Properties

Label 5.3e10_19e4.6t15.2c1
Dimension 5
Group $A_6$
Conductor $ 3^{10} \cdot 19^{4}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$7695324729= 3^{10} \cdot 19^{4} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_6$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 11 + 42\cdot 311 + 169\cdot 311^{2} + 275\cdot 311^{3} + 188\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 178 + 66\cdot 311 + 83\cdot 311^{2} + 306\cdot 311^{3} + 283\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 224 + 261\cdot 311 + 143\cdot 311^{2} + 74\cdot 311^{3} + 94\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 261 + 310\cdot 311 + 19\cdot 311^{2} + 227\cdot 311^{3} + 110\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 271 + 309\cdot 311 + 283\cdot 311^{2} + 257\cdot 311^{3} + 5\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 302 + 252\cdot 311 + 232\cdot 311^{2} + 102\cdot 311^{3} + 249\cdot 311^{4} +O\left(311^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2,3)$
$(1,2)(3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$2$
$40$$3$$(1,2,3)$$-1$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$72$$5$$(1,2,3,4,5)$$0$
$72$$5$$(1,3,4,5,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.