Properties

Label 2.3800.12t18.d.a
Dimension $2$
Group $C_6\times S_3$
Conductor $3800$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(3800\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 19 \)
Artin stem field: Galois closure of 12.0.533794816000000.1
Galois orbit size: $2$
Smallest permutation container: $C_6\times S_3$
Parity: odd
Determinant: 1.152.6t1.c.b
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.2888.1

Defining polynomial

$f(x)$$=$ \( x^{12} - 2 x^{11} - x^{10} + 10 x^{9} + 13 x^{8} - 42 x^{7} - 6 x^{6} + 50 x^{5} + 45 x^{4} - 24 x^{3} - 16 x^{2} + 12 x + 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{6} + 2x^{4} + 10x^{2} + 3x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a^{5} + 8 a^{4} + 4 a^{3} + 5 a^{2} + 5 a + 1 + \left(6 a^{5} + 2 a^{4} + 16 a + 4\right)\cdot 17 + \left(11 a^{5} + 16 a^{4} + 12 a^{3} + 12 a^{2} + 2 a + 9\right)\cdot 17^{2} + \left(16 a^{5} + 2 a^{4} + a^{3} + 6 a^{2} + 10 a + 16\right)\cdot 17^{3} + \left(3 a^{5} + 16 a^{3} + 12 a^{2} + 11 a + 4\right)\cdot 17^{4} + \left(15 a^{5} + 13 a^{4} + 14 a^{3} + 16 a^{2} + 2 a + 5\right)\cdot 17^{5} + \left(16 a^{5} + 3 a^{4} + 5 a^{3} + 13 a^{2} + 5 a + 9\right)\cdot 17^{6} + \left(11 a^{5} + 11 a^{4} + 2 a^{3} + 11 a^{2} + 14 a + 15\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a^{5} + 12 a^{4} + 11 a^{3} + a^{2} + 15 a + 14 + \left(12 a^{4} + 6 a^{3} + 7 a^{2} + 15 a + 2\right)\cdot 17 + \left(4 a^{5} + a^{4} + 10 a^{3} + 15 a^{2} + 5 a + 4\right)\cdot 17^{2} + \left(13 a^{5} + 8 a^{4} + 4 a^{3} + 6 a^{2} + 15 a + 7\right)\cdot 17^{3} + \left(2 a^{5} + 12 a^{4} + 5 a^{2} + 15 a + 3\right)\cdot 17^{4} + \left(9 a^{5} + 9 a^{4} + 12 a^{3} + a^{2} + 6 a + 16\right)\cdot 17^{5} + \left(3 a^{5} + a^{4} + a^{3} + 10 a^{2} + 14 a + 6\right)\cdot 17^{6} + \left(16 a^{5} + 11 a^{4} + 16 a^{3} + 14 a^{2} + 6 a + 7\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 5 a^{5} + 11 a^{4} + 12 a^{3} + 9 a^{2} + 12 a + 15 + \left(2 a^{5} + 14 a^{4} + 15 a^{3} + a + 4\right)\cdot 17 + \left(13 a^{5} + 5 a^{4} + 14 a^{3} + 15 a^{2} + 5 a + 3\right)\cdot 17^{2} + \left(14 a^{5} + 2 a^{4} + a^{3} + 4 a^{2} + 9 a + 10\right)\cdot 17^{3} + \left(3 a^{5} + 12 a^{4} + 3 a^{3} + 9 a^{2} + 9 a + 12\right)\cdot 17^{4} + \left(10 a^{5} + 8 a^{4} + 13 a^{3} + 4 a^{2} + 14 a + 12\right)\cdot 17^{5} + \left(9 a^{5} + 12 a^{4} + 4 a^{3} + 8\right)\cdot 17^{6} + \left(3 a^{5} + 2 a^{2} + 9 a + 8\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 14 a^{5} + 9 a^{4} + 5 a^{3} + 10 a^{2} + 5 a + 2 + \left(8 a^{5} + 11 a^{4} + 10 a^{3} + 6 a^{2} + 2 a\right)\cdot 17 + \left(16 a^{4} + 15 a^{3} + 12 a^{2} + 9 a + 5\right)\cdot 17^{2} + \left(9 a^{5} + 6 a^{4} + 9 a^{3} + 10 a^{2} + 3 a + 10\right)\cdot 17^{3} + \left(9 a^{5} + 2 a^{4} + 8 a^{3} + 9 a^{2} + 3 a\right)\cdot 17^{4} + \left(2 a^{5} + 9 a^{4} + 13 a^{3} + 6 a^{2} + 15 a\right)\cdot 17^{5} + \left(9 a^{5} + 3 a^{4} + 8 a^{3} + 5 a^{2} + 12 a\right)\cdot 17^{6} + \left(4 a^{5} + 8 a^{4} + 2 a^{3} + a^{2} + 13 a + 5\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 15 a^{5} + a^{4} + 2 a^{3} + a^{2} + 6 a + 4 + \left(11 a^{5} + 9 a^{4} + 8 a^{3} + 15 a^{2} + 2 a\right)\cdot 17 + \left(16 a^{5} + 10 a^{3} + 10 a^{2} + 15 a + 15\right)\cdot 17^{2} + \left(2 a^{5} + 4 a^{4} + 12 a^{3} + 12 a^{2} + 8 a + 11\right)\cdot 17^{3} + \left(13 a^{5} + 6 a^{4} + a^{3} + 12 a^{2} + 2 a + 6\right)\cdot 17^{4} + \left(2 a^{5} + 11 a^{4} + 13 a^{3} + 10 a^{2} + 7 a + 12\right)\cdot 17^{5} + \left(15 a^{5} + 16 a^{4} + 9 a^{3} + 15 a^{2} + 3 a + 9\right)\cdot 17^{6} + \left(15 a^{5} + a^{4} + 6 a^{3} + 8 a^{2} + a + 16\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 a^{5} + 5 a^{4} + 2 a^{3} + 11 a^{2} + 5 a + 10 + \left(11 a^{5} + 9 a^{3} + 6 a^{2} + 9 a + 15\right)\cdot 17 + \left(10 a^{5} + 4 a^{4} + 8 a^{3} + 15 a^{2} + 2 a + 15\right)\cdot 17^{2} + \left(5 a^{4} + 10 a^{3} + 7 a + 15\right)\cdot 17^{3} + \left(6 a^{5} + a^{4} + 13 