Properties

Label 35.354...976.70.a.a
Dimension $35$
Group $A_7$
Conductor $3.546\times 10^{56}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $A_7$
Conductor: \(354\!\cdots\!976\)\(\medspace = 2^{30} \cdot 3^{50} \cdot 7^{28}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.112021056.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_7$
Projective stem field: Galois closure of 7.3.112021056.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 3x^{6} + 3x^{5} + 3x^{4} - 9x^{3} + 3x^{2} + x - 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 659 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 659 }$: \( x^{2} + 655x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 84 a + 206 + \left(212 a + 614\right)\cdot 659 + \left(642 a + 490\right)\cdot 659^{2} + \left(642 a + 529\right)\cdot 659^{3} + \left(484 a + 480\right)\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 385 + 268\cdot 659 + 129\cdot 659^{2} + 397\cdot 659^{3} + 485\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 158 + 292\cdot 659 + 202\cdot 659^{2} + 538\cdot 659^{3} + 347\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 575 a + 542 + \left(446 a + 60\right)\cdot 659 + \left(16 a + 212\right)\cdot 659^{2} + \left(16 a + 482\right)\cdot 659^{3} + \left(174 a + 459\right)\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 141 a + 604 + \left(154 a + 480\right)\cdot 659 + \left(465 a + 24\right)\cdot 659^{2} + \left(510 a + 317\right)\cdot 659^{3} + \left(575 a + 322\right)\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 518 a + 509 + \left(504 a + 297\right)\cdot 659 + \left(193 a + 413\right)\cdot 659^{2} + \left(148 a + 576\right)\cdot 659^{3} + \left(83 a + 137\right)\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 235 + 621\cdot 659 + 503\cdot 659^{2} + 453\cdot 659^{3} + 401\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display

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$$1$$()$$35$
$105$$2$$(1,2)(3,4)$$-1$
$70$$3$$(1,2,3)$$-1$
$280$$3$$(1,2,3)(4,5,6)$$-1$
$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$$(1,2,3,4,5,6,7)$$0$
$360$$7$$(1,3,4,5,6,7,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.