Properties

Label 1.2e3_5_251.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 2^{3} \cdot 5 \cdot 251 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$10040= 2^{3} \cdot 5 \cdot 251 $
Artin number field: Splitting field of $f= x^{4} + 2510 x^{2} + 1260020 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{10040}(1003,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 8.
Roots:
$r_{ 1 }$ $=$ $ 1 + 4\cdot 19 + 6\cdot 19^{2} + 4\cdot 19^{3} + 19^{4} + 15\cdot 19^{5} + 2\cdot 19^{6} + 8\cdot 19^{7} +O\left(19^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 4 + 18\cdot 19 + 4\cdot 19^{2} + 8\cdot 19^{3} + 6\cdot 19^{4} + 6\cdot 19^{5} + 6\cdot 19^{6} + 7\cdot 19^{7} +O\left(19^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 15 + 14\cdot 19^{2} + 10\cdot 19^{3} + 12\cdot 19^{4} + 12\cdot 19^{5} + 12\cdot 19^{6} + 11\cdot 19^{7} +O\left(19^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 18 + 14\cdot 19 + 12\cdot 19^{2} + 14\cdot 19^{3} + 17\cdot 19^{4} + 3\cdot 19^{5} + 16\cdot 19^{6} + 10\cdot 19^{7} +O\left(19^{ 8 }\right)$

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

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

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$$4$$(1,2,4,3)$$-\zeta_{4}$
$1$$4$$(1,3,4,2)$$\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.