Normalized defining polynomial

\( x^{23} - 7 x^{22} + 2 x^{21} + 88 x^{20} - 150 x^{19} - 422 x^{18} + 1148 x^{17} + 862 x^{16} - 4167 x^{15} + \cdots - 1 \) x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 18*x^3 + 39*x^2 - 2*x - 1

Invariants

Degree:		$23$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[11, 6]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(1105964546020085080320539114269649147026432\) 1105964546020085080320539114269649147026432 \(\medspace = 2^{10}\cdot 22483\cdot 2527336563667297\cdot 19007452217635529293\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(67.30\)	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:		\(2\), \(22483\), \(2527336563667297\), \(19007452217635529293\) 2, 22483, 2527336563667297, 19007452217635529293	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		$\Q(\sqrt{10800\!\cdots\!45143}$)
$\card{ \Aut(K/\Q) }$:		$1$	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:		$16$	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:		$a$, $a^{21}-7a^{20}+2a^{19}+88a^{18}-150a^{17}-422a^{16}+1148a^{15}+862a^{14}-4167a^{13}-105a^{12}+8562a^{11}-2932a^{10}-10385a^{9}+5893a^{8}+7263a^{7}-5316a^{6}-2638a^{5}+2371a^{4}+349a^{3}-450a^{2}+14a+18$, $a^{21}-6a^{20}-4a^{19}+84a^{18}-66a^{17}-488a^{16}+660a^{15}+1522a^{14}-2645a^{13}-2750a^{12}+5813a^{11}+2878a^{10}-7514a^{9}-1597a^{8}+5682a^{7}+298a^{6}-2351a^{5}+101a^{4}+445a^{3}-41a^{2}-20a+1$, $5a^{22}-35a^{21}+11a^{20}+434a^{19}-753a^{18}-2031a^{17}+5667a^{16}+3888a^{15}-20183a^{14}+635a^{13}+40381a^{12}-16350a^{11}-46963a^{10}+30588a^{9}+30388a^{8}-26604a^{7}-9056a^{6}+11532a^{5}+148a^{4}-2135a^{3}+363a^{2}+79a-18$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+862a^{15}-4167a^{14}-105a^{13}+8563a^{12}-2935a^{11}-10393a^{10}+5920a^{9}+7285a^{8}-5405a^{7}-2661a^{6}+2502a^{5}+356a^{4}-533a^{3}+11a^{2}+35a+2$, $4a^{22}-30a^{21}+22a^{20}+347a^{19}-771a^{18}-1380a^{17}+5365a^{16}+1155a^{15}-17972a^{14}+7677a^{13}+33123a^{12}-27796a^{11}-33225a^{10}+42192a^{9}+14669a^{8}-33561a^{7}+1567a^{6}+13673a^{5}-3713a^{4}-2312a^{3}+952a^{2}+35a-34$, $5a^{22}-31a^{21}-16a^{20}+436a^{19}-406a^{18}-2541a^{17}+3916a^{16}+7915a^{15}-16010a^{14}-14108a^{13}+36766a^{12}+14047a^{11}-50988a^{10}-6418a^{9}+43015a^{8}-440a^{7}-21235a^{6}+1640a^{5}+5556a^{4}-535a^{3}-627a^{2}+33a+19$, $8a^{22}-54a^{21}+3a^{20}+701a^{19}-1022a^{18}-3588a^{17}+8190a^{16}+8772a^{15}-30474a^{14}-8320a^{13}+64234a^{12}-6520a^{11}-80892a^{10}+24122a^{9}+60595a^{8}-24047a^{7}-25606a^{6}+10971a^{5}+5379a^{4}-2143a^{3}-452a^{2}+102a+18$, $5a^{22}-36a^{21}+17a^{20}+438a^{19}-838a^{18}-1959a^{17}+6157a^{16}+3156a^{15}-21635a^{14}+3624a^{13}+42610a^{12}-22993a^{11}-48232a^{10}+39153a^{9}+29391a^{8}-32919a^{7}-7104a^{6}+13969a^{5}-940a^{4}-2509a^{3}+584a^{2}+78a-26$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+861a^{15}-4162a^{14}-102a^{13}+8516a^{12}-2908a^{11}-10225a^{10}+5760a^{9}+6996a^{8}-5080a^{7}-2408a^{6}+2197a^{5}+244a^{4}-399a^{3}+36a^{2}+12a-1$, $12a^{22}-79a^{21}-9a^{20}+1054a^{19}-1369a^{18}-5638a^{17}+11525a^{16}+15026a^