Properties

Label 4.5e3_11e2_13e2.6t10.2c1
Dimension 4
Group $C_3^2:C_4$
Conductor $ 5^{3} \cdot 11^{2} \cdot 13^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:C_4$
Conductor:$2556125= 5^{3} \cdot 11^{2} \cdot 13^{2} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - 6 x^{4} + 19 x^{3} - x^{2} - 37 x + 31 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:C_4$
Parity: Even
Determinant: 1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 4 + 6\cdot 29 + 26\cdot 29^{2} + 22\cdot 29^{3} + 21\cdot 29^{4} + 9\cdot 29^{5} + 6\cdot 29^{6} + 9\cdot 29^{7} + 7\cdot 29^{8} + 7\cdot 29^{9} + 26\cdot 29^{10} + 13\cdot 29^{11} + 27\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 2 }$ $=$ $ 20 a + 14 + \left(26 a + 8\right)\cdot 29 + \left(10 a + 20\right)\cdot 29^{2} + \left(23 a + 28\right)\cdot 29^{3} + \left(11 a + 3\right)\cdot 29^{4} + \left(28 a + 2\right)\cdot 29^{5} + \left(5 a + 16\right)\cdot 29^{6} + 7\cdot 29^{7} + \left(3 a + 11\right)\cdot 29^{8} + \left(16 a + 9\right)\cdot 29^{9} + \left(11 a + 12\right)\cdot 29^{10} + \left(10 a + 1\right)\cdot 29^{11} + \left(15 a + 1\right)\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 3 }$ $=$ $ 9 a + 27 + \left(2 a + 5\right)\cdot 29 + \left(18 a + 19\right)\cdot 29^{2} + \left(5 a + 18\right)\cdot 29^{3} + \left(17 a + 10\right)\cdot 29^{4} + 16\cdot 29^{5} + \left(23 a + 17\right)\cdot 29^{6} + \left(28 a + 2\right)\cdot 29^{7} + \left(25 a + 26\right)\cdot 29^{8} + \left(12 a + 28\right)\cdot 29^{9} + \left(17 a + 24\right)\cdot 29^{10} + \left(18 a + 12\right)\cdot 29^{11} + \left(13 a + 9\right)\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 4 }$ $=$ $ 18 + 14\cdot 29 + 18\cdot 29^{2} + 10\cdot 29^{3} + 14\cdot 29^{4} + 10\cdot 29^{5} + 24\cdot 29^{6} + 18\cdot 29^{7} + 20\cdot 29^{8} + 19\cdot 29^{9} + 20\cdot 29^{10} + 14\cdot 29^{11} + 18\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 5 }$ $=$ $ 14 a + 7 + \left(12 a + 16\right)\cdot 29 + \left(10 a + 10\right)\cdot 29^{2} + \left(5 a + 9\right)\cdot 29^{3} + \left(a + 3\right)\cdot 29^{4} + \left(16 a + 28\right)\cdot 29^{5} + \left(19 a + 13\right)\cdot 29^{6} + \left(17 a + 4\right)\cdot 29^{7} + \left(18 a + 2\right)\cdot 29^{8} + \left(13 a + 15\right)\cdot 29^{9} + \left(23 a + 7\right)\cdot 29^{10} + \left(25 a + 27\right)\cdot 29^{11} + 25\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 6 }$ $=$ $ 15 a + 19 + \left(16 a + 6\right)\cdot 29 + \left(18 a + 21\right)\cdot 29^{2} + \left(23 a + 25\right)\cdot 29^{3} + \left(27 a + 3\right)\cdot 29^{4} + \left(12 a + 20\right)\cdot 29^{5} + \left(9 a + 8\right)\cdot 29^{6} + \left(11 a + 15\right)\cdot 29^{7} + \left(10 a + 19\right)\cdot 29^{8} + \left(15 a + 6\right)\cdot 29^{9} + \left(5 a + 24\right)\cdot 29^{10} + \left(3 a + 16\right)\cdot 29^{11} + \left(28 a + 4\right)\cdot 29^{12} +O\left(29^{ 13 }\right)$

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

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

Character values on conjugacy classes

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