Properties

Label 2.3757.8t7.a.a
Dimension $2$
Group $C_8:C_2$
Conductor $3757$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(3757\)\(\medspace = 13 \cdot 17^{2} \)
Artin stem field: Galois closure of 8.0.69347235737.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: even
Determinant: 1.221.4t1.a.b
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{13}, \sqrt{17})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} + 10x^{6} - 11x^{5} + 15x^{4} + 7x^{3} + 24x^{2} + 21x + 103 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 6 + 48\cdot 101 + 11\cdot 101^{2} + 56\cdot 101^{3} + 88\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 47 + 85\cdot 101 + 71\cdot 101^{2} + 10\cdot 101^{3} + 41\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 51 + 97\cdot 101 + 62\cdot 101^{2} + 56\cdot 101^{3} + 10\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 62 + 30\cdot 101 + 60\cdot 101^{2} + 14\cdot 101^{3} + 66\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 79 + 38\cdot 101 + 77\cdot 101^{2} + 10\cdot 101^{3} + 92\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 80 + 63\cdot 101 + 76\cdot 101^{2} + 15\cdot 101^{3} + 35\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 82 + 18\cdot 101 + 37\cdot 101^{2} + 7\cdot 101^{3} + 12\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 99 + 20\cdot 101 + 6\cdot 101^{2} + 30\cdot 101^{3} + 58\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,5,8,2)(3,4,7,6)$
$(1,7,2,6,8,3,5,4)$
$(3,7)(4,6)$
$(1,8)(2,5)(3,7)(4,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,5)(3,7)(4,6)$$-2$
$2$$2$$(3,7)(4,6)$$0$
$1$$4$$(1,2,8,5)(3,4,7,6)$$2 \zeta_{4}$
$1$$4$$(1,5,8,2)(3,6,7,4)$$-2 \zeta_{4}$
$2$$4$$(1,5,8,2)(3,4,7,6)$$0$
$2$$8$$(1,7,2,6,8,3,5,4)$$0$
$2$$8$$(1,6,5,7,8,4,2,3)$$0$
$2$$8$$(1,6,2,3,8,4,5,7)$$0$
$2$$8$$(1,3,5,6,8,7,2,4)$$0$

The blue line marks the conjugacy class containing complex conjugation.