Properties

Label 1.112.4t1.b.a
Dimension $1$
Group $C_4$
Conductor $112$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(112\)\(\medspace = 2^{4} \cdot 7 \)
Artin field: Galois closure of 4.0.100352.5
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{112}(69,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 28x^{2} + 98 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 8 + 21\cdot 41 + 31\cdot 41^{2} + 4\cdot 41^{3} + 31\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 20 + 20\cdot 41 + 12\cdot 41^{2} + 5\cdot 41^{3} + 11\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 21 + 20\cdot 41 + 28\cdot 41^{2} + 35\cdot 41^{3} + 29\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 33 + 19\cdot 41 + 9\cdot 41^{2} + 36\cdot 41^{3} + 9\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.