Properties

Label 4.2e6_3e2_41.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{6} \cdot 3^{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:$23616= 2^{6} \cdot 3^{2} \cdot 41 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} + 2 x^{3} - 4 x^{2} + 4 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:
$r_{ 1 }$ $=$ $ 9 + 18\cdot 31 + 28\cdot 31^{2} + 20\cdot 31^{3} + 18\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 + 9\cdot 31 + 5\cdot 31^{2} + 5\cdot 31^{3} + 30\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ a + 26 + \left(19 a + 18\right)\cdot 31 + \left(20 a + 5\right)\cdot 31^{2} + \left(12 a + 18\right)\cdot 31^{3} + \left(11 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 15 + \left(12 a + 17\right)\cdot 31 + \left(11 a + 7\right)\cdot 31^{2} + \left(a + 17\right)\cdot 31^{3} + \left(2 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 30 a + 28 + \left(11 a + 24\right)\cdot 31 + \left(10 a + 27\right)\cdot 31^{2} + \left(18 a + 22\right)\cdot 31^{3} + \left(19 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 24 a + 29 + \left(18 a + 3\right)\cdot 31 + \left(19 a + 18\right)\cdot 31^{2} + \left(29 a + 8\right)\cdot 31^{3} + \left(28 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

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