Properties

Label 3.7e2_19e2.4t4.1c1
Dimension 3
Group $A_4$
Conductor $ 7^{2} \cdot 19^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_4$
Conductor:$17689= 7^{2} \cdot 19^{2} $
Artin number field: Splitting field of $f= x^{4} + 3 x^{2} - 7 x + 4 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_4$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 109 + 18\cdot 163 + 38\cdot 163^{2} + 114\cdot 163^{3} + 15\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 117 + 87\cdot 163 + 157\cdot 163^{2} + 15\cdot 163^{3} + 62\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 119 + 122\cdot 163 + 70\cdot 163^{2} + 88\cdot 163^{3} + 108\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 144 + 96\cdot 163 + 59\cdot 163^{2} + 107\cdot 163^{3} + 139\cdot 163^{4} +O\left(163^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$3$
$3$$2$$(1,2)(3,4)$$-1$
$4$$3$$(1,2,3)$$0$
$4$$3$$(1,3,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.