a^{3} + 10 a^{2} + 2 a + 8\right)\cdot 17^{4} + \left(13 a^{5} + 16 a^{4} + 12 a^{2} + a + 8\right)\cdot 17^{5} + \left(6 a^{5} + 6 a^{4} + 14 a^{3} + 7 a^{2} + 13 a + 2\right)\cdot 17^{6} + \left(8 a^{5} + 12 a^{4} + 16 a^{3} + 9 a^{2} + 8 a + 14\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 12 a^{5} + 11 a^{4} + 16 a^{3} + 4 a^{2} + 14 a + 15 + \left(9 a^{5} + 11 a^{4} + 16 a^{3} + 16 a^{2} + 11 a\right)\cdot 17 + \left(4 a^{5} + 13 a^{4} + a^{3} + 8 a^{2} + 12 a + 11\right)\cdot 17^{2} + \left(a^{5} + 13 a^{4} + 12 a^{3} + 7 a^{2} + 14 a + 16\right)\cdot 17^{3} + \left(11 a^{4} + 3 a^{3} + 2 a^{2} + 8 a + 10\right)\cdot 17^{4} + \left(9 a^{5} + 13 a^{4} + 10 a^{3} + a^{2} + 16 a + 5\right)\cdot 17^{5} + \left(9 a^{5} + 2 a^{4} + 11 a^{3} + 13 a^{2} + 15 a + 16\right)\cdot 17^{6} + \left(11 a^{5} + 11 a^{4} + 11 a^{3} + 3 a^{2} + 7 a + 5\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 12 a^{5} + 14 a^{4} + 16 a^{3} + 5 a^{2} + 15 a + 9 + \left(16 a^{4} + 3 a^{3} + 8 a^{2} + 10 a + 3\right)\cdot 17 + \left(15 a^{5} + 6 a^{4} + 16 a^{3} + 7 a^{2} + 9 a + 6\right)\cdot 17^{2} + \left(13 a^{5} + 15 a^{4} + 4 a^{3} + 7 a^{2} + a + 11\right)\cdot 17^{3} + \left(3 a^{5} + 7 a^{4} + a^{3} + 9 a^{2} + 12\right)\cdot 17^{4} + \left(15 a^{5} + 9 a^{4} + 11 a^{3} + a^{2} + 3 a + 1\right)\cdot 17^{5} + \left(13 a^{5} + 4 a^{4} + 6 a^{3} + 5 a^{2} + 14 a\right)\cdot 17^{6} + \left(16 a^{5} + 3 a^{4} + 6 a^{3} + 10 a^{2} + 2 a + 4\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 4 a^{5} + 6 a^{4} + 8 a^{3} + 6 a^{2} + 4 a + 15 + \left(6 a^{5} + 13 a^{4} + 2 a^{3} + 16 a^{2} + 5 a + 9\right)\cdot 17 + \left(12 a^{4} + 5 a^{3} + 9 a^{2} + 15 a + 10\right)\cdot 17^{2} + \left(2 a^{5} + 16 a^{4} + 11 a^{3} + 12 a^{2} + 10 a + 3\right)\cdot 17^{3} + \left(12 a^{5} + a^{4} + 10 a^{3} + 13 a^{2} + 14 a + 13\right)\cdot 17^{4} + \left(5 a^{5} + 11 a^{4} + 14 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 17^{5} + \left(7 a^{5} + 3 a^{4} + 13 a^{3} + 8 a^{2} + 3 a + 12\right)\cdot 17^{6} + \left(5 a^{5} + 6 a^{4} + 10 a^{3} + 16 a^{2} + 7 a + 6\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 14 a^{5} + 14 a^{4} + 16 a^{3} + 9 a^{2} + 14 a + 11 + \left(7 a^{5} + 12 a^{4} + 12 a^{3} + 12 a^{2} + 10 a + 5\right)\cdot 17 + \left(15 a^{5} + 12 a^{4} + 11 a^{3} + 11 a^{2} + 10 a + 15\right)\cdot 17^{2} + \left(13 a^{5} + 3 a^{4} + 11 a^{2} + 5 a + 5\right)\cdot 17^{3} + \left(13 a^{5} + 16 a^{4} + 4 a^{3} + 11 a^{2} + 11 a + 1\right)\cdot 17^{4} + \left(16 a^{5} + 12 a^{4} + 6 a^{3} + 6 a^{2} + 6 a + 5\right)\cdot 17^{5} + \left(14 a^{5} + 16 a^{3} + 3 a^{2} + 5 a + 1\right)\cdot 17^{6} + \left(6 a^{5} + 6 a^{4} + 11 a^{3} + 9 a^{2} + 5 a + 13\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 3 a^{5} + a^{4} + 6 a^{3} + 16 a^{2} + 1 + \left(5 a^{5} + 8 a^{4} + 6 a^{3} + 6 a^{2} + 14 a + 1\right)\cdot 17 + \left(13 a^{5} + 10 a^{4} + 10 a^{3} + 15 a^{2} + 6 a + 3\right)\cdot 17^{2} + \left(8 a^{5} + 11 a^{4} + 14 a^{3} + 2 a^{2} + 3 a + 12\right)\cdot 17^{3} + \left(3 a^{5} + 8 a^{4} + 13 a^{3} + 2 a^{2} + 2\right)\cdot 17^{4} + \left(3 a^{5} + 7 a^{4} + 3 a^{3} + 9 a^{2} + 2 a + 3\right)\cdot 17^{5} + \left(16 a^{5} + 8 a^{4} + 8 a^{3} + 15 a^{2} + 12 a + 5\right)\cdot 17^{6} + \left(3 a^{5} + 9 a^{4} + 3 a^{3} + 3 a^{2} + 6 a + 9\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 11 a^{5} + 10 a^{4} + 4 a^{3} + 8 a^{2} + 7 a + 7 + \left(13 a^{5} + 5 a^{4} + 9 a^{3} + 5 a^{2} + a + 2\right)\cdot 17 + \left(13 a^{5} + a^{3} + a^{2} + 6 a + 3\right)\cdot 17^{2} + \left(4 a^{5} + 11 a^{4} + 11 a + 14\right)\cdot 17^{3} + \left(12 a^{5} + 3 a^{4} + 8 a^{3} + 3 a^{2} + 4 a + 6\right)\cdot 17^{4} + \left(15 a^{5} + 13 a^{4} + 5 a^{3} + 5 a^{2} + 3 a + 8\right)\cdot 17^{5} + \left(12 a^{5} + 2 a^{4} + 3 a^{2} + a + 12\right)\cdot 17^{6} + \left(13 a^{5} + 3 a^{4} + 13 a^{3} + 10 a^{2} + a + 12\right)\cdot 17^{7} +O(17^{8})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

