Properties

Label 2.3275.10t3.a.b
Dimension $2$
Group $D_{10}$
Conductor $3275$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{10}$
Conductor: \(3275\)\(\medspace = 5^{2} \cdot 131 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 10.0.120560905159375.1
Galois orbit size: $2$
Smallest permutation container: $D_{10}$
Parity: odd
Determinant: 1.131.2t1.a.a
Projective image: $D_5$
Projective stem field: 5.1.17161.1

Defining polynomial

$f(x)$$=$\(x^{10} - 2 x^{9} + 7 x^{8} - 13 x^{7} - x^{6} + 67 x^{5} + 54 x^{4} - 55 x^{3} - 340 x^{2} - 276 x + 1424\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{5} + 10 x^{2} + 9\)  Toggle raw display

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,3)(4,10)(5,6)(7,9)$$-2$
$5$$2$$(1,3)(2,8)(4,7)(9,10)$$0$
$5$$2$$(1,2)(3,8)(4,9)(5,6)(7,10)$$0$
$2$$5$$(1,3,4,5,7)(2,10,6,9,8)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$$5$$(1,4,7,3,5)(2,6,8,10,9)$$\zeta_{5}^{3} + \zeta_{5}^{2}$
$2$$10$$(1,2,4,6,7,8,3,10,5,9)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$2$$10$$(1,6,3,9,4,8,5,2,7,10)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.