Properties

Label 2.5952.4t3.e.a
Dimension $2$
Group $D_{4}$
Conductor $5952$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(5952\)\(\medspace = 2^{6} \cdot 3 \cdot 31 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.738048.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.372.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{31})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 4x^{2} - 27 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 29 + 55\cdot 79 + 60\cdot 79^{2} + 44\cdot 79^{3} + 9\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 36 + 42\cdot 79 + 64\cdot 79^{2} + 11\cdot 79^{3} + 73\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 + 36\cdot 79 + 14\cdot 79^{2} + 67\cdot 79^{3} + 5\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 50 + 23\cdot 79 + 18\cdot 79^{2} + 34\cdot 79^{3} + 69\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.