Properties

Label 1.75.10t1.a.d
Dimension $1$
Group $C_{10}$
Conductor $75$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_{10}$
Conductor: \(75\)\(\medspace = 3 \cdot 5^{2} \)
Artin field: Galois closure of 10.0.37078857421875.1
Galois orbit size: $4$
Smallest permutation container: $C_{10}$
Parity: odd
Dirichlet character: \(\chi_{75}(71,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{10} + 10x^{8} - 10x^{7} + 90x^{6} - 49x^{5} + 125x^{4} + 70x^{3} + 95x^{2} + 10x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{5} + 4x + 11 \) Copy content Toggle raw display

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

Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.