Properties

Label 4.5e2_821.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 5^{2} \cdot 821 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$20525= 5^{2} \cdot 821 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} + x^{3} - x^{2} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.821.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,2)(3,4)(5,6)$$0$
$6$$2$$(3,6)$$2$
$9$$2$$(3,6)(4,5)$$0$
$4$$3$$(1,4,5)$$1$
$4$$3$$(1,4,5)(2,3,6)$$-2$
$18$$4$$(1,2)(3,5,6,4)$$0$
$12$$6$$(1,3,4,6,5,2)$$0$
$12$$6$$(1,4,5)(3,6)$$-1$
The blue line marks the conjugacy class containing complex conjugation.