Properties

Label 4.7842780125.6t10.b.a
Dimension $4$
Group $C_3^2:C_4$
Conductor $7842780125$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:C_4$
Conductor: \(7842780125\)\(\medspace = 5^{3} \cdot 89^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.39213900625.1
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 6.2.39213900625.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} - 56x^{4} + 28x^{3} + 686x^{2} + 1925x - 8345 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.