Properties

Label 2.2e4_5_19.4t3.7c1
Dimension 2
Group $D_4$
Conductor $ 2^{4} \cdot 5 \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$1520= 2^{4} \cdot 5 \cdot 19 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} - 38 x^{6} + 2 x^{5} + 458 x^{4} + 562 x^{3} - 1438 x^{2} - 3402 x - 1899 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Even
Determinant: 1.2e2_5_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 12\cdot 61 + 52\cdot 61^{2} + 52\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 + 57\cdot 61 + 35\cdot 61^{2} + 57\cdot 61^{3} + 42\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 + 49\cdot 61 + 31\cdot 61^{2} + 19\cdot 61^{3} + 32\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 29 + 21\cdot 61 + 9\cdot 61^{2} + 24\cdot 61^{3} + 26\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 42 + 47\cdot 61 + 2\cdot 61^{2} + 51\cdot 61^{3} + 32\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 45 + 36\cdot 61 + 19\cdot 61^{2} + 44\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 53 + 2\cdot 61 + 25\cdot 61^{2} + 18\cdot 61^{3} + 34\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 55 + 16\cdot 61 + 6\cdot 61^{2} + 37\cdot 61^{3} + 11\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

Character values on conjugacy classes

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