_{3} - 1$
$1$$6$$(1,2,5,6,3,4)$$\zeta_{3} + 1$
$1$$6$$(1,4,3,6,5,2)$$-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.