Properties

Label 2.2300.12t11.a
Dimension $2$
Group $S_3 \times C_4$
Conductor $2300$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$S_3 \times C_4$
Conductor:\(2300\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 23 \)
Artin number field: Galois closure of 12.0.139920500000000.1
Galois orbit size: $2$
Smallest permutation container: $S_3 \times C_4$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.460.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{4} + 3x^{2} + 12x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 5 a^{3} + 8 a^{2} + 4 a + 11 + \left(12 a^{3} + 6 a^{2} + 9 a + 2\right)\cdot 13 + \left(5 a^{3} + 3 a^{2} + 4 a + 2\right)\cdot 13^{2} + \left(4 a^{3} + 10 a^{2} + 9 a + 2\right)\cdot 13^{3} + \left(a^{3} + 6 a^{2}\right)\cdot 13^{4} + \left(12 a^{3} + 5 a^{2} + 11 a + 1\right)\cdot 13^{5} + \left(10 a^{3} + 3 a^{2} + a + 5\right)\cdot 13^{6} + \left(4 a^{3} + 12 a^{2} + 5 a + 5\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 a^{3} + 3 a^{2} + 2 a + 1 + \left(2 a^{3} + 5 a^{2} + 10 a + 11\right)\cdot 13 + \left(12 a^{3} + 2 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(11 a^{3} + 9 a^{2} + 11 a + 9\right)\cdot 13^{3} + \left(12 a^{3} + 9 a^{2} + a\right)\cdot 13^{4} + \left(9 a^{3} + a^{2} + 12 a + 8\right)\cdot 13^{5} + \left(10 a^{3} + 6 a^{2} + 11 a + 9\right)\cdot 13^{6} + \left(9 a^{3} + 8 a^{2} + 3\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a^{3} + a + 12 + \left(6 a^{3} + 9 a^{2} + 6 a + 9\right)\cdot 13 + \left(5 a^{2} + 3 a + 5\right)\cdot 13^{2} + \left(3 a^{3} + 4 a^{2} + 3 a\right)\cdot 13^{3} + \left(12 a^{3} + 9 a^{2} + 10 a\right)\cdot 13^{4} + \left(8 a^{3} + 3 a^{2} + 4\right)\cdot 13^{5} + \left(12 a^{3} + 11 a^{2} + 11 a + 4\right)\cdot 13^{6} + \left(4 a^{3} + 3 a^{2} + 6 a + 6\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 5 a^{3} + 10 a + 1 + \left(4 a^{3} + 7 a^{2} + a + 8\right)\cdot 13 + \left(a^{3} + 2 a^{2} + 11 a + 11\right)\cdot 13^{2} + \left(a^{3} + 10 a^{2} + 4 a + 4\right)\cdot 13^{3} + \left(6 a^{3} + a^{2} + 3 a + 5\right)\cdot 13^{4} + \left(9 a^{2} + 6 a + 11\right)\cdot 13^{5} + \left(3 a^{3} + 10 a^{2} + 8 a + 11\right)\cdot 13^{6} + \left(2 a^{3} + 2 a^{2} + 2 a + 4\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a^{3} + 10 a^{2} + 7 a + 9 + \left(8 a^{3} + 8 a^{2} + 6 a + 7\right)\cdot 13 + \left(2 a^{3} + 11 a^{2} + a + 10\right)\cdot 13^{2} + \left(10 a^{3} + 7 a^{2} + 7 a + 4\right)\cdot 13^{3} + \left(3 a^{3} + a + 6\right)\cdot 13^{4} + \left(a^{3} + 2 a^{2} + 11 a + 2\right)\cdot 13^{5} + \left(8 a^{3} + 9 a^{2} + 4 a + 1\right)\cdot 13^{6} + \left(11 a^{3} + 2 a^{2} + 6 a\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 a^{3} + a^{2} + 10 a + 2 + \left(8 a^{3} + 3 a^{2} + 12 a + 11\right)\cdot 13 + \left(12 a^{3} + 2 a^{2} + 8 a + 5\right)\cdot 13^{2} + \left(2 a^{3} + 11 a^{2} + 3\right)\cdot 13^{3} + \left(10 a^{2} + 8 a + 11\right)\cdot 13^{4} + \left(10 a^{3} + 10 a^{2} + 2 a + 4\right)\cdot 13^{5} + \left(4 a^{3} + 12 a^{2} + 8 a\right)\cdot 13^{6} + \left(10 a^{3} + a^{2} + 5 a\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 12 a^{3} + 2 a^{2} + 6 a + 3 + \left(7 a^{3} + 4 a^{2} + 10 a + 7\right)\cdot 13 + \left(9 a^{3} + 12 a^{2} + 