Cycle notation
$(1,6)(2,11)(3,10)(4,9)(5,8)(7,12)$
$(3,11,6)(5,12,9)$
$(1,8,10,7,2,4)(3,5,6,9,11,12)$
$(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 12 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$$-2$
$3$$2$$(1,6)(2,11)(3,10)(4,9)(5,8)(7,12)$$0$
$3$$2$$(1,9)(2,12)(3,7)(4,11)(5,10)(6,8)$$0$
$1$$3$$(1,10,2)(3,11,6)(4,8,7)(5,12,9)$$-2 \zeta_{3} - 2$
$1$$3$$(1,2,10)(3,6,11)(4,7,8)(5,9,12)$$2 \zeta_{3}$
$2$$3$$(1,10,2)(3,6,11)(4,8,7)(5,9,12)$$-1$
$2$$3$$(3,11,6)(5,12,9)$$-\zeta_{3}$
$2$$3$$(3,6,11)(5,9,12)$$\zeta_{3} + 1$
$1$$6$$(1,8,10,7,2,4)(3,12,11,9,6,5)$$-2 \zeta_{3}$
$1$$6$$(1,4,2,7,10,8)(3,5,6,9,11,12)$$2 \zeta_{3} + 2$
$2$$6$$(1,8,10,7,2,4)(3,5,6,9,11,12)$$1$
$2$$6$$(1,7)(2,8)(3,5,6,9,11,12)(4,10)$$\zeta_{3}$
$2$$6$$(1,7)(2,8)(3,12,11,9,6,5)(4,10)$$-\zeta_{3} - 1$
$3$$6$$(1,3,10,11,2,6)(4,5,8,12,7,9)$$0$
$3$$6$$(1,6,2,11,10,3)(4,9,7,12,8,5)$$0$
$3$$6$$(1,9,10,5,2,12)(3,4,11,8,6,7)$$0$
$3$$6$$(1,12,2,5,10,9)(3,7,6,8,11,4)$$0$

The blue line marks the conjugacy class containing complex conjugation.