Properties

Label 1.37.9t1.1c1
Dimension 1
Group $C_9$
Conductor $ 37 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_9$
Conductor:$37 $
Artin number field: Splitting field of $f= x^{9} - x^{8} - 16 x^{7} + 11 x^{6} + 66 x^{5} - 32 x^{4} - 73 x^{3} + 7 x^{2} + 7 x - 1 $ over $\Q$
Size of Galois orbit: 6
Smallest containing permutation representation: $C_9$
Parity: Even
Corresponding Dirichlet character: \(\chi_{37}(16,\cdot)\)

Galois action

Roots of defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

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

Character values on conjugacy classes

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