Properties

Label 2.3600.4t3.f.a
Dimension $2$
Group $D_{4}$
Conductor $3600$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.10800.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\zeta_{12})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 15x^{2} + 75 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 4 + 13 + 2\cdot 13^{2} + 8\cdot 13^{3} + 5\cdot 13^{4} + 12\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 12\cdot 13 + 5\cdot 13^{2} + 12\cdot 13^{3} + 7\cdot 13^{4} + 11\cdot 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 + 7\cdot 13^{2} + 5\cdot 13^{4} + 13^{5} +O(13^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 + 11\cdot 13 + 10\cdot 13^{2} + 4\cdot 13^{3} + 7\cdot 13^{4} +O(13^{6})\) Copy content 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.