Normalized defining polynomial

\( x^{23} - 3 x^{21} - 6 x^{20} + x^{19} + 17 x^{18} + 18 x^{17} - 10 x^{16} - 43 x^{15} - 25 x^{14} + \cdots - 1 \) x^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 11*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 46*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^2 + x - 1

Invariants

Degree:		$23$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[1, 11]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(-100388389417020021246100166894420339\) -100388389417020021246100166894420339 \(\medspace = -\,167\cdot 30727\cdot 499717\cdot 39149185505838967853863\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(33.25\)	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:		$167^{1/2}30727^{1/2}499717^{1/2}39149185505838967853863^{1/2}\approx 3.168412684878976e+17$
Ramified primes:		\(167\), \(30727\), \(499717\), \(39149185505838967853863\) 167, 30727, 499717, 39149185505838967853863	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		$\Q(\sqrt{-10038\!\cdots\!20339}$)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$ 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, a^22

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:		$a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+46a^{6}+17a^{5}-17a^{4}-16a^{3}+6a+1$, $a^{20}-3a^{18}-5a^{17}+a^{16}+14a^{15}+13a^{14}-9a^{13}-29a^{12}-12a^{11}+24a^{10}+31a^{9}-a^{8}-30a^{7}-16a^{6}+11a^{5}+16a^{4}+a^{3}-6a^{2}+1$, $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+46a^{6}+17a^{5}-17a^{4}-16a^{3}+5a+1$, $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+46a^{6}+17a^{5}-17a^{4}-16a^{3}+6a$, $a^{22}+a^{21}-2a^{20}-8a^{19}-7a^{18}+10a^{17}+28a^{16}+18a^{15}-25a^{14}-50a^{13}-17a^{12}+43a^{11}+54a^{10}-47a^{8}-35a^{7}+11a^{6}+28a^{5}+12a^{4}-4a^{3}-6a^{2}-a$, $11a^{22}+13a^{21}-25a^{20}-98a^{19}-86a^{18}+135a^{17}+372a^{16}+227a^{15}-364a^{14}-717a^{13}-231a^{12}+639a^{11}+804a^{10}-4a^{9}-727a^{8}-536a^{7}+178a^{6}+467a^{5}+183a^{4}-93a^{3}-103a^{2}-5a+17$, $3a^{22}+2a^{21}-7a^{20}-23a^{19}-14a^{18}+39a^{17}+81a^{16}+33a^{15}-101a^{14}-148a^{13}-17a^{12}+161a^{11}+156a^{10}-40a^{9}-166a^{8}-93a^{7}+64a^{6}+99a^{5}+25a^{4}-28a^{3}-22a^{2}+2a+4$, $2a^{22}-a^{21}-6a^{20}-9a^{19}+7a^{18}+34a^{17}+21a^{16}-34a^{15}-80a^{14}-19a^{13}+84a^{12}+98a^{11}-13a^{10}-116a^{9}-60a^{8}+46a^{7}+87a^{6}+12a^{5}-40a^{4}-22a^{3}+a^{2}+14a-3$, $4a^{22}+4a^{21}-10a^{20}-34a^{19}-25a^{18}+55a^{17}+127a^{16}+60a^{15}-147a^{14}-242a^{13}-44a^{12}+247a^{11}+265a^{10}-40a^{9}-271a^{8}-170a^{7}+89a^{6}+174a^{5}+54a^{4}-45a^{3}-42a^{2}-a+8$, $10a^{22}+4a^{21}-28a^{20}-71a^{19}-20a^{18}+159a^{17}+244a^{16}+7a^{15}-417a^{14}-424a^{13}+134a^{12}+639a^{11}+391a^{10}-343a^{9}-603a^{8}-163a^{7}+360a^{6}+326a^{5}-7a^{4}-151a^{3}-73a^{2}+22a+20$, $3a^{22}+a^{21}-9a^{20}-21a^{19}-3a^{18}+51a^{17}+71a^{16}-9a^{15}-134a^{14}-119a^{13}+60a^{12}+201a^{11}+102a^{10}-124a^{9}-185a^{8}-35a^{7}+124a^{6}+100a^{5}-10a^{4}-54a^{3}-24a^{2}+8a+7$ a^22 - 3*a^20 - 6*a^19 + a^18 + 17*a^17 + 18*a^16 - 10*a^15 - 43*a^14 - 25*a^13 + 33*a^12 + 60*a^11 + 11*a^10 - 54*a^9 - 47*a^8 + 12*a^7 + 46*a^6 + 17*a^5 - 17*a^4 - 16*a^3 + 6*a + 1, a^20 - 3*a^18 - 5*a^17 + a^16 + 14*a^15 + 13*a^14 - 9*a^13 - 29*a^12 - 12*a^11 + 24*a^10 + 31*a^9 - a^8 - 30*a^7 - 16*a^6 + 11*a^5 + 16*a^4 + a^3 - 6*a^2 + 1, a^22 - 3*a^20 - 6*a^19 + a^18 + 17*a^17 + 18*a^16 - 10*a^15 - 43*a^14 - 25*a^13 + 33*a^12 + 60*a^11 + 11*a^10 - 54*a^9 - 47*a^8 + 12*a^7 + 46*a^6 + 17*a^5 - 17*a^4 - 16*a^3 + 5*a + 1, a^22 - 3*a^20 - 6*a^19 + a^18 + 17*a^17 + 18*a^16 - 10*a^15 - 43*a^14 - 25*a^13 + 33*a^12 + 60*a^11 + 11*a^10 - 54*a^9 - 47*a^8 + 12*a^7 + 46*a^6 + 17*a^5 - 17*a^4 - 16*a^3 + 6*a, a^22 + a^21 - 2*a^20 - 8*a^19 - 7*a^18 + 10*a^17 + 28*a^16 + 18*a^15 - 25*a^14 - 50*a^13 - 17*a^12 + 43*a^11 + 54*a^10 - 47*a^8 - 35*a^7 + 11*a^6 + 28*a^5 + 12*a^4 - 4*a^3 - 6*a^2 - a, 11*a^22 + 13*a^21 - 25*a^20 - 98*a^19 - 86*a^18 + 135*a^17 + 372*a^16 + 227*a^15 - 364*a^14 - 717*a^13 - 231*a^12 + 639*a^11 + 804*a^10 - 4*a^9 - 727*a^8 - 536*a^7 + 178*a^6 + 467*a^5 + 183*a^4 - 93*a^3 - 103*a^2 - 5*a + 17, 3*a^22 + 2*a^21 - 7*a^20 - 23*a^19 - 14*a^18 + 39*a^17 + 81*a^16 + 33*a^15 - 101*a^14 - 148*a^13 - 17*a^12 + 161*a^11 + 156*a^10 - 40*a^9 - 166*a^8 - 93*a^7 + 64*a^6 + 99*a^5 + 25*a^4 - 28*a^3 - 22*a^2 + 2*a + 4, 2*a^22 - a^21 - 6*a^20 - 9*a^19 + 7*a^18 + 34*a^17 + 21*a^16 - 34*a^15 - 80*a^14 - 19*a^13 + 84*a^12 + 98*a^11 - 13*a^10 - 116*a^9 - 60*a^8 + 46*a^7 + 87*a^6 + 12*a^5 - 40*a^4 - 22*a^3 + a^2 + 14*a - 3, 4*a^22 + 4*a^21 - 10*a^20 - 34*a^19 - 25*a^18 + 55*a^17 + 127*a^16 + 60*a^15 - 147*a^14 - 242*a^13 - 44*a^12 + 247*a^11 + 265*a^10 - 40*a^9 - 271*a^8 - 170*a^7 + 89*a^6 + 174*a^5 + 54*a^4 - 45*a^3 - 42*a^2 - a + 8, 10*a^22 + 4*a^21 - 28*a^20 - 71*a^19 - 20*a^18 + 159*a^17 + 244*a^16 + 7*a^15 - 417*a^14 - 424*a^13 + 134*a^12 + 639*a^11 + 391*a^10 - 343*a^9 - 603*a^8 - 163*a^7 + 360*a^6 + 326*a^5 - 7*a^4 - 151*a^3 - 73*a^2 + 22*a + 20, 3*a^22 + a^21 - 9*a^20 - 21*a^19 - 3*a^18 + 51*a^17 + 71*a^16 - 9*a^15 - 134*a^14 - 119*a^13 + 60*a^12 + 201*a^11 + 102*a^10 - 124*a^9 - 185*a^8 - 35*a^7 + 124*a^6 + 100*a^5 - 10*a^4 - 54*a^3 - 24*a^2 + 8*a + 7 (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:		\( 139979317.774 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 139979317.774 \cdot 1}{2\cdot\sqrt{100388389417020021246100166894420339}}\cr\approx \mathstrut & 0.266195482608 \end{aligned}\] (assuming GRH)

Galois group

$S_{23}$ (as 23T7):

Intermediate fields

Sibling fields

Frobenius cycle types

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
\(167\) 167	$\Q_{167}$	$x + 162$	$1$	$1$	$0$	Trivial	$[\ ]$
	167.2.1.1	$x^{2} + 835$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	167.20.0.1	$x^{20} - x + 71$	$1$	$20$	$0$	20T1	$[\ ]^{20}$
\(30727\) 30727	$\Q_{30727}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{30727}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $13$	$1$	$13$	$0$	$C_{13}$	$[\ ]^{13}$
\(499717\) 499717		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
		Deg $13$	$1$	$13$	$0$	$C_{13}$	$[\ ]^{13}$
\(391\!\cdots\!863\) 39149185505838967853863	$\Q_{39\!\cdots\!63}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $20$	$1$	$20$	$0$	20T1	$[\ ]^{20}$