Properties

Label 4.1865461525.12t34.a.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $1865461525$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(1865461525\)\(\medspace = 5^{2} \cdot 421^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.373092305.1
Galois orbit size: $1$
Smallest permutation container: 12T34
Parity: even
Determinant: 1.421.2t1.a.a
Projective image: $\SOPlus(4,2)$
Projective stem field: Galois closure of 6.2.373092305.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + x^{4} - 21x^{3} + 21x^{2} + 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 7 a + 5 + \left(25 a + 5\right)\cdot 31 + \left(5 a + 6\right)\cdot 31^{2} + \left(16 a + 8\right)\cdot 31^{3} + \left(27 a + 24\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a + 19 + \left(5 a + 17\right)\cdot 31 + \left(25 a + 23\right)\cdot 31^{2} + \left(14 a + 3\right)\cdot 31^{3} + \left(3 a + 1\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 + 8\cdot 31 + 31^{2} + 19\cdot 31^{3} + 5\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 16 a + 13 + \left(15 a + 28\right)\cdot 31 + \left(18 a + 15\right)\cdot 31^{2} + \left(15 a + 6\right)\cdot 31^{3} + \left(28 a + 13\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 5 + 21\cdot 31 + 8\cdot 31^{2} + 5\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 15 a + 14 + \left(15 a + 12\right)\cdot 31 + \left(12 a + 6\right)\cdot 31^{2} + \left(15 a + 19\right)\cdot 31^{3} + \left(2 a + 23\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,4)(2,5)(3,6)$$-2$
$6$$2$$(1,2)$$0$
$9$$2$$(1,2)(4,5)$$0$
$4$$3$$(1,2,3)(4,5,6)$$1$
$4$$3$$(1,2,3)$$-2$
$18$$4$$(1,5,2,4)(3,6)$$0$
$12$$6$$(1,5,2,6,3,4)$$1$
$12$$6$$(1,2)(4,5,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.