Properties

Label 4.13e2_113.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 13^{2} \cdot 113 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$19097= 13^{2} \cdot 113 $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 4 x^{4} - 2 x^{3} - 2 x^{2} + 4 x - 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.113.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 35 + 19\cdot 61 + 2\cdot 61^{2} + 2\cdot 61^{3} + 33\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 a + 47 + \left(44 a + 16\right)\cdot 61 + \left(a + 56\right)\cdot 61^{2} + \left(8 a + 15\right)\cdot 61^{3} + \left(35 a + 57\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 43 a + 11 + \left(30 a + 24\right)\cdot 61 + \left(48 a + 56\right)\cdot 61^{2} + \left(57 a + 50\right)\cdot 61^{3} + \left(14 a + 5\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 18 a + 54 + \left(30 a + 11\right)\cdot 61 + \left(12 a + 13\right)\cdot 61^{2} + \left(3 a + 60\right)\cdot 61^{3} + \left(46 a + 23\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 41 + 41\cdot 61^{2} + 31\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 49 a + 59 + \left(16 a + 48\right)\cdot 61 + \left(59 a + 13\right)\cdot 61^{2} + \left(52 a + 22\right)\cdot 61^{3} + \left(25 a + 23\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

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