Properties

 Label 1.280.4t1.b.b Dimension $1$ Group $C_4$ Conductor $280$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$280$$$$\medspace = 2^{3} \cdot 5 \cdot 7$$ Artin field: 4.0.392000.2 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{280}(27,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

Defining polynomial

 $f(x)$ $=$ $$x^{4} + 70 x^{2} + 980$$  .

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 7.

Roots:
 $r_{ 1 }$ $=$ $$2 + 4\cdot 11 + 9\cdot 11^{4} + 8\cdot 11^{5} +O(11^{7})$$ $r_{ 2 }$ $=$ $$5 + 3\cdot 11 + 8\cdot 11^{2} + 3\cdot 11^{3} + 5\cdot 11^{4} + 6\cdot 11^{6} +O(11^{7})$$ $r_{ 3 }$ $=$ $$6 + 7\cdot 11 + 2\cdot 11^{2} + 7\cdot 11^{3} + 5\cdot 11^{4} + 10\cdot 11^{5} + 4\cdot 11^{6} +O(11^{7})$$ $r_{ 4 }$ $=$ $$9 + 6\cdot 11 + 10\cdot 11^{2} + 10\cdot 11^{3} + 11^{4} + 2\cdot 11^{5} + 10\cdot 11^{6} +O(11^{7})$$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,4)(2,3)$ $(1,2,4,3)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,4)(2,3)$ $-1$ $1$ $4$ $(1,2,4,3)$ $-\zeta_{4}$ $1$ $4$ $(1,3,4,2)$ $\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.