Properties

Label 3.1101.6t11.b.a
Dimension $3$
Group $S_4\times C_2$
Conductor $1101$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(1101\)\(\medspace = 3 \cdot 367 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.404067.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: even
Determinant: 1.1101.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.3303.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{4} + 2x^{2} + 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 a + 12 + \left(2 a + 12\right)\cdot 13 + \left(a + 6\right)\cdot 13^{2} + \left(2 a + 12\right)\cdot 13^{3} + \left(2 a + 12\right)\cdot 13^{4} + \left(8 a + 9\right)\cdot 13^{6} + \left(4 a + 1\right)\cdot 13^{7} + \left(2 a + 1\right)\cdot 13^{8} + \left(a + 7\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 a + 11 + \left(8 a + 10\right)\cdot 13 + \left(7 a + 6\right)\cdot 13^{2} + \left(7 a + 6\right)\cdot 13^{3} + \left(9 a + 5\right)\cdot 13^{4} + \left(9 a + 6\right)\cdot 13^{5} + \left(7 a + 7\right)\cdot 13^{6} + \left(10 a + 11\right)\cdot 13^{7} + \left(6 a + 1\right)\cdot 13^{8} + \left(11 a + 4\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 + 7\cdot 13 + 11\cdot 13^{2} + 8\cdot 13^{3} + 8\cdot 13^{4} + 13^{5} + 7\cdot 13^{6} + 10\cdot 13^{7} + 4\cdot 13^{8} + 6\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 a + 1 + 10 a\cdot 13 + \left(11 a + 6\right)\cdot 13^{2} + 10 a\cdot 13^{3} + 10 a\cdot 13^{4} + \left(12 a + 12\right)\cdot 13^{5} + \left(4 a + 3\right)\cdot 13^{6} + \left(8 a + 11\right)\cdot 13^{7} + \left(10 a + 11\right)\cdot 13^{8} + \left(11 a + 5\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 9 a + 2 + \left(4 a + 2\right)\cdot 13 + \left(5 a + 6\right)\cdot 13^{2} + \left(5 a + 6\right)\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} + \left(3 a + 6\right)\cdot 13^{5} + \left(5 a + 5\right)\cdot 13^{6} + \left(2 a + 1\right)\cdot 13^{7} + \left(6 a + 11\right)\cdot 13^{8} + \left(a + 8\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 + 5\cdot 13 + 13^{2} + 4\cdot 13^{3} + 4\cdot 13^{4} + 11\cdot 13^{5} + 5\cdot 13^{6} + 2\cdot 13^{7} + 8\cdot 13^{8} + 6\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$1$$2$$(1,4)(2,5)(3,6)$$-3$
$3$$2$$(1,4)$$1$
$3$$2$$(1,4)(3,6)$$-1$
$6$$2$$(2,3)(5,6)$$1$
$6$$2$$(1,4)(2,3)(5,6)$$-1$
$8$$3$$(1,3,2)(4,6,5)$$0$
$6$$4$$(1,6,4,3)$$1$
$6$$4$$(1,4)(2,6,5,3)$$-1$
$8$$6$$(1,6,5,4,3,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.