# Properties

 Label 10.141...528.70.a.b Dimension $10$ Group $A_7$ Conductor $1.412\times 10^{16}$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $10$ Group: $A_7$ Conductor: $$14117306610774528$$$$\medspace = 2^{9} \cdot 3^{14} \cdot 7^{8}$$ Artin stem field: Galois closure of 7.3.112021056.1 Galois orbit size: $2$ 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

 $f(x)$ $=$ $$x^{7} - 3x^{6} + 3x^{5} + 3x^{4} - 9x^{3} + 3x^{2} + x - 3$$ x^7 - 3*x^6 + 3*x^5 + 3*x^4 - 9*x^3 + 3*x^2 + x - 3 .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 659 }$: $$x^{2} + 655x + 2$$

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

## 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

 Size Order Action 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.