Properties

Label 4.2e4_3e2_103.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{4} \cdot 3^{2} \cdot 103 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$14832= 2^{4} \cdot 3^{2} \cdot 103 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} - 2 x^{3} + x^{2} + 3 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.2e2_103.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 24 a + 1 + \left(19 a + 21\right)\cdot 31 + \left(25 a + 16\right)\cdot 31^{2} + \left(3 a + 12\right)\cdot 31^{3} + \left(8 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 a + 18 + \left(11 a + 5\right)\cdot 31 + \left(5 a + 17\right)\cdot 31^{2} + \left(27 a + 25\right)\cdot 31^{3} + \left(22 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 30 a + 7 + \left(28 a + 10\right)\cdot 31 + \left(20 a + 27\right)\cdot 31^{2} + \left(15 a + 4\right)\cdot 31^{3} + \left(22 a + 3\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ a + 5 + \left(2 a + 7\right)\cdot 31 + \left(10 a + 9\right)\cdot 31^{2} + \left(15 a + 15\right)\cdot 31^{3} + \left(8 a + 1\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 14 + 30\cdot 31 + 2\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 18 + 18\cdot 31 + 21\cdot 31^{2} + 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

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$$2$$(2,6)$$2$
$9$$2$$(2,6)(4,5)$$0$
$4$$3$$(1,2,6)(3,4,5)$$-2$
$4$$3$$(1,2,6)$$1$
$18$$4$$(1,3)(2,5,6,4)$$0$
$12$$6$$(1,4,2,5,6,3)$$0$
$12$$6$$(2,6)(3,4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.