Properties

Label 4.962948.6t10.a.a
Dimension 4
Group $C_3^2:C_4$
Conductor $ 2^{2} \cdot 7^{2} \cdot 17^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:C_4$
Conductor:$962948= 2^{2} \cdot 7^{2} \cdot 17^{3} $
Artin number field: Splitting field of 6.2.16370116.1 defined by $f= x^{6} - 2 x^{5} + 7 x^{4} + 2 x^{3} + x^{2} + 24 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:C_4$
Parity: Even
Determinant: 1.17.2t1.a.a
Projective image: $C_3:S_3.C_2$
Projective field: Galois closure of 6.2.16370116.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 18.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 7 a + 5 + \left(7 a + 7\right)\cdot 13 + \left(8 a + 7\right)\cdot 13^{2} + 7 a\cdot 13^{3} + \left(3 a + 3\right)\cdot 13^{4} + \left(11 a + 4\right)\cdot 13^{5} + \left(12 a + 12\right)\cdot 13^{6} + \left(6 a + 4\right)\cdot 13^{7} + \left(6 a + 6\right)\cdot 13^{8} + \left(4 a + 11\right)\cdot 13^{9} + 6 a\cdot 13^{10} + \left(3 a + 6\right)\cdot 13^{11} + \left(4 a + 5\right)\cdot 13^{12} + \left(7 a + 1\right)\cdot 13^{13} + \left(12 a + 8\right)\cdot 13^{14} + \left(6 a + 12\right)\cdot 13^{15} + \left(8 a + 5\right)\cdot 13^{16} + 13^{17} +O\left(13^{ 18 }\right)$
$r_{ 2 }$ $=$ $ 5 + 9\cdot 13 + 4\cdot 13^{2} + 11\cdot 13^{3} + 2\cdot 13^{4} + 3\cdot 13^{6} + 5\cdot 13^{7} + 11\cdot 13^{8} + 9\cdot 13^{9} + 2\cdot 13^{11} + 10\cdot 13^{12} + 7\cdot 13^{13} + 8\cdot 13^{14} + 2\cdot 13^{15} + 9\cdot 13^{16} + 9\cdot 13^{17} +O\left(13^{ 18 }\right)$
$r_{ 3 }$ $=$ $ 10 + 10\cdot 13 + 9\cdot 13^{2} + 12\cdot 13^{3} + 10\cdot 13^{4} + 9\cdot 13^{5} + 12\cdot 13^{6} + 8\cdot 13^{7} + 5\cdot 13^{9} + 9\cdot 13^{10} + 3\cdot 13^{11} + 13^{12} + 7\cdot 13^{13} + 4\cdot 13^{14} + 6\cdot 13^{15} + 12\cdot 13^{16} + 4\cdot 13^{17} +O\left(13^{ 18 }\right)$
$r_{ 4 }$ $=$ $ 10 a + 6 + \left(3 a + 11\right)\cdot 13 + \left(11 a + 6\right)\cdot 13^{2} + 11 a\cdot 13^{3} + \left(9 a + 6\right)\cdot 13^{4} + \left(a + 10\right)\cdot 13^{5} + \left(4 a + 3\right)\cdot 13^{6} + 11 a\cdot 13^{7} + \left(11 a + 7\right)\cdot 13^{8} + \left(7 a + 3\right)\cdot 13^{9} + \left(12 a + 10\right)\cdot 13^{10} + \left(2 a + 3\right)\cdot 13^{11} + \left(3 a + 1\right)\cdot 13^{12} + \left(5 a + 8\right)\cdot 13^{13} + \left(9 a + 6\right)\cdot 13^{14} + \left(10 a + 4\right)\cdot 13^{15} + \left(a + 6\right)\cdot 13^{16} + \left(a + 8\right)\cdot 13^{17} +O\left(13^{ 18 }\right)$
$r_{ 5 }$ $=$ $ 6 a + 12 + \left(5 a + 7\right)\cdot 13 + \left(4 a + 8\right)\cdot 13^{2} + \left(5 a + 12\right)\cdot 13^{3} + \left(9 a + 11\right)\cdot 13^{4} + \left(a + 11\right)\cdot 13^{5} + \left(6 a + 12\right)\cdot 13^{7} + \left(6 a + 5\right)\cdot 13^{8} + \left(8 a + 9\right)\cdot 13^{9} + \left(6 a + 2\right)\cdot 13^{10} + \left(9 a + 3\right)\cdot 13^{11} + \left(8 a + 6\right)\cdot 13^{12} + \left(5 a + 4\right)\cdot 13^{13} + \left(6 a + 7\right)\cdot 13^{15} + \left(4 a + 7\right)\cdot 13^{16} + \left(12 a + 6\right)\cdot 13^{17} +O\left(13^{ 18 }\right)$
$r_{ 6 }$ $=$ $ 3 a + 3 + \left(9 a + 5\right)\cdot 13 + \left(a + 1\right)\cdot 13^{2} + \left(a + 1\right)\cdot 13^{3} + \left(3 a + 4\right)\cdot 13^{4} + \left(11 a + 2\right)\cdot 13^{5} + \left(8 a + 6\right)\cdot 13^{6} + \left(a + 7\right)\cdot 13^{7} + \left(a + 7\right)\cdot 13^{8} + \left(5 a + 12\right)\cdot 13^{9} + 13^{10} + \left(10 a + 7\right)\cdot 13^{11} + \left(9 a + 1\right)\cdot 13^{12} + \left(7 a + 10\right)\cdot 13^{13} + \left(3 a + 10\right)\cdot 13^{14} + \left(2 a + 5\right)\cdot 13^{15} + \left(11 a + 10\right)\cdot 13^{16} + \left(11 a + 7\right)\cdot 13^{17} +O\left(13^{ 18 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$9$$2$$(1,3)(2,4)$$0$
$4$$3$$(1,3,5)$$1$
$4$$3$$(1,3,5)(2,4,6)$$-2$
$9$$4$$(1,2,3,4)(5,6)$$0$
$9$$4$$(1,4,3,2)(5,6)$$0$
The blue line marks the conjugacy class containing complex conjugation.