Properties

Label 10.2e9_7e6_73e6.70.1c1
Dimension 10
Group $A_7$
Conductor $ 2^{9} \cdot 7^{6} \cdot 73^{6}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$10$
Group:$A_7$
Conductor:$9115812039001375232= 2^{9} \cdot 7^{6} \cdot 73^{6} $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 70
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$: $ x^{2} + 220 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 128 + 168\cdot 227 + 137\cdot 227^{2} + 129\cdot 227^{3} + 178\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 55 + 149\cdot 227 + 170\cdot 227^{2} + 56\cdot 227^{3} + 186\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 + 157\cdot 227 + 55\cdot 227^{2} + 95\cdot 227^{3} + 2\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 77 a + 157 + \left(121 a + 116\right)\cdot 227 + \left(167 a + 170\right)\cdot 227^{2} + \left(3 a + 69\right)\cdot 227^{3} + \left(33 a + 29\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 150 a + 15 + \left(105 a + 208\right)\cdot 227 + \left(59 a + 86\right)\cdot 227^{2} + \left(223 a + 155\right)\cdot 227^{3} + \left(193 a + 29\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 190 a + 59 + \left(137 a + 7\right)\cdot 227 + \left(131 a + 92\right)\cdot 227^{2} + \left(194 a + 39\right)\cdot 227^{3} + \left(133 a + 210\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 37 a + 27 + \left(89 a + 101\right)\cdot 227 + \left(95 a + 194\right)\cdot 227^{2} + \left(32 a + 134\right)\cdot 227^{3} + \left(93 a + 44\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(3,4,5,6,7)$
$(1,2,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$10$
$105$$2$$(1,2)(3,4)$$-2$
$70$$3$$(1,2,3)$$1$
$280$$3$$(1,2,3)(4,5,6)$$1$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$0$
$210$$6$$(1,2,3)(4,5)(6,7)$$1$
$360$$7$$(1,2,3,4,5,6,7)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
$360$$7$$(1,3,4,5,6,7,2)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
The blue line marks the conjugacy class containing complex conjugation.