Normalized defining polynomial

\( x^{22} + 3x^{20} - 11x^{16} - 24x^{14} - 24x^{12} + x^{10} + 11x^{8} - 9x^{6} - 5x^{4} + 5x^{2} - 1 \) x^22 + 3*x^20 - 11*x^16 - 24*x^14 - 24*x^12 + x^10 + 11*x^8 - 9*x^6 - 5*x^4 + 5*x^2 - 1

Invariants

Degree:		$22$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[2, 10]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(3029890053010200585803137024\) 3029890053010200585803137024 \(\medspace = 2^{22}\cdot 1583^{2}\cdot 2731^{2}\cdot 6217^{2}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(17.75\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		not computed
Ramified primes:		\(2\), \(1583\), \(2731\), \(6217\) 2, 1583, 2731, 6217	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$2$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, a^18, a^19, a^20, a^21

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$11$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		$12a^{20}+20a^{18}-24a^{16}-95a^{14}-166a^{12}-89a^{10}+90a^{8}-14a^{6}-71a^{4}+35a^{2}-5$, $a$, $a^{19}+2a^{17}-2a^{15}-9a^{13}-15a^{11}-9a^{9}+10a^{7}+a^{5}-10a^{3}+4a$, $22a^{20}+35a^{18}-49a^{16}-172a^{14}-285a^{12}-129a^{10}+195a^{8}-45a^{6}-143a^{4}+90a^{2}-17$, $13a^{20}+19a^{18}-32a^{16}-99a^{14}-155a^{12}-50a^{10}+133a^{8}-34a^{6}-84a^{4}+60a^{2}-11$, $11a^{20}+19a^{18}-25a^{16}-90a^{14}-147a^{12}-71a^{10}+109a^{8}-20a^{6}-86a^{4}+54a^{2}-10$, $15a^{21}+26a^{19}-30a^{17}-122a^{15}-212a^{13}-116a^{11}+122a^{9}-7a^{7}-94a^{5}+47a^{3}-5a+1$, $13a^{21}+a^{20}+19a^{19}+3a^{18}-32a^{17}-a^{16}-99a^{15}-11a^{14}-155a^{13}-22a^{12}-50a^{11}-20a^{10}+133a^{9}+8a^{8}-34a^{7}+8a^{6}-84a^{5}-9a^{4}+60a^{3}+a^{2}-12a+1$, $7a^{21}+15a^{19}+2a^{18}-8a^{17}+5a^{16}-61a^{15}-3a^{14}-124a^{13}-20a^{12}-103a^{11}-37a^{10}+20a^{9}-29a^{8}+12a^{7}+18a^{6}-38a^{5}+9a^{4}+6a^{3}-19a^{2}+a+5$, $6a^{21}+4a^{20}+11a^{19}+8a^{18}-9a^{17}-7a^{16}-48a^{15}-35a^{14}-93a^{13}-63a^{12}-67a^{11}-42a^{10}+21a^{9}+31a^{8}-8a^{7}+4a^{6}-35a^{5}-30a^{4}+9a^{3}+11a^{2}-a-1$, $6a^{21}-5a^{20}+10a^{19}-7a^{18}-11a^{17}+11a^{16}-47a^{15}+37a^{14}-85a^{13}+62a^{12}-50a^{11}+23a^{10}+36a^{9}-41a^{8}-7a^{7}+8a^{6}-32a^{5}+24a^{4}+16a^{3}-11a^{2}-a+1$ 12*a^20 + 20*a^18 - 24*a^16 - 95*a^14 - 166*a^12 - 89*a^10 + 90*a^8 - 14*a^6 - 71*a^4 + 35*a^2 - 5, a, a^19 + 2*a^17 - 2*a^15 - 9*a^13 - 15*a^11 - 9*a^9 + 10*a^7 + a^5 - 10*a^3 + 4*a, 22*a^20 + 35*a^18 - 49*a^16 - 172*a^14 - 285*a^12 - 129*a^10 + 195*a^8 - 45*a^6 - 143*a^4 + 90*a^2 - 17, 13*a^20 + 19*a^18 - 32*a^16 - 99*a^14 - 155*a^12 - 50*a^10 + 133*a^8 - 34*a^6 - 84*a^4 + 60*a^2 - 11, 11*a^20 + 19*a^18 - 25*a^16 - 90*a^14 - 147*a^12 - 71*a^10 + 109*a^8 - 20*a^6 - 86*a^4 + 54*a^2 - 10, 15*a^21 + 26*a^19 - 30*a^17 - 122*a^15 - 212*a^13 - 116*a^11 + 122*a^9 - 7*a^7 - 94*a^5 + 47*a^3 - 5*a + 1, 13*a^21 + a^20 + 19*a^19 + 3*a^18 - 32*a^17 - a^16 - 99*a^15 - 11*a^14 - 155*a^13 - 22*a^12 - 50*a^11 - 20*a^10 + 133*a^9 + 8*a^8 - 34*a^7 + 8*a^6 - 84*a^5 - 9*a^4 + 60*a^3 + a^2 - 12*a + 1, 7*a^21 + 15*a^19 + 2*a^18 - 8*a^17 + 5*a^16 - 61*a^15 - 3*a^14 - 124*a^13 - 20*a^12 - 103*a^11 - 37*a^10 + 20*a^9 - 29*a^8 + 12*a^7 + 18*a^6 - 38*a^5 + 9*a^4 + 6*a^3 - 19*a^2 + a + 5, 6*a^21 + 4*a^20 + 11*a^19 + 8*a^18 - 9*a^17 - 7*a^16 - 48*a^15 - 35*a^14 - 93*a^13 - 63*a^12 - 67*a^11 - 42*a^10 + 21*a^9 + 31*a^8 - 8*a^7 + 4*a^6 - 35*a^5 - 30*a^4 + 9*a^3 + 11*a^2 - a - 1, 6*a^21 - 5*a^20 + 10*a^19 - 7*a^18 - 11*a^17 + 11*a^16 - 47*a^15 + 37*a^14 - 85*a^13 + 62*a^12 - 50*a^11 + 23*a^10 + 36*a^9 - 41*a^8 - 7*a^7 + 8*a^6 - 32*a^5 + 24*a^4 + 16*a^3 - 11*a^2 - a + 1 (assuming GRH)	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 45090.7591736 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 45090.7591736 \cdot 1}{2\cdot\sqrt{3029890053010200585803137024}}\cr\approx \mathstrut & 0.157109624982 \end{aligned}\] (assuming GRH)

