Properties

Label 4.3e2_853.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 3^{2} \cdot 853 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$7677= 3^{2} \cdot 853 $
Artin number field: Splitting field of $f= x^{6} + x^{4} - x^{3} + x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.853.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 26 a + \left(16 a + 7\right)\cdot 31 + \left(21 a + 8\right)\cdot 31^{2} + \left(4 a + 4\right)\cdot 31^{3} + \left(4 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 a + 21 + \left(14 a + 14\right)\cdot 31 + \left(9 a + 3\right)\cdot 31^{2} + \left(26 a + 23\right)\cdot 31^{3} + \left(26 a + 11\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 + 28\cdot 31 + 13\cdot 31^{2} + 9\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 15 a + 6 + \left(12 a + 27\right)\cdot 31 + \left(3 a + 26\right)\cdot 31^{2} + \left(20 a + 7\right)\cdot 31^{3} + \left(2 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 10 + 9\cdot 31 + 19\cdot 31^{2} + 3\cdot 31^{3} + 11\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 a + 5 + \left(18 a + 6\right)\cdot 31 + \left(27 a + 21\right)\cdot 31^{2} + \left(10 a + 13\right)\cdot 31^{3} + \left(28 a + 3\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,3)(2,4)(5,6)$$0$
$6$$2$$(2,5)$$2$
$9$$2$$(2,5)(4,6)$$0$
$4$$3$$(1,2,5)(3,4,6)$$-2$
$4$$3$$(1,2,5)$$1$
$18$$4$$(1,3)(2,6,5,4)$$0$
$12$$6$$(1,4,2,6,5,3)$$0$
$12$$6$$(2,5)(3,4,6)$$-1$
The blue line marks the conjugacy class containing complex conjugation.