Properties

Label 4.5e2_449.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 5^{2} \cdot 449 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$11225= 5^{2} \cdot 449 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{3} - 3 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.449.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 10 a + 19 + \left(19 a + 3\right)\cdot 41 + \left(28 a + 6\right)\cdot 41^{2} + \left(4 a + 35\right)\cdot 41^{3} + \left(36 a + 26\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 35 a + 32 + \left(25 a + 6\right)\cdot 41 + \left(30 a + 39\right)\cdot 41^{2} + \left(18 a + 20\right)\cdot 41^{3} + \left(3 a + 14\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 21 + 8\cdot 41 + 23\cdot 41^{2} + 36\cdot 41^{3} + 34\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 30 + 3\cdot 41 + 4\cdot 41^{3} + 33\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 31 a + 8 + \left(21 a + 10\right)\cdot 41 + \left(12 a + 31\right)\cdot 41^{2} + \left(36 a + 20\right)\cdot 41^{3} + \left(4 a + 7\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 6 a + 14 + \left(15 a + 8\right)\cdot 41 + \left(10 a + 23\right)\cdot 41^{2} + \left(22 a + 5\right)\cdot 41^{3} + \left(37 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

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)$$1$
$4$$3$$(1,3,5)(2,4,6)$$-2$
$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.