Properties

Label 3.7e2_379.6t6.1c1
Dimension 3
Group $A_4\times C_2$
Conductor $ 7^{2} \cdot 379 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_4\times C_2$
Conductor:$18571= 7^{2} \cdot 379 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 7 x^{4} - 12 x^{3} + 21 x^{2} - 15 x + 13 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_4\times C_2$
Parity: Odd
Determinant: 1.379.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 17 a + 23 + \left(17 a + 38\right)\cdot 41 + \left(36 a + 13\right)\cdot 41^{2} + \left(21 a + 19\right)\cdot 41^{3} + \left(11 a + 8\right)\cdot 41^{4} + \left(26 a + 8\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 24 a + 33 + \left(23 a + 32\right)\cdot 41 + \left(4 a + 23\right)\cdot 41^{2} + \left(19 a + 7\right)\cdot 41^{3} + \left(29 a + 21\right)\cdot 41^{4} + \left(14 a + 34\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 39 a + 39 + \left(3 a + 25\right)\cdot 41 + \left(23 a + 2\right)\cdot 41^{2} + \left(37 a + 18\right)\cdot 41^{3} + \left(33 a + 9\right)\cdot 41^{4} + \left(17 a + 17\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 23 + 31\cdot 41 + 7\cdot 41^{2} + 36\cdot 41^{3} + 25\cdot 41^{4} + 18\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 15 + 36\cdot 41 + 6\cdot 41^{2} + 16\cdot 41^{3} + 25\cdot 41^{4} + 7\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 2 a + 33 + \left(37 a + 39\right)\cdot 41 + \left(17 a + 26\right)\cdot 41^{2} + \left(3 a + 25\right)\cdot 41^{3} + \left(7 a + 32\right)\cdot 41^{4} + \left(23 a + 36\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$

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

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

Character values on conjugacy classes

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