Properties

Label 1.341.15t1.a.f
Dimension $1$
Group $C_{15}$
Conductor $341$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_{15}$
Conductor: \(341\)\(\medspace = 11 \cdot 31 \)
Artin field: Galois closure of 15.15.2572344674223769220522333521.1
Galois orbit size: $8$
Smallest permutation container: $C_{15}$
Parity: even
Dirichlet character: \(\chi_{341}(180,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + \cdots + 47081 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{5} + 3x + 27 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a^{4} + 9 a^{3} + 3 a^{2} + 21 a + 11 + \left(21 a^{4} + 19 a^{3} + 9 a^{2} + 1\right)\cdot 29 + \left(4 a^{4} + 27 a^{3} + 25 a^{2} + 2 a + 16\right)\cdot 29^{2} + \left(26 a^{4} + 24 a^{3} + 11 a^{2} + 9 a + 7\right)\cdot 29^{3} + \left(8 a^{4} + 28 a^{2} + 14 a + 8\right)\cdot 29^{4} + \left(3 a^{4} + 15 a^{3} + 11 a^{2} + 19 a + 16\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 a^{4} + 9 a^{3} + 3 a^{2} + 21 a + 13 + \left(21 a^{4} + 19 a^{3} + 9 a^{2} + 13\right)\cdot 29 + \left(4 a^{4} + 27 a^{3} + 25 a^{2} + 2 a\right)\cdot 29^{2} + \left(26 a^{4} + 24 a^{3} + 11 a^{2} + 9 a + 20\right)\cdot 29^{3} + \left(8 a^{4} + 28 a^{2} + 14 a + 16\right)\cdot 29^{4} + \left(3 a^{4} + 15 a^{3} + 11 a^{2} + 19 a + 11\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 5 a^{4} + 9 a^{3} + 3 a^{2} + 21 a + 24 + \left(21 a^{4} + 19 a^{3} + 9 a^{2} + 15\right)\cdot 29 + \left(4 a^{4} + 27 a^{3} + 25 a^{2} + 2 a + 17\right)\cdot 29^{2} + \left(26 a^{4} + 24 a^{3} + 11 a^{2} + 9 a + 21\right)\cdot 29^{3} + \left(8 a^{4} + 28 a^{2} + 14 a + 21\right)\cdot 29^{4} + \left(3 a^{4} + 15 a^{3} + 11 a^{2} + 19 a + 1\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 a^{4} + 12 a^{3} + 13 a^{2} + 5 a + 10 + \left(5 a^{4} + 24 a^{3} + 19 a^{2} + 22 a + 15\right)\cdot 29 + \left(4 a^{4} + 28 a^{3} + 13 a^{2} + 5 a + 20\right)\cdot 29^{2} + \left(12 a^{4} + 28 a + 8\right)\cdot 29^{3} + \left(24 a^{4} + 13 a^{3} + 2 a^{2} + 22\right)\cdot 29^{4} + \left(23 a^{4} + 13 a^{3} + 22 a^{2} + 28 a + 1\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 7 a^{4} + 12 a^{3} + 13 a^{2} + 5 a + 12 + \left(5 a^{4} + 24 a^{3} + 19 a^{2} + 22 a + 27\right)\cdot 29 + \left(4 a^{4} + 28 a^{3} + 13 a^{2} + 5 a + 4\right)\cdot 29^{2} + \left(12 a^{4} + 28 a + 21\right)\cdot 29^{3} + \left(24 a^{4} + 13 a^{3} + 2 a^{2} + 1\right)\cdot 29^{4} + \left(23 a^{4} + 13 a^{3} + 22 a^{2} + 28 a + 26\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 a^{4} + 12 a^{3} + 13 a^{2} + 5 a + 23 + \left(5 a^{4} + 24 a^{3} + 19 a^{2} + 22 a\right)\cdot 29 + \left(4 a^{4} + 28 a^{3} + 13 a^{2} + 