Properties

Label 2.4900.12t11.a.a
Dimension $2$
Group $S_3 \times C_4$
Conductor $4900$
Root number not computed
Indicator $0$

Related objects

Learn more about

Basic invariants

Dimension: $2$
Group: $S_3 \times C_4$
Conductor: \(4900\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 7^{2}\)
Artin stem field: 12.4.46118408000000000.2
Galois orbit size: $2$
Smallest permutation container: $S_3 \times C_4$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.980.1

Defining polynomial

$f(x)$$=$\(x^{12} - 2 x^{11} - 10 x^{10} + 40 x^{9} - 9 x^{8} - 150 x^{7} + 231 x^{6} + 50 x^{5} - 409 x^{4} + 1190 x^{2} - 988 x - 239\)  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 $\mathbb{Q}_{ 13 }$: \(x^{4} + 3 x^{2} + 12 x + 2\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.