{15}-44161a^{14}-18781a^{13}+95664a^{12}+2014a^{11}-124054a^{10}+23351a^{9}+96052a^{8}-28011a^{7}-42194a^{6}+13661a^{5}+9318a^{4}-2728a^{3}-840a^{2}+123a+27$, $38a^{22}-253a^{21}-11a^{20}+3343a^{19}-4555a^{18}-17633a^{17}+37631a^{16}+45849a^{15}-143037a^{14}-53554a^{13}+308599a^{12}-5053a^{11}-400094a^{10}+87738a^{9}+311393a^{8}-99613a^{7}-138708a^{6}+48796a^{5}+31605a^{4}-10210a^{3}-3072a^{2}+562a+126$, $18a^{22}-120a^{21}-4a^{20}+1583a^{19}-2174a^{18}-8323a^{17}+17912a^{16}+21483a^{15}-67970a^{14}-24463a^{13}+146362a^{12}-4386a^{11}-189240a^{10}+44183a^{9}+146668a^{8}-49220a^{7}-64887a^{6}+23976a^{5}+14615a^{4}-5000a^{3}-1400a^{2}+276a+58$, $a^{22}-11a^{21}+29a^{20}+86a^{19}-498a^{18}+94a^{17}+2902a^{16}-3241a^{15}-8279a^{14}+15032a^{13}+11676a^{12}-34409a^{11}-4704a^{10}+44585a^{9}-8250a^{8}-32967a^{7}+12315a^{6}+12913a^{5}-6469a^{4}-2112a^{3}+1330a^{2}+37a-55$, $a^{21}-6a^{20}-4a^{19}+84a^{18}-66a^{17}-487a^{16}+655a^{15}+1517a^{14}-2588a^{13}-2771a^{12}+5553a^{11}+3085a^{10}-6903a^{9}-2181a^{8}+4888a^{7}+1050a^{6}-1790a^{5}-324a^{4}+262a^{3}+28a^{2}-11a-1$, $3a^{22}-21a^{21}+6a^{20}+264a^{19}-450a^{18}-1266a^{17}+3443a^{16}+2590a^{15}-12491a^{14}-368a^{13}+25658a^{12}-8520a^{11}-31177a^{10}+16962a^{9}+22031a^{8}-14973a^{7}-8345a^{6}+6448a^{5}+1348a^{4}-1158a^{3}-32a^{2}+45a+2$ a, a^21 - 7*a^20 + 2*a^19 + 88*a^18 - 150*a^17 - 422*a^16 + 1148*a^15 + 862*a^14 - 4167*a^13 - 105*a^12 + 8562*a^11 - 2932*a^10 - 10385*a^9 + 5893*a^8 + 7263*a^7 - 5316*a^6 - 2638*a^5 + 2371*a^4 + 349*a^3 - 450*a^2 + 14*a + 18, a^21 - 6*a^20 - 4*a^19 + 84*a^18 - 66*a^17 - 488*a^16 + 660*a^15 + 1522*a^14 - 2645*a^13 - 2750*a^12 + 5813*a^11 + 2878*a^10 - 7514*a^9 - 1597*a^8 + 5682*a^7 + 298*a^6 - 2351*a^5 + 101*a^4 + 445*a^3 - 41*a^2 - 20*a + 1, 5*a^22 - 35*a^21 + 11*a^20 + 434*a^19 - 753*a^18 - 2031*a^17 + 5667*a^16 + 3888*a^15 - 20183*a^14 + 635*a^13 + 40381*a^12 - 16350*a^11 - 46963*a^10 + 30588*a^9 + 30388*a^8 - 26604*a^7 - 9056*a^6 + 11532*a^5 + 148*a^4 - 2135*a^3 + 363*a^2 + 79*a - 18, a^22 - 7*a^21 + 2*a^20 + 88*a^19 - 150*a^18 - 422*a^17 + 1148*a^16 + 862*a^15 - 4167*a^14 - 105*a^13 + 8563*a^12 - 2935*a^11 - 10393*a^10 + 5920*a^9 + 7285*a^8 - 5405*a^7 - 2661*a^6 + 2502*a^5 + 356*a^4 - 533*a^3 + 11*a^2 + 35*a + 2, 4*a^22 - 30*a^21 + 22*a^20 + 347*a^19 - 771*a^18 - 1380*a^17 + 5365*a^16 + 1155*a^15 - 17972*a^14 + 7677*a^13 + 33123*a^12 - 27796*a^11 - 33225*a^10 + 42192*a^9 + 14669*a^8 - 33561*a^7 + 1567*a^6 + 13673*a^5 - 3713*a^4 - 2312*a^3 + 952*a^2 + 35*a - 34, 5*a^22 - 31*a^21 - 16*a^20 + 436*a^19 - 406*a^18 - 2541*a^17 + 3916*a^16 + 7915*a^15 - 16010*a^14 - 14108*a^13 + 36766*a^12 + 14047*a^11 - 50988*a^10 - 6418*a^9 + 43015*a^8 - 440*a^7 - 21235*a^6 + 1640*a^5 + 5556*a^4 - 535*a^3 - 627*a^2 + 33*a + 19, 8*a^22 - 54*a^21 + 3*a^20 + 701*a^19 - 1022*a^18 - 3588*a^17 + 8190*a^16 + 8772*a^15 - 30474*a^14 - 8320*a^13 + 64234*a^12 - 6520*a^11 - 80892*a^10 + 24122*a^9 + 60595*a^8 - 24047*a^7 - 25606*a^6 + 10971*a^5 + 5379*a^4 - 2143*a^3 - 452*a^2 + 102*a + 18, 5*a^22 - 36*a^21 + 17*a^20 + 438*a^19 - 838*a^18 - 1959*a^17 + 