Properties

Label 3.11e2_401e2.42t37.1c1
Dimension 3
Group $\GL(3,2)$
Conductor $ 11^{2} \cdot 401^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$19456921= 11^{2} \cdot 401^{2} $
Artin number field: Splitting field of $f= x^{7} - 3 x^{6} + 5 x^{4} - 2 x^{3} - 2 x^{2} + 4 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $\PSL(2,7)$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{3} + x + 28 $
Roots:
$r_{ 1 }$ $=$ $ 12 a^{2} + 26 a + 7 + \left(2 a^{2} + 2 a + 15\right)\cdot 31 + \left(3 a^{2} + 20 a + 5\right)\cdot 31^{2} + \left(10 a^{2} + 24 a + 6\right)\cdot 31^{3} + \left(22 a^{2} + 28 a + 3\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 a^{2} + 10 a + 4 + \left(15 a^{2} + 24 a + 1\right)\cdot 31 + \left(11 a^{2} + 12 a + 11\right)\cdot 31^{2} + \left(7 a^{2} + 25 a + 10\right)\cdot 31^{3} + \left(9 a^{2} + 5 a + 1\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 17\cdot 31 + 10\cdot 31^{2} + 16\cdot 31^{3} + 18\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 16 a^{2} + 27 a + 20 + \left(10 a^{2} + a + 20\right)\cdot 31 + \left(10 a^{2} + 20 a + 20\right)\cdot 31^{2} + \left(15 a^{2} + a + 9\right)\cdot 31^{3} + \left(5 a^{2} + 3 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 10 a^{2} + 11 + \left(15 a^{2} + 17 a + 11\right)\cdot 31 + \left(5 a^{2} + 8 a + 17\right)\cdot 31^{2} + \left(24 a^{2} + 17 a + 21\right)\cdot 31^{3} + \left(26 a^{2} + 10 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 6 a^{2} + 21 a + 29 + \left(20 a + 21\right)\cdot 31 + \left(14 a^{2} + 9 a + 12\right)\cdot 31^{2} + \left(30 a^{2} + 19 a + 15\right)\cdot 31^{3} + \left(25 a^{2} + 14 a + 12\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 3 a^{2} + 9 a + 1 + \left(18 a^{2} + 26 a + 5\right)\cdot 31 + \left(17 a^{2} + 21 a + 15\right)\cdot 31^{2} + \left(5 a^{2} + 4 a + 13\right)\cdot 31^{3} + \left(3 a^{2} + 30 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$3$
$21$$2$$(1,4)(5,6)$$-1$
$56$$3$$(1,7,6)(2,5,3)$$0$
$42$$4$$(1,2,7,5)(3,4)$$1$
$24$$7$$(1,2,7,6,5,4,3)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$24$$7$$(1,6,3,7,4,2,5)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
The blue line marks the conjugacy class containing complex conjugation.