Properties

Label 1.95.12t1.a.b
Dimension 1
Group $C_{12}$
Conductor $ 5 \cdot 19 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_{12}$
Conductor:$95= 5 \cdot 19 $
Artin number field: Splitting field of 12.0.33171021564453125.1 defined by $f= x^{12} - x^{11} + 7 x^{10} - 6 x^{9} + 41 x^{8} - 62 x^{7} + 266 x^{6} - 351 x^{5} + 1513 x^{4} - 1757 x^{3} + 2107 x^{2} - 2058 x + 2401 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_{12}$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{95}(87,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{4} + 6 x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 28 a^{3} + 4 a^{2} + 12 a + \left(26 a^{3} + 2 a^{2} + 23 a + 30\right)\cdot 37 + \left(25 a^{3} + 24 a^{2} + 12 a + 28\right)\cdot 37^{2} + \left(32 a^{3} + 5 a^{2} + 27 a + 15\right)\cdot 37^{3} + \left(22 a^{3} + 21 a^{2} + 7 a + 12\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 30 a^{3} + 24 a^{2} + 3 a + 31 + \left(34 a^{2} + 34 a + 13\right)\cdot 37 + \left(6 a^{3} + 15 a^{2} + 5 a + 16\right)\cdot 37^{2} + \left(13 a^{3} + 34 a^{2} + 25 a + 23\right)\cdot 37^{3} + \left(26 a^{3} + 34 a^{2} + 13 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 a^{3} + 28 a^{2} + 7 a + 18 + \left(13 a^{3} + 36 a^{2} + a + 16\right)\cdot 37 + \left(4 a^{3} + 32 a^{2} + 17 a\right)\cdot 37^{2} + \left(6 a^{3} + 29 a^{2} + 26 a + 24\right)\cdot 37^{3} + \left(32 a^{3} + 12 a^{2} + 18 a + 1\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 29 a^{3} + a^{2} + 14 a + 9 + \left(16 a^{3} + 29 a^{2} + 28 a + 5\right)\cdot 37 + \left(29 a^{3} + 8 a^{2} + 22 a + 13\right)\cdot 37^{2} + \left(24 a^{3} + 14 a^{2} + 11 a + 10\right)\cdot 37^{3} + \left(28 a^{3} + 33 a^{2} + 6 a + 5\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 8 a^{3} + 5 a^{2} + 27 a + 22 + \left(32 a^{3} + 35 a^{2} + 35 a + 25\right)\cdot 37 + \left(28 a^{3} + 21 a^{2} + 31 a + 1\right)\cdot 37^{2} + \left(21 a^{3} + 20 a^{2} + 23 a + 26\right)\cdot 37^{3} + \left(23 a^{3} + 9 a^{2} + 30 a + 5\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 25 a^{3} + 30 a^{2} + 16 a + \left(14 a^{3} + 36 a^{2} + 34 a + 3\right)\cdot 37 + \left(27 a^{3} + 12 a^{2} + 23 a + 22\right)\cdot 37^{2} + \left(31 a^{3} + 14 a^{2} + 25 a + 31\right)\cdot 37^{3} + \left(10 a^{3} + 10 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 25 a^{3} + 11 a^{2} + 15 a + 4 + \left(15 a^{3} + 24 a^{2} + 14 a + 8\right)\cdot 37 + \left(3 a^{3} + 11 a + 1\right)\cdot 37^{2} + \left(18 a^{3} + 18 a^{2} + 36 a + 12\right)\cdot 37^{3} + \left(24 a^{3} + 19 a^{2} + 33 a + 36\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 30 a^{3} + 23 a^{2} + 15 a + 28 + \left(26 a^{3} + 11 a^{2} + 25 a + 5\right)\cdot 37 + \left(30 a^{3} + 28 a^{2} + 4 a + 17\right)\cdot 37^{2} + \left(19 a^{3} + a^{2} + 32 a + 8\right)\cdot 37^{3} + \left(22 a^{3} + 19 a^{2} + 3 a + 15\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 9 }$ $=$ $ 14 a^{3} + 26 a^{2} + 31 a + 12 + \left(28 a^{3} + 28 a^{2} + 8 a + 3\right)\cdot 37 + \left(27 a^{3} + 11 a^{2} + 25\right)\cdot 37^{2} + \left(12 a^{3} + 10 a^{2} + 29 a + 10\right)\cdot 37^{3} + \left(7 a^{3} + 36 a^{2} + 18 a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 10 }$ $=$ $ 29 a^{3} + 21 a^{2} + 33 a + 32 + \left(14 a^{3} + 18 a^{2} + 7 a + 11\right)\cdot 37 + \left(15 a^{3} + 3 a^{2} + 27 a + 3\right)\cdot 37^{2} + \left(35 a^{3} + 36 a^{2} + 35 a + 8\right)\cdot 37^{3} + \left(34 a^{3} + 36 a^{2} + 25 a + 18\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 11 }$ $=$ $ 6 a^{3} + 22 a^{2} + 29 a + \left(14 a^{3} + 29 a^{2} + 15 a + 17\right)\cdot 37 + \left(8 a^{3} + 7 a^{2} + 31 a + 34\right)\cdot 37^{2} + \left(19 a^{3} + 17 a^{2} + 29 a + 6\right)\cdot 37^{3} + \left(a^{3} + 10 a^{2} + 25 a + 18\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 12 }$ $=$ $ 21 a^{3} + 27 a^{2} + 20 a + 30 + \left(17 a^{3} + 8 a^{2} + 29 a + 7\right)\cdot 37 + \left(14 a^{3} + 16 a^{2} + 32 a + 21\right)\cdot 37^{2} + \left(23 a^{3} + 19 a^{2} + 29 a + 7\right)\cdot 37^{3} + \left(23 a^{3} + 24 a^{2} + 25 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

Character values on conjugacy classes

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