6157*a^16 + 3156*a^15 - 21635*a^14 + 3624*a^13 + 42610*a^12 - 22993*a^11 - 48232*a^10 + 39153*a^9 + 29391*a^8 - 32919*a^7 - 7104*a^6 + 13969*a^5 - 940*a^4 - 2509*a^3 + 584*a^2 + 78*a - 26, a^22 - 7*a^21 + 2*a^20 + 88*a^19 - 150*a^18 - 422*a^17 + 1148*a^16 + 861*a^15 - 4162*a^14 - 102*a^13 + 8516*a^12 - 2908*a^11 - 10225*a^10 + 5760*a^9 + 6996*a^8 - 5080*a^7 - 2408*a^6 + 2197*a^5 + 244*a^4 - 399*a^3 + 36*a^2 + 12*a - 1, 12*a^22 - 79*a^21 - 9*a^20 + 1054*a^19 - 1369*a^18 - 5638*a^17 + 11525*a^16 + 15026*a^15 - 44161*a^14 - 18781*a^13 + 95664*a^12 + 2014*a^11 - 124054*a^10 + 23351*a^9 + 96052*a^8 - 28011*a^7 - 42194*a^6 + 13661*a^5 + 9318*a^4 - 2728*a^3 - 840*a^2 + 123*a + 27, 38*a^22 - 253*a^21 - 11*a^20 + 3343*a^19 - 4555*a^18 - 17633*a^17 + 37631*a^16 + 45849*a^15 - 143037*a^14 - 53554*a^13 + 308599*a^12 - 5053*a^11 - 400094*a^10 + 87738*a^9 + 311393*a^8 - 99613*a^7 - 138708*a^6 + 48796*a^5 + 31605*a^4 - 10210*a^3 - 3072*a^2 + 562*a + 126, 18*a^22 - 120*a^21 - 4*a^20 + 1583*a^19 - 2174*a^18 - 8323*a^17 + 17912*a^16 + 21483*a^15 - 67970*a^14 - 24463*a^13 + 146362*a^12 - 4386*a^11 - 189240*a^10 + 44183*a^9 + 146668*a^8 - 49220*a^7 - 64887*a^6 + 23976*a^5 + 14615*a^4 - 5000*a^3 - 1400*a^2 + 276*a + 58, a^22 - 11*a^21 + 29*a^20 + 86*a^19 - 498*a^18 + 94*a^17 + 2902*a^16 - 3241*a^15 - 8279*a^14 + 15032*a^13 + 11676*a^12 - 34409*a^11 - 4704*a^10 + 44585*a^9 - 8250*a^8 - 32967*a^7 + 12315*a^6 + 12913*a^5 - 6469*a^4 - 2112*a^3 + 1330*a^2 + 37*a - 55, a^21 - 6*a^20 - 4*a^19 + 84*a^18 - 66*a^17 - 487*a^16 + 655*a^15 + 1517*a^14 - 2588*a^13 - 2771*a^12 + 5553*a^11 + 3085*a^10 - 6903*a^9 - 2181*a^8 + 4888*a^7 + 1050*a^6 - 1790*a^5 - 324*a^4 + 262*a^3 + 28*a^2 - 11*a - 1, 3*a^22 - 21*a^21 + 6*a^20 + 264*a^19 - 450*a^18 - 1266*a^17 + 3443*a^16 + 2590*a^15 - 12491*a^14 - 368*a^13 + 25658*a^12 - 8520*a^11 - 31177*a^10 + 16962*a^9 + 22031*a^8 - 14973*a^7 - 8345*a^6 + 6448*a^5 + 1348*a^4 - 1158*a^3 - 32*a^2 + 45*a + 2 (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:		\( 11886760766000 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{6}\cdot 11886760766000 \cdot 1}{2\cdot\sqrt{1105964546020085080320539114269649147026432}}\cr\approx \mathstrut & 0.712150996690076 \end{aligned}\] (assuming GRH)

Galois group

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

Intermediate fields

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	2.4.4.5	$x^{4} + 2 x + 2$	$4$	$1$	$4$	$S_4$	$[4/3, 4/3]_{3}^{2}$
	2.6.6.6	$x^{6} - 4 x^{5} + 30 x^{4} - 16 x^{3} + 164 x^{2} + 160 x + 88$	$2$	$3$	$6$	$A_4\times C_2$	$[2, 2, 2]^{3}$
	2.13.0.1	$x^{13} + x^{4} + x^{3} + x + 1$	$1$	$13$	$0$	$C_{13}$	$[\ ]^{13}$
\(22483\) 22483		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $16$	$1$	$16$	$0$	$C_{16}$	$[\ ]^{16}$
\(2527336563667297\) 2527336563667297	$\Q_{2527336563667297}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $20$	$1$	$20$	$0$	20T1	$[\ ]^{20}$
\(19007452217635529293\) 19007452217635529293	$\Q_{19007452217635529293}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
		Deg $15$	$1$	$15$	$0$	$C_{15}$	$[\ ]^{15}$