Properties

Label 4.5e2_409.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 5^{2} \cdot 409 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$10225= 5^{2} \cdot 409 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.409.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 35 + 32\cdot 109 + 72\cdot 109^{2} + 55\cdot 109^{3} + 82\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 98 a + 49 + \left(47 a + 68\right)\cdot 109 + \left(80 a + 78\right)\cdot 109^{2} + \left(48 a + 91\right)\cdot 109^{3} + \left(80 a + 73\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 71 a + 2 + \left(80 a + 88\right)\cdot 109 + \left(33 a + 41\right)\cdot 109^{2} + \left(74 a + 6\right)\cdot 109^{3} + \left(44 a + 28\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 23 + 22\cdot 109 + 28\cdot 109^{2} + 66\cdot 109^{3} + 38\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 a + 38 + \left(61 a + 18\right)\cdot 109 + \left(28 a + 2\right)\cdot 109^{2} + \left(60 a + 60\right)\cdot 109^{3} + \left(28 a + 105\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 38 a + 73 + \left(28 a + 97\right)\cdot 109 + \left(75 a + 103\right)\cdot 109^{2} + \left(34 a + 46\right)\cdot 109^{3} + \left(64 a + 107\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$

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

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

Character values on conjugacy classes

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