5 a + 22\right)\cdot 29^{2} + \left(12 a^{4} + 28 a + 22\right)\cdot 29^{3} + \left(24 a^{4} + 13 a^{3} + 2 a^{2} + 6\right)\cdot 29^{4} + \left(23 a^{4} + 13 a^{3} + 22 a^{2} + 28 a + 16\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 9 a^{4} + 18 a^{3} + 6 a^{2} + 20 a + 9 + \left(24 a^{4} + 5 a^{3} + 5 a^{2} + 10 a + 26\right)\cdot 29 + \left(15 a^{4} + 23 a^{3} + 16 a^{2} + 27 a + 7\right)\cdot 29^{2} + \left(14 a^{4} + 14 a^{3} + 23 a^{2} + 28 a + 20\right)\cdot 29^{3} + \left(21 a^{4} + 2 a^{3} + 20 a^{2} + 6 a + 3\right)\cdot 29^{4} + \left(28 a^{4} + 23 a^{3} + 15 a^{2} + 14 a + 25\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 9 a^{4} + 18 a^{3} + 6 a^{2} + 20 a + 11 + \left(24 a^{4} + 5 a^{3} + 5 a^{2} + 10 a + 9\right)\cdot 29 + \left(15 a^{4} + 23 a^{3} + 16 a^{2} + 27 a + 21\right)\cdot 29^{2} + \left(14 a^{4} + 14 a^{3} + 23 a^{2} + 28 a + 3\right)\cdot 29^{3} + \left(21 a^{4} + 2 a^{3} + 20 a^{2} + 6 a + 12\right)\cdot 29^{4} + \left(28 a^{4} + 23 a^{3} + 15 a^{2} + 14 a + 20\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 9 a^{4} + 18 a^{3} + 6 a^{2} + 20 a + 22 + \left(24 a^{4} + 5 a^{3} + 5 a^{2} + 10 a + 11\right)\cdot 29 + \left(15 a^{4} + 23 a^{3} + 16 a^{2} + 27 a + 9\right)\cdot 29^{2} + \left(14 a^{4} + 14 a^{3} + 23 a^{2} + 28 a + 5\right)\cdot 29^{3} + \left(21 a^{4} + 2 a^{3} + 20 a^{2} + 6 a + 17\right)\cdot 29^{4} + \left(28 a^{4} + 23 a^{3} + 15 a^{2} + 14 a + 10\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 13 a^{4} + 5 a^{3} + 2 a^{2} + 11 a + 7 + \left(9 a^{3} + 9 a^{2} + 25 a + 21\right)\cdot 29 + \left(28 a^{4} + 4 a^{3} + 27 a^{2} + 13 a + 19\right)\cdot 29^{2} + \left(a^{4} + 25 a^{3} + 16 a^{2} + 12 a + 1\right)\cdot 29^{3} + \left(6 a^{4} + 11 a^{3} + 23 a^{2} + a + 13\right)\cdot 29^{4} + \left(2 a^{4} + 3 a^{3} + 15 a^{2} + 28 a + 19\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 13 a^{4} + 5 a^{3} + 2 a^{2} + 11 a + 9 + \left(9 a^{3} + 9 a^{2} + 25 a + 4\right)\cdot 29 + \left(28 a^{4} + 4 a^{3} + 27 a^{2} + 13 a + 4\right)\cdot 29^{2} + \left(a^{4} + 25 a^{3} + 16 a^{2} + 12 a + 14\right)\cdot 29^{3} + \left(6 a^{4} + 11 a^{3} + 23 a^{2} + a + 21\right)\cdot 29^{4} + \left(2 a^{4} + 3 a^{3} + 15 a^{2} + 28 a + 14\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 13 a^{4} + 5 a^{3} + 2 a^{2} + 11 a + 20 + \left(9 a^{3} + 9 a^{2} + 25 a + 6\right)\cdot 29 + \left(28 a^{4} + 4 a^{3} + 27 a^{2} + 13 a + 21\right)\cdot 29^{2} + \left(a^{4} + 25 a^{3} + 16 a^{2} + 12 a + 15\right)\cdot 29^{3} + \left(6 a^{4} + 11 a^{3} + 23 a^{2} + a + 26\right)\cdot 29^{4} + \left(2 a^{4} + 3 a^{3} + 15 a^{2} + 28 a + 