Properties

Label 1.35.12t1.a.a
Dimension $1$
Group $C_{12}$
Conductor $35$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_{12}$
Conductor: \(35\)\(\medspace = 5 \cdot 7 \)
Artin field: 12.0.11259376953125.1
Galois orbit size: $4$
Smallest permutation container: $C_{12}$
Parity: odd
Dirichlet character: \(\chi_{35}(2,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 9 x^{8} + 2 x^{7} + 12 x^{6} + x^{5} + 25 x^{4} - 11 x^{3} + 5 x^{2} - 2 x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{4} + 3 x^{2} + 12 x + 2\)  Toggle raw display

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

Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

Cycle notation
$(1,2,3,9,7,11)(4,12,5,6,10,8)$
$(1,4,9,6)(2,12,7,10)(3,5,11,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 12 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,9)(2,7)(3,11)(4,6)(5,8)(10,12)$$-1$
$1$$3$$(1,3,7)(2,9,11)(4,5,10)(6,8,12)$$-\zeta_{12}^{2}$
$1$$3$$(1,7,3)(2,11,9)(4,10,5)(6,12,8)$$\zeta_{12}^{2} - 1$
$1$$4$$(1,4,9,6)(2,12,7,10)(3,5,11,8)$$\zeta_{12}^{3}$
$1$$4$$(1,6,9,4)(2,10,7,12)(3,8,11,5)$$-\zeta_{12}^{3}$
$1$$6$$(1,2,3,9,7,11)(4,12,5,6,10,8)$$-\zeta_{12}^{2} + 1$
$1$$6$$(1,11,7,9,3,2)(4,8,10,6,5,12)$$\zeta_{12}^{2}$
$1$$12$$(1,12,11,4,7,8,9,10,3,6,2,5)$$\zeta_{12}$
$1$$12$$(1,8,2,4,3,12,9,5,7,6,11,10)$$\zeta_{12}^{3} - \zeta_{12}$
$1$$12$$(1,10,11,6,7,5,9,12,3,4,2,8)$$-\zeta_{12}$
$1$$12$$(1,5,2,6,3,10,9,8,7,4,11,12)$$-\zeta_{12}^{3} + \zeta_{12}$

The blue line marks the conjugacy class containing complex conjugation.