Properties

 Label 2.8100.12t11.c.b Dimension $2$ Group $S_3 \times C_4$ Conductor $8100$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $2$ Group: $S_3 \times C_4$ Conductor: $$8100$$$$\medspace = 2^{2} \cdot 3^{4} \cdot 5^{2}$$ Artin stem field: 12.0.21523360500000000.3 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: 3.1.1620.1

Defining polynomial

 $f(x)$ $=$ $$x^{12} + 3 x^{10} - 8 x^{9} + 9 x^{8} + 72 x^{7} + 91 x^{6} + 144 x^{5} - 303 x^{4} - 296 x^{3} + 576 x^{2} + 1536 x + 4096$$  .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{4} + 3 x^{2} + 12 x + 2$$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.