Properties

Label 4.2e4_5e2_41.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{4} \cdot 5^{2} \cdot 41 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$16400= 2^{4} \cdot 5^{2} \cdot 41 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - x^{3} + 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: 1.41.2t1.1c1

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:
$r_{ 1 }$ $=$ $ a + 25 + \left(23 a + 16\right)\cdot 31 + \left(10 a + 8\right)\cdot 31^{2} + \left(25 a + 15\right)\cdot 31^{3} + \left(13 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 30 a + 27 + \left(7 a + 30\right)\cdot 31 + \left(20 a + 6\right)\cdot 31^{2} + \left(5 a + 24\right)\cdot 31^{3} + \left(17 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 10 a + 4 + \left(4 a + 6\right)\cdot 31 + \left(23 a + 29\right)\cdot 31^{2} + \left(8 a + 5\right)\cdot 31^{3} + \left(21 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 21 a + 24 + \left(26 a + 4\right)\cdot 31 + \left(7 a + 9\right)\cdot 31^{2} + 22 a\cdot 31^{3} + \left(9 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 + 13\cdot 31 + 11\cdot 31^{2} + 24\cdot 31^{3} + 13\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 23 + 20\cdot 31 + 27\cdot 31^{2} + 22\cdot 31^{3} + 10\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.