Properties

Label 2.135.3t2.a.a
Dimension $2$
Group $S_3$
Conductor $135$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(135\)\(\medspace = 3^{3} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin field: Galois closure of 6.0.273375.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.15.2t1.a.a
Projective image: $S_3$
Projective field: Galois closure of 6.0.273375.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.