Properties

Label 3.1421.6t6.a.a
Dimension $3$
Group $A_4\times C_2$
Conductor $1421$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $3$
Group: $A_4\times C_2$
Conductor: \(1421\)\(\medspace = 7^{2} \cdot 29 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.69629.1
Galois orbit size: $1$
Smallest permutation container: $A_4\times C_2$
Parity: even
Determinant: 1.29.2t1.a.a
Projective image: $A_4$
Projective stem field: Galois closure of 4.0.41209.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + x^{4} + 3x^{3} - 4x^{2} + 3x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.