Properties

Label 2.5e2_11.4t3.1c1
Dimension 2
Group $D_{4}$
Conductor $ 5^{2} \cdot 11 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$275= 5^{2} \cdot 11 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + x^{2} + 4 x - 4 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.11.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 39\cdot 71 + 35\cdot 71^{2} + 23\cdot 71^{3} + 19\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 + 57\cdot 71 + 48\cdot 71^{2} + 54\cdot 71^{3} + 24\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 + 40\cdot 71 + 63\cdot 71^{2} + 52\cdot 71^{3} + 22\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 44 + 5\cdot 71 + 65\cdot 71^{2} + 10\cdot 71^{3} + 4\cdot 71^{4} +O\left(71^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

Character values on conjugacy classes

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