Properties

Label 15.2e12_3e26_11e8.42t294.1c1
Dimension 15
Group $A_7$
Conductor $ 2^{12} \cdot 3^{26} \cdot 11^{8}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$15$
Group:$A_7$
Conductor:$2231793723853793443221504= 2^{12} \cdot 3^{26} \cdot 11^{8} $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 2 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_7$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 21 a + 28 + \left(22 a + 5\right)\cdot 103 + \left(42 a + 60\right)\cdot 103^{2} + \left(13 a + 62\right)\cdot 103^{3} + \left(17 a + 70\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 67 + 7\cdot 103 + 16\cdot 103^{2} + 44\cdot 103^{3} + 46\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 84 + 21\cdot 103 + 32\cdot 103^{2} + 43\cdot 103^{3} + 59\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 82 a + 49 + \left(80 a + 6\right)\cdot 103 + \left(60 a + 80\right)\cdot 103^{2} + \left(89 a + 33\right)\cdot 103^{3} + \left(85 a + 74\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 66 + 36\cdot 103 + 56\cdot 103^{2} + 11\cdot 103^{3} + 73\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 77 a + 73 + \left(68 a + 16\right)\cdot 103 + \left(42 a + 45\right)\cdot 103^{2} + \left(20 a + 16\right)\cdot 103^{3} + \left(58 a + 25\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 26 a + 47 + \left(34 a + 8\right)\cdot 103 + \left(60 a + 19\right)\cdot 103^{2} + \left(82 a + 97\right)\cdot 103^{3} + \left(44 a + 62\right)\cdot 103^{4} +O\left(103^{ 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$$1$$()$$15$
$105$$2$$(1,2)(3,4)$$-1$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$630$$4$$(1,2,3,4)(5,6)$$-1$
$504$$5$$(1,2,3,4,5)$$0$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$360$$7$$(1,2,3,4,5,6,7)$$1$
$360$$7$$(1,3,4,5,6,7,2)$$1$
The blue line marks the conjugacy class containing complex conjugation.