Properties

Label 2.2888.12t18.f.a
Dimension $2$
Group $C_6\times S_3$
Conductor $2888$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_6\times S_3$
Conductor: \(2888\)\(\medspace = 2^{3} \cdot 19^{2}\)
Artin stem field: Galois closure of 12.0.4452139149819904.6
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)$$=$ \( x^{12} - 4 x^{11} + 10 x^{10} - 8 x^{9} + 16 x^{8} - 26 x^{7} + 74 x^{6} - 12 x^{5} + 72 x^{4} - 4 x^{3} + 207 x^{2} + 154 x + 139 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{6} + 10x^{3} + 11x^{2} + 11x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.