Properties

Label 1.7_11_13.4t1.1
Dimension 1
Group $C_4$
Conductor $ 7 \cdot 11 \cdot 13 $
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$1001= 7 \cdot 11 \cdot 13 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 249 x^{2} + 251 x + 4943 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 25 + 35\cdot 43 + 11\cdot 43^{2} + 29\cdot 43^{3} + 11\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 30 + 5\cdot 43 + 29\cdot 43^{2} + 37\cdot 43^{3} + 12\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 34 + 4\cdot 43 + 40\cdot 43^{2} + 23\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 41 + 39\cdot 43 + 4\cdot 43^{2} + 38\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $2$ $(1,2)(3,4)$ $-1$ $-1$
$1$ $4$ $(1,3,2,4)$ $\zeta_{4}$ $-\zeta_{4}$
$1$ $4$ $(1,4,2,3)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.