Properties

Label 1.7_13.12t1.1c2
Dimension 1
Group $C_{12}$
Conductor $ 7 \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_{12}$
Conductor:$91= 7 \cdot 13 $
Artin number field: Splitting field of $f= x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_{12}$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{91}(44,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{4} + 23 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 38 a^{3} + 24 a^{2} + \left(19 a^{3} + 8 a^{2} + 22 a + 32\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 38\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 27\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 38 a^{3} + 24 a^{2} + 34 + \left(19 a^{3} + 8 a^{2} + 22 a + 29\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 12\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 19\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 16\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 38 a^{3} + 24 a^{2} + 16 + \left(19 a^{3} + 8 a^{2} + 22 a + 26\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 30\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 3\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 16 a^{3} + 34 a^{2} + 20 a + 3 + \left(10 a^{3} + 15 a^{2} + 37 a + 19\right)\cdot 41 + \left(a^{2} + 37 a + 20\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 38\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 16 a^{3} + 34 a^{2} + 20 a + 10 + \left(10 a^{3} + 15 a^{2} + 37 a + 21\right)\cdot 41 + \left(a^{2} + 37 a + 5\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 6\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 a^{3} + 34 a^{2} + 20 a + 26 + \left(10 a^{3} + 15 a^{2} + 37 a + 15\right)\cdot 41 + \left(a^{2} + 37 a + 38\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 22\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 24 a^{3} + 36 a^{2} + 30 a + 25 + \left(13 a^{3} + 12 a^{2} + a + 4\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a + 9\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 32\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 34\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 24 a^{3} + 36 a^{2} + 30 a + \left(13 a^{3} + 12 a^{2} + a + 40\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 8\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 9 }$ $=$ $ 24 a^{3} + 36 a^{2} + 30 a + 18 + \left(13 a^{3} + 12 a^{2} + a + 2\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a + 24\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 23\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 10 }$ $=$ $ 4 a^{3} + 29 a^{2} + 32 a + 1 + \left(38 a^{3} + 3 a^{2} + 20 a + 5\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 7\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 37\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 11 }$ $=$ $ 4 a^{3} + 29 a^{2} + 32 a + 8 + \left(38 a^{3} + 3 a^{2} + 20 a + 7\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 33\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 4\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 7\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 12 }$ $=$ $ 4 a^{3} + 29 a^{2} + 32 a + 24 + \left(38 a^{3} + 3 a^{2} + 20 a + 1\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 25\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 21\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 19\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

Character values on conjugacy classes

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