Properties

Label 3.2461.6t11.a.a
Dimension $3$
Group $S_4\times C_2$
Conductor $2461$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(2461\)\(\medspace = 23 \cdot 107 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.56603.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: even
Determinant: 1.2461.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.263327.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + 3x^{4} + x^{2} + 3x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 27 a + 34 + \left(14 a + 4\right)\cdot 37 + \left(27 a + 18\right)\cdot 37^{2} + \left(21 a + 16\right)\cdot 37^{3} + \left(16 a + 22\right)\cdot 37^{4} + \left(10 a + 6\right)\cdot 37^{5} + \left(12 a + 25\right)\cdot 37^{6} + \left(23 a + 35\right)\cdot 37^{7} +O(37^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 a + 8 + \left(17 a + 12\right)\cdot 37 + \left(31 a + 20\right)\cdot 37^{2} + \left(17 a + 2\right)\cdot 37^{3} + \left(a + 12\right)\cdot 37^{4} + \left(32 a + 22\right)\cdot 37^{5} + \left(36 a + 26\right)\cdot 37^{6} + \left(10 a + 14\right)\cdot 37^{7} +O(37^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 26 + 20\cdot 37 + 11\cdot 37^{2} + 37^{3} + 12\cdot 37^{4} + 20\cdot 37^{5} + 20\cdot 37^{6} + 8\cdot 37^{7} +O(37^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + 31 + \left(22 a + 36\right)\cdot 37 + \left(9 a + 1\right)\cdot 37^{2} + \left(15 a + 2\right)\cdot 37^{3} + \left(20 a + 30\right)\cdot 37^{4} + \left(26 a + 31\right)\cdot 37^{5} + \left(24 a + 26\right)\cdot 37^{6} + \left(13 a + 5\right)\cdot 37^{7} +O(37^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 19 + 35\cdot 37 + 3\cdot 37^{2} + 9\cdot 37^{3} + 34\cdot 37^{4} + 28\cdot 37^{5} + 17\cdot 37^{6} + 24\cdot 37^{7} +O(37^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 31 a + 32 + 19 a\cdot 37 + \left(5 a + 18\right)\cdot 37^{2} + \left(19 a + 5\right)\cdot 37^{3} + 35 a\cdot 37^{4} + \left(4 a + 1\right)\cdot 37^{5} + 31\cdot 37^{6} + \left(26 a + 21\right)\cdot 37^{7} +O(37^{8})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.