Properties

Label 1.2e2_13.6t1.2c2
Dimension 1
Group $C_6$
Conductor $ 2^{2} \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$52= 2^{2} \cdot 13 $
Artin number field: Splitting field of $f= x^{6} + 13 x^{4} + 26 x^{2} + 13 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{52}(43,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 70 a + 41 + \left(62 a + 13\right)\cdot 73 + \left(24 a + 67\right)\cdot 73^{2} + \left(47 a + 50\right)\cdot 73^{3} + \left(15 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 63 a + 15 + \left(35 a + 14\right)\cdot 73 + \left(11 a + 37\right)\cdot 73^{2} + \left(46 a + 9\right)\cdot 73^{3} + \left(59 a + 43\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 37 a + 54 + \left(46 a + 21\right)\cdot 73 + \left(59 a + 43\right)\cdot 73^{2} + \left(55 a + 55\right)\cdot 73^{3} + \left(4 a + 20\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 32 + \left(10 a + 59\right)\cdot 73 + \left(48 a + 5\right)\cdot 73^{2} + \left(25 a + 22\right)\cdot 73^{3} + \left(57 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 10 a + 58 + \left(37 a + 58\right)\cdot 73 + \left(61 a + 35\right)\cdot 73^{2} + \left(26 a + 63\right)\cdot 73^{3} + \left(13 a + 29\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 36 a + 19 + \left(26 a + 51\right)\cdot 73 + \left(13 a + 29\right)\cdot 73^{2} + \left(17 a + 17\right)\cdot 73^{3} + \left(68 a + 52\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

Character values on conjugacy classes

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