Properties

Label 1.680.4t1.h.b
Dimension 1
Group $C_4$
Conductor $ 2^{3} \cdot 5 \cdot 17 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$680= 2^{3} \cdot 5 \cdot 17 $
Artin number field: Splitting field of 4.0.2312000.1 defined by $f= x^{4} + 170 x^{2} + 5780 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{680}(373,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 8.
Roots:
$r_{ 1 }$ $=$ $ 1 + 3\cdot 11 + 7\cdot 11^{3} + 2\cdot 11^{4} + 3\cdot 11^{5} + 4\cdot 11^{6} + 8\cdot 11^{7} +O\left(11^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 4 + 4\cdot 11 + 10\cdot 11^{2} + 9\cdot 11^{3} + 11^{4} + 10\cdot 11^{5} + 3\cdot 11^{6} + 10\cdot 11^{7} +O\left(11^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 7 + 6\cdot 11 + 11^{3} + 9\cdot 11^{4} + 7\cdot 11^{6} +O\left(11^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 10 + 7\cdot 11 + 10\cdot 11^{2} + 3\cdot 11^{3} + 8\cdot 11^{4} + 7\cdot 11^{5} + 6\cdot 11^{6} + 2\cdot 11^{7} +O\left(11^{ 8 }\right)$

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.