Properties

Label 2.3_7_19.4t3.5c1
Dimension 2
Group $D_4$
Conductor $ 3 \cdot 7 \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$399= 3 \cdot 7 \cdot 19 $
Artin number field: Splitting field of $f= x^{8} + 11 x^{6} + 124 x^{4} - 33 x^{2} + 9 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.3_7_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 49 + 116\cdot 223 + 74\cdot 223^{2} + 150\cdot 223^{3} + 169\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 53 + 111\cdot 223 + 101\cdot 223^{2} + 223^{3} + 89\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 96 + 112\cdot 223 + 157\cdot 223^{2} + 128\cdot 223^{3} + 70\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 110 + 201\cdot 223 + 112\cdot 223^{2} + 59\cdot 223^{3} + 103\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 113 + 21\cdot 223 + 110\cdot 223^{2} + 163\cdot 223^{3} + 119\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 127 + 110\cdot 223 + 65\cdot 223^{2} + 94\cdot 223^{3} + 152\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 170 + 111\cdot 223 + 121\cdot 223^{2} + 221\cdot 223^{3} + 133\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 174 + 106\cdot 223 + 148\cdot 223^{2} + 72\cdot 223^{3} + 53\cdot 223^{4} +O\left(223^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$2$$2$$(1,3)(2,4)(5,7)(6,8)$$0$
$2$$2$$(1,4)(2,6)(3,7)(5,8)$$0$
$2$$4$$(1,2,8,7)(3,5,6,4)$$0$
The blue line marks the conjugacy class containing complex conjugation.