Properties

Label 4.2e4_5_149.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{4} \cdot 5 \cdot 149 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$11920= 2^{4} \cdot 5 \cdot 149 $
Artin number field: Splitting field of $f= x^{6} + x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.5_149.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 5\cdot 13 + 3\cdot 13^{2} + 2\cdot 13^{3} + 12\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 1 + 7\cdot 13 + \left(11 a + 11\right)\cdot 13^{2} + \left(3 a + 1\right)\cdot 13^{3} + \left(7 a + 6\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 9 + 5\cdot 13^{2} + 3\cdot 13^{3} + 10\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 5 a + 3 + \left(9 a + 8\right)\cdot 13 + \left(7 a + 5\right)\cdot 13^{2} + \left(10 a + 10\right)\cdot 13^{3} + \left(a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 8 + \left(3 a + 12\right)\cdot 13 + \left(5 a + 3\right)\cdot 13^{2} + 2 a\cdot 13^{3} + \left(11 a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 11 a + 3 + \left(12 a + 5\right)\cdot 13 + \left(a + 9\right)\cdot 13^{2} + \left(9 a + 7\right)\cdot 13^{3} + \left(5 a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

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

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

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