Galois group

$C_2^{10}.S_{11}$ (as 22T51):

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Frobenius cycle types

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(2\) 2	2.10.10.9	$x^{10} + 10 x^{9} + 22 x^{8} - 8 x^{7} + 72 x^{6} + 1328 x^{5} + 4496 x^{4} + 3680 x^{3} - 3696 x^{2} - 96 x - 32$	$2$	$5$	$10$	$C_2 \times (C_2^4 : C_5)$	$[2, 2, 2, 2, 2]^{5}$
	2.12.12.5	$x^{12} + 4 x^{11} + 24 x^{10} - 8 x^{9} + 140 x^{8} - 1488 x^{7} + 832 x^{6} - 10208 x^{5} + 20336 x^{4} + 12160 x^{3} + 113280 x^{2} + 139520 x + 45376$	$2$	$6$	$12$	12T51	$[2, 2, 2, 2]^{6}$
\(1583\) 1583	$\Q_{1583}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{1583}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $4$	$2$	$2$	$2$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
\(2731\) 2731	$\Q_{2731}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2731}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2731}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2731}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $4$	$2$	$2$	$2$
		Deg $12$	$1$	$12$	$0$	$C_{12}$	$[\ ]^{12}$
\(6217\) 6217		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $4$	$2$	$2$	$2$
		Deg $14$	$1$	$14$	$0$	$C_{14}$	$[\ ]^{14}$