Properties

Label 4.7260125.12t34.a.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $7260125$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(7260125\)\(\medspace = 5^{3} \cdot 241^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.30125.1
Galois orbit size: $1$
Smallest permutation container: 12T34
Parity: even
Determinant: 1.5.2t1.a.a
Projective image: $S_3\wr C_2$
Projective field: Galois closure of 6.2.30125.1

Defining polynomial

$f(x)$$=$$ x^{6} - x^{5} + x^{4} - 2 x^{2} + x - 1 $.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.