4\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 13 }$ $=$ \( 24 a^{4} + 14 a^{3} + 5 a^{2} + a + \left(6 a^{4} + 28 a^{3} + 15 a^{2} + 28 a + 22\right)\cdot 29 + \left(5 a^{4} + 2 a^{3} + 4 a^{2} + 8 a + 18\right)\cdot 29^{2} + \left(3 a^{4} + 21 a^{3} + 5 a^{2} + 8 a + 18\right)\cdot 29^{3} + \left(26 a^{4} + 12 a^{2} + 5 a + 16\right)\cdot 29^{4} + \left(28 a^{4} + 3 a^{3} + 21 a^{2} + 26 a + 22\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 14 }$ $=$ \( 24 a^{4} + 14 a^{3} + 5 a^{2} + a + 16 + \left(6 a^{4} + 28 a^{3} + 15 a^{2} + 28 a + 7\right)\cdot 29 + \left(5 a^{4} + 2 a^{3} + 4 a^{2} + 8 a + 17\right)\cdot 29^{2} + \left(3 a^{4} + 21 a^{3} + 5 a^{2} + 8 a + 4\right)\cdot 29^{3} + \left(26 a^{4} + 12 a^{2} + 5 a + 3\right)\cdot 29^{4} + \left(28 a^{4} + 3 a^{3} + 21 a^{2} + 26 a + 8\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display
$r_{ 15 }$ $=$ \( 24 a^{4} + 14 a^{3} + 5 a^{2} + a + 18 + \left(6 a^{4} + 28 a^{3} + 15 a^{2} + 28 a + 19\right)\cdot 29 + \left(5 a^{4} + 2 a^{3} + 4 a^{2} + 8 a + 1\right)\cdot 29^{2} + \left(3 a^{4} + 21 a^{3} + 5 a^{2} + 8 a + 17\right)\cdot 29^{3} + \left(26 a^{4} + 12 a^{2} + 5 a + 11\right)\cdot 29^{4} + \left(28 a^{4} + 3 a^{3} + 21 a^{2} + 26 a + 3\right)\cdot 29^{5} +O(29^{6})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

Cycle notation
$(1,10,7,14,4)(2,11,8,15,5)(3,12,9,13,6)$
$(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 15 }$ Character value
$1$$1$$()$$1$
$1$$3$$(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)$$-\zeta_{15}^{5} - 1$
$1$$3$$(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)$$\zeta_{15}^{5}$
$1$$5$$(1,10,7,14,4)(2,11,8,15,5)(3,12,9,13,6)$$\zeta_{15}^{6}$
$1$$5$$(1,7,4,10,14)(2,8,5,11,15)(3,9,6,12,13)$$-\zeta_{15}^{7} - \zeta_{15}^{2}$
$1$$5$$(1,14,10,4,7)(2,15,11,5,8)(3,13,12,6,9)$$\zeta_{15}^{3}$
$1$$5$$(1,4,14,7,10)(2,5,15,8,11)(3,6,13,9,12)$$\zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$
$1$$15$$(1,11,9,14,5,3,10,8,13,4,2,12,7,15,6)$$\zeta_{15}$
$1$$15$$(1,9,5,10,13,2,7,6,11,14,3,8,4,12,15)$$\zeta_{15}^{2}$
$1$$15$$(1,5,13,7,11,3,4,15,9,10,2,6,14,8,12)$$\zeta_{15}^{4}$
$1$$15$$(1,8,6,10,15,3,7,5,12,14,2,9,4,11,13)$$\zeta_{15}^{7}$
$1$$15$$(1,13,11,4,9,2,14,12,5,7,3,15,10,6,8)$$\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$
$1$$15$$(1,12,8,14,6,2,10,9,15,4,3,11,7,13,5)$$-\zeta_{15}^{6} - \zeta_{15}$
$1$$15$$(1,15,12,4,8,3,14,11,6,7,2,13,10,5,9)$$-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$
$1$$15$$(1,6,15,7,12,2,4,13,8,10,3,5,14,9,11)$$-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$

The blue line marks the conjugacy class containing complex conjugation.