7 a + 6\right)\cdot 13^{2} + \left(7 a^{3} + 8 a^{2} + 10 a + 10\right)\cdot 13^{3} + \left(5 a^{3} + 8 a^{2} + 3 a + 10\right)\cdot 13^{4} + \left(9 a^{2} + 11 a + 7\right)\cdot 13^{5} + \left(6 a^{3} + 11 a^{2} + 10 a + 3\right)\cdot 13^{6} + \left(3 a^{3} + 2 a^{2} + 7 a + 4\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 3 a^{3} + 7 a^{2} + 3 a + 11 + \left(8 a^{3} + a^{2} + 9 a + 9\right)\cdot 13 + \left(10 a^{3} + 2 a^{2} + 7 a + 9\right)\cdot 13^{2} + \left(6 a^{3} + a + 1\right)\cdot 13^{3} + \left(6 a^{3} + 4 a^{2} + 10 a + 10\right)\cdot 13^{4} + \left(a^{3} + 6 a^{2} + 10\right)\cdot 13^{5} + \left(a^{3} + a^{2} + 7 a + 4\right)\cdot 13^{6} + \left(9 a^{3} + 12 a^{2} + 2 a + 4\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 2 a^{3} + a^{2} + 12 a + 6 + \left(10 a^{3} + 9 a^{2} + 12\right)\cdot 13 + \left(6 a^{3} + 8 a^{2} + 12 a + 10\right)\cdot 13^{2} + \left(4 a^{3} + 7 a^{2} + 7 a + 11\right)\cdot 13^{3} + \left(a^{3} + a^{2} + 11\right)\cdot 13^{4} + \left(11 a^{3} + 12 a^{2} + 3 a + 1\right)\cdot 13^{5} + \left(5 a^{3} + 11 a^{2} + 12 a + 11\right)\cdot 13^{6} + \left(11 a^{2} + 11 a + 10\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 4 a^{3} + 12 a^{2} + 6 a + 10 + \left(2 a^{3} + 2 a^{2} + 5 a + 2\right)\cdot 13 + 4\cdot 13^{2} + \left(8 a^{3} + a^{2} + 10 a + 1\right)\cdot 13^{3} + \left(10 a^{3} + 10 a^{2}\right)\cdot 13^{4} + \left(3 a^{3} + a^{2} + 3 a + 11\right)\cdot 13^{5} + \left(12 a^{3} + 6 a^{2} + 11 a + 3\right)\cdot 13^{6} + \left(9 a^{3} + 8 a^{2} + 12 a + 11\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 11 a^{3} + 10 a^{2} + 9 a + 6 + \left(2 a^{3} + 9 a^{2} + 9 a + 2\right)\cdot 13 + \left(3 a^{3} + 5 a^{2} + 5 a + 4\right)\cdot 13^{2} + \left(12 a^{3} + a^{2} + 11 a + 7\right)\cdot 13^{3} + \left(7 a^{3} + 10 a^{2} + 3 a + 1\right)\cdot 13^{4} + \left(6 a^{3} + a^{2} + 11 a + 6\right)\cdot 13^{5} + \left(2 a^{3} + 3 a^{2} + 8 a + 4\right)\cdot 13^{6} + \left(7 a^{3} + 4 a^{2} + 5 a + 7\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 6 a^{3} + 11 a^{2} + 8 a + 7 + \left(3 a^{3} + 10 a^{2} + 8 a + 5\right)\cdot 13 + \left(12 a^{3} + 7 a^{2} + 8 a + 7\right)\cdot 13^{2} + \left(4 a^{3} + 5 a^{2} + 12 a + 6\right)\cdot 13^{3} + \left(9 a^{3} + 4 a^{2} + 6 a + 6\right)\cdot 13^{4} + \left(11 a^{3} + 4 a + 8\right)\cdot 13^{5} + \left(12 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 13^{6} + \left(3 a^{3} + 6 a^{2} + 9 a + 6\right)\cdot 13^{7} +O(13^{8})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 12 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,8)(2,12)(3,7)(4,10)(5,9)(6,11)$ $-2$ $-2$
$3$ $2$ $(1,11)(2,5)(3,7)(4,10)(6,8)(9,12)$ $0$ $0$
$3$ $2$ $(2,7)(3,12)(4,6)(10,11)$ $0$ $0$
$2$ $3$ $(1,6,4)(2,7,9)(3,5,12)(8,11,10)$ $-1$ $-1$
$1$ $4$ $(1,9,8,5)(2,11,12,6)(3,4,7,10)$ $-2 \zeta_{4}$ $2 \zeta_{4}$
$1$ $4$ $(1,5,8,9)(2,6,12,11)(3,10,7,4)$ $2 \zeta_{4}$ $-2 \zeta_{4}$
$3$ $4$ $(1,7,8,3)(2,11,12,6)(4,9,10,5)$ $0$ $0$
$3$ $4$ $(1,3,8,7)(2,6,12,11)(4,5,10,9)$ $0$ $0$
$2$ $6$ $(1,10,6,8,4,11)(2,5,7,12,9,3)$ $1$ $1$
$2$ $12$ $(1,12,10,9,6,3,8,2,4,5,11,7)$ $-\zeta_{4}$ $\zeta_{4}$
$2$ $12$ $(1,2,10,5,6,7,8,12,4,9,11,3)$ $\zeta_{4}$ $-\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.