Properties

Label 2.2475.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $2475$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(2475\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 11 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.12375.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.11.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 4x^{2} - 11x - 29 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 2 + 42\cdot 59 + 21\cdot 59^{2} + 23\cdot 59^{3} +O(59^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 + 11\cdot 59 + 4\cdot 59^{2} + 22\cdot 59^{3} + 36\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 41 + 49\cdot 59 + 13\cdot 59^{2} + 50\cdot 59^{3} + 10\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 52 + 14\cdot 59 + 19\cdot 59^{2} + 22\cdot 59^{3} + 11\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display

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.