Properties

Label 2.16075.6t3.b.a
Dimension 2
Group $D_{6}$
Conductor $ 5^{2} \cdot 643 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$16075= 5^{2} \cdot 643 $
Artin number field: Splitting field of 6.0.33230963375.1 defined by $f= x^{6} - 16 x^{4} - 55 x^{3} + 64 x^{2} + 440 x + 1560 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{6}$
Parity: Odd
Determinant: 1.643.2t1.a.a
Projective image: $S_3$
Projective field: Galois closure of 3.1.643.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 8\cdot 13 + 4\cdot 13^{2} + 8\cdot 13^{3} + 11\cdot 13^{4} + 2\cdot 13^{5} + 11\cdot 13^{6} + 10\cdot 13^{7} + 6\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 2 }$ $=$ $ a + 6 + \left(8 a + 5\right)\cdot 13 + \left(3 a + 6\right)\cdot 13^{2} + \left(5 a + 1\right)\cdot 13^{3} + \left(8 a + 12\right)\cdot 13^{4} + \left(5 a + 12\right)\cdot 13^{5} + \left(7 a + 12\right)\cdot 13^{6} + \left(4 a + 8\right)\cdot 13^{7} + \left(2 a + 10\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 4 a + 3 + \left(5 a + 2\right)\cdot 13 + \left(4 a + 2\right)\cdot 13^{2} + \left(4 a + 12\right)\cdot 13^{3} + \left(10 a + 1\right)\cdot 13^{4} + \left(4 a + 5\right)\cdot 13^{5} + 12 a\cdot 13^{6} + 9\cdot 13^{7} + \left(11 a + 11\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 3 + 7\cdot 13 + 9\cdot 13^{2} + 13^{3} + 3\cdot 13^{4} + 8\cdot 13^{5} + 4\cdot 13^{6} + 6\cdot 13^{7} + 5\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 12 a + 7 + \left(4 a + 12\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(7 a + 3\right)\cdot 13^{3} + \left(4 a + 2\right)\cdot 13^{4} + \left(7 a + 10\right)\cdot 13^{5} + \left(5 a + 1\right)\cdot 13^{6} + \left(8 a + 6\right)\cdot 13^{7} + \left(10 a + 8\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 9 a + 7 + \left(7 a + 3\right)\cdot 13 + \left(8 a + 1\right)\cdot 13^{2} + \left(8 a + 12\right)\cdot 13^{3} + \left(2 a + 7\right)\cdot 13^{4} + \left(8 a + 12\right)\cdot 13^{5} + 7\cdot 13^{6} + \left(12 a + 10\right)\cdot 13^{7} + \left(a + 8\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,4)(2,3)(5,6)$$-2$
$3$$2$$(2,5)(3,6)$$0$
$3$$2$$(1,3)(2,4)(5,6)$$0$
$2$$3$$(1,2,5)(3,6,4)$$-1$
$2$$6$$(1,3,5,4,2,6)$$1$
The blue line marks the conjugacy class containing complex conjugation.