Properties

Label 4.3e2_1777.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 3^{2} \cdot 1777 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$15993= 3^{2} \cdot 1777 $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 2 x^{4} - x^{3} + x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.1777.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 9 a + 11 + \left(4 a + 5\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(9 a + 9\right)\cdot 13^{3} + \left(8 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 + 7\cdot 13 + 8\cdot 13^{2} + 11\cdot 13^{3} + 9\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 1 + \left(12 a + 10\right)\cdot 13 + 12\cdot 13^{2} + \left(12 a + 1\right)\cdot 13^{3} + 9 a\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a + 7 + \left(8 a + 1\right)\cdot 13 + \left(a + 2\right)\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(4 a + 3\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 7 + 3\cdot 13 + \left(12 a + 1\right)\cdot 13^{2} + \left(3 a + 11\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 10 + 10\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 9\cdot 13^{4} +O\left(13^{ 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$$(1,3)(2,5)(4,6)$$0$
$6$$2$$(2,4)$$2$
$9$$2$$(2,4)(5,6)$$0$
$4$$3$$(1,2,4)$$1$
$4$$3$$(1,2,4)(3,5,6)$$-2$
$18$$4$$(1,3)(2,6,4,5)$$0$
$12$$6$$(1,5,2,6,4,3)$$0$
$12$$6$$(2,4)(3,5,6)$$-1$
The blue line marks the conjugacy class containing complex conjugation.