Properties

Label 10.149e6_211e6.70.1c2
Dimension 10
Group $A_7$
Conductor $ 149^{6} \cdot 211^{6}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$10$
Group:$A_7$
Conductor:$965633540591074128803235361= 149^{6} \cdot 211^{6} $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 16 x^{3} - 14 x^{2} - 11 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 $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 44 + 46\cdot 53 + 14\cdot 53^{2} + 25\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 a + 28 + \left(18 a + 26\right)\cdot 53 + \left(41 a + 19\right)\cdot 53^{2} + \left(29 a + 22\right)\cdot 53^{3} + \left(14 a + 17\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 49 + 42\cdot 53 + 16\cdot 53^{2} + 6\cdot 53^{3} + 22\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 48 a + 48 + \left(34 a + 40\right)\cdot 53 + \left(11 a + 7\right)\cdot 53^{2} + \left(23 a + 47\right)\cdot 53^{3} + \left(38 a + 45\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 16 a + 30 + \left(13 a + 11\right)\cdot 53 + \left(26 a + 41\right)\cdot 53^{2} + \left(33 a + 29\right)\cdot 53^{3} + \left(36 a + 3\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 37 a + 41 + \left(39 a + 48\right)\cdot 53 + \left(26 a + 26\right)\cdot 53^{2} + \left(19 a + 31\right)\cdot 53^{3} + \left(16 a + 10\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 27 + 47\cdot 53 + 31\cdot 53^{2} + 49\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 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}$
$360$$7$$(1,3,4,5,6,7,2)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
The blue line marks the conjugacy class containing complex conjugation.