Properties

Label 2.2736.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $2736$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(2736\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.8208.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.19.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-3}, \sqrt{-19})\)

Defining polynomial

$f(x)$$=$\(x^{4} + 15 x^{2} + 57\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 8.

Roots:
$r_{ 1 }$ $=$ \( 2 + 2\cdot 7 + 4\cdot 7^{2} + 3\cdot 7^{3} + 4\cdot 7^{4} + 4\cdot 7^{6} + 7^{7} +O(7^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 5\cdot 7 + 2\cdot 7^{2} + 7^{3} + 4\cdot 7^{4} + 6\cdot 7^{5} + 6\cdot 7^{6} + 2\cdot 7^{7} +O(7^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 4 + 7 + 4\cdot 7^{2} + 5\cdot 7^{3} + 2\cdot 7^{4} + 4\cdot 7^{7} +O(7^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 5 + 4\cdot 7 + 2\cdot 7^{2} + 3\cdot 7^{3} + 2\cdot 7^{4} + 6\cdot 7^{5} + 2\cdot 7^{6} + 5\cdot 7^{7} +O(7^{8})\)  Toggle raw display

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

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

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.