Properties

Label 4.2e2_7e2_53.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{2} \cdot 7^{2} \cdot 53 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$10388= 2^{2} \cdot 7^{2} \cdot 53 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - x^{3} + 2 x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.53.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $ x^{2} + 7 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 9 a + 5 + \left(3 a + 8\right)\cdot 11 + \left(2 a + 1\right)\cdot 11^{2} + a\cdot 11^{3} + \left(a + 3\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 + 5\cdot 11 + 9\cdot 11^{2} + 11^{3} + 10\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 8 a + 5 a\cdot 11 + 2\cdot 11^{2} + \left(a + 10\right)\cdot 11^{3} + \left(2 a + 4\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 10 + \left(5 a + 3\right)\cdot 11 + \left(10 a + 9\right)\cdot 11^{2} + \left(9 a + 2\right)\cdot 11^{3} + \left(8 a + 1\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 3 + 3\cdot 11^{2} + 4\cdot 11^{3} + 7\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 2 a + 8 + \left(7 a + 3\right)\cdot 11 + \left(8 a + 7\right)\cdot 11^{2} + \left(9 a + 2\right)\cdot 11^{3} + \left(9 a + 6\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$

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

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

Character values on conjugacy classes

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