Properties

Label 4.435125.6t10.b.a
Dimension $4$
Group $C_3^2:C_4$
Conductor $435125$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:C_4$
Conductor: \(435125\)\(\medspace = 5^{3} \cdot 59^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.7573350625.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:C_4$
Parity: even
Determinant: 1.5.2t1.a.a
Projective image: $C_3^2:C_4$
Projective stem field: Galois closure of 6.2.7573350625.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + 41x^{4} - 51x^{3} + 411x^{2} - 220x - 4321 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 10\cdot 29 + 11\cdot 29^{2} + 14\cdot 29^{3} + 28\cdot 29^{4} + 29^{5} + 18\cdot 29^{7} + 12\cdot 29^{8} + 26\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 27 + \left(15 a + 25\right)\cdot 29 + 14\cdot 29^{2} + \left(17 a + 26\right)\cdot 29^{3} + \left(25 a + 13\right)\cdot 29^{4} + \left(16 a + 9\right)\cdot 29^{5} + \left(12 a + 15\right)\cdot 29^{6} + \left(15 a + 16\right)\cdot 29^{7} + \left(26 a + 2\right)\cdot 29^{8} + \left(18 a + 17\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 24 + 19\cdot 29 + 11\cdot 29^{2} + 7\cdot 29^{3} + 6\cdot 29^{4} + 9\cdot 29^{5} + 10\cdot 29^{6} + 18\cdot 29^{7} + 22\cdot 29^{8} + 13\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 22 + \left(a + 22\right)\cdot 29 + \left(5 a + 25\right)\cdot 29^{2} + \left(25 a + 4\right)\cdot 29^{3} + \left(25 a + 6\right)\cdot 29^{4} + \left(18 a + 8\right)\cdot 29^{5} + \left(19 a + 18\right)\cdot 29^{6} + \left(16 a + 2\right)\cdot 29^{7} + \left(2 a + 10\right)\cdot 29^{8} + \left(4 a + 21\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 27 a + 8 + \left(13 a + 12\right)\cdot 29 + \left(28 a + 2\right)\cdot 29^{2} + \left(11 a + 24\right)\cdot 29^{3} + \left(3 a + 8\right)\cdot 29^{4} + \left(12 a + 10\right)\cdot 29^{5} + \left(16 a + 3\right)\cdot 29^{6} + \left(13 a + 23\right)\cdot 29^{7} + \left(2 a + 3\right)\cdot 29^{8} + \left(10 a + 27\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 26 a + 8 + \left(27 a + 25\right)\cdot 29 + \left(23 a + 20\right)\cdot 29^{2} + \left(3 a + 9\right)\cdot 29^{3} + \left(3 a + 23\right)\cdot 29^{4} + \left(10 a + 18\right)\cdot 29^{5} + \left(9 a + 10\right)\cdot 29^{6} + \left(12 a + 8\right)\cdot 29^{7} + \left(26 a + 6\right)\cdot 29^{8} + \left(24 a + 10\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$9$$2$$(1,4)(2,3)$$0$
$4$$3$$(2,3,5)$$-2$
$4$$3$$(1,4,6)(2,3,5)$$1$
$9$$4$$(1,3,4,2)(5,6)$$0$
$9$$4$$(1,2,4,3)(5,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.