Properties

Label 1.148.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $148$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(148\)\(\medspace = 2^{2} \cdot 37 \)
Artin field: Galois closure of 6.0.4438013248.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{148}(11,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} + 37x^{4} + 74x^{2} + 37 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 6 a + 14 + \left(13 a + 13\right)\cdot 29 + \left(26 a + 27\right)\cdot 29^{2} + \left(8 a + 19\right)\cdot 29^{3} + \left(9 a + 24\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 26 + \left(19 a + 27\right)\cdot 29 + \left(7 a + 4\right)\cdot 29^{2} + \left(15 a + 9\right)\cdot 29^{3} + \left(27 a + 11\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 a + 11 + \left(12 a + 4\right)\cdot 29 + \left(12 a + 4\right)\cdot 29^{2} + \left(6 a + 19\right)\cdot 29^{3} + \left(18 a + 15\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 23 a + 15 + \left(15 a + 15\right)\cdot 29 + \left(2 a + 1\right)\cdot 29^{2} + \left(20 a + 9\right)\cdot 29^{3} + \left(19 a + 4\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 a + 3 + \left(9 a + 1\right)\cdot 29 + \left(21 a + 24\right)\cdot 29^{2} + \left(13 a + 19\right)\cdot 29^{3} + \left(a + 17\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 16 a + 18 + \left(16 a + 24\right)\cdot 29 + \left(16 a + 24\right)\cdot 29^{2} + \left(22 a + 9\right)\cdot 29^{3} + \left(10 a + 13\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.