LMFDB - Number field 18.10.36163754150128780430177325000000000000.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 384*x^14 - 672*x^13 + 749*x^12 - 204*x^11 - 1008*x^10 + 2191*x^9 - 2253*x^8 + 876*x^7 + 684*x^6 - 1107*x^5 + 414*x^4 + 269*x^3 - 180*x^2 - 24*x + 16)

gp:K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 156*y^15 + 384*y^14 - 672*y^13 + 749*y^12 - 204*y^11 - 1008*y^10 + 2191*y^9 - 2253*y^8 + 876*y^7 + 684*y^6 - 1107*y^5 + 414*y^4 + 269*y^3 - 180*y^2 - 24*y + 16, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 384*x^14 - 672*x^13 + 749*x^12 - 204*x^11 - 1008*x^10 + 2191*x^9 - 2253*x^8 + 876*x^7 + 684*x^6 - 1107*x^5 + 414*x^4 + 269*x^3 - 180*x^2 - 24*x + 16);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 384*x^14 - 672*x^13 + 749*x^12 - 204*x^11 - 1008*x^10 + 2191*x^9 - 2253*x^8 + 876*x^7 + 684*x^6 - 1107*x^5 + 414*x^4 + 269*x^3 - 180*x^2 - 24*x + 16)

$ x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 384 x^{14} - 672 x^{13} + 749 x^{12} - 204 x^{11} + \cdots + 16 $ x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 384*x^14 - 672*x^13 + 749*x^12 - 204*x^11 - 1008*x^10 + 2191*x^9 - 2253*x^8 + 876*x^7 + 684*x^6 - 1107*x^5 + 414*x^4 + 269*x^3 - 180*x^2 - 24*x + 16

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$18$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[10, 4]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$36163754150128780430177325000000000000$ 36163754150128780430177325000000000000 $\medspace = 2^{12}\cdot 3^{24}\cdot 5^{14}\cdot 13^{6}\cdot 1061117$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$122.06$	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:	$2$, $3$, $5$, $13$, $1061117$ 2, 3, 5, 13, 1061117	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{1061117}) $
$\Aut(K/\Q)$:	$C_2$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{10}-\frac{1}{5}a^{8}-\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}$, $\frac{1}{10}a^{14}-\frac{1}{10}a^{13}+\frac{1}{10}a^{11}+\frac{2}{5}a^{10}-\frac{1}{10}a^{9}+\frac{1}{10}a^{8}-\frac{1}{2}a^{7}-\frac{1}{10}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{10}a+\frac{2}{5}$, $\frac{1}{10}a^{15}-\frac{1}{10}a^{13}-\frac{1}{10}a^{12}-\frac{3}{10}a^{11}+\frac{1}{10}a^{10}-\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{3}{10}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{10}a^{2}-\frac{3}{10}a+\frac{1}{5}$, $\frac{1}{100}a^{16}+\frac{1}{50}a^{15}+\frac{1}{100}a^{14}+\frac{3}{100}a^{13}-\frac{9}{100}a^{12}-\frac{29}{100}a^{11}-\frac{9}{25}a^{10}-\frac{3}{50}a^{9}+\frac{11}{50}a^{8}-\frac{11}{100}a^{7}+\frac{17}{50}a^{6}+\frac{9}{25}a^{5}-\frac{11}{25}a^{4}+\frac{13}{100}a^{3}+\frac{1}{100}a^{2}-\frac{19}{50}a-\frac{6}{25}$, $\frac{1}{100}a^{17}-\frac{3}{100}a^{15}+\frac{1}{100}a^{14}+\frac{1}{20}a^{13}+\frac{9}{100}a^{12}+\frac{1}{50}a^{11}+\frac{3}{50}a^{10}+\frac{17}{50}a^{9}+\frac{1}{4}a^{8}+\frac{9}{25}a^{7}-\frac{3}{25}a^{6}+\frac{11}{25}a^{5}+\frac{1}{100}a^{4}-\frac{1}{20}a^{3}-\frac{1}{5}a^{2}+\frac{8}{25}a-\frac{3}{25}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/5*a^12 - 1/5*a^11 + 1/5*a^10 - 1/5*a^7 + 1/5*a^6 - 1/5*a^5 + 1/5*a^2 - 1/5*a + 1/5, 1/5*a^13 + 1/5*a^10 - 1/5*a^8 - 1/5*a^5 + 1/5*a^3 + 1/5, 1/10*a^14 - 1/10*a^13 + 1/10*a^11 + 2/5*a^10 - 1/10*a^9 + 1/10*a^8 - 1/2*a^7 - 1/10*a^6 - 2/5*a^5 - 2/5*a^4 + 2/5*a^3 + 1/10*a + 2/5, 1/10*a^15 - 1/10*a^13 - 1/10*a^12 - 3/10*a^11 + 1/10*a^10 - 2/5*a^8 - 2/5*a^7 + 3/10*a^6 + 2/5*a^5 + 2/5*a^3 - 1/10*a^2 - 3/10*a + 1/5, 1/100*a^16 + 1/50*a^15 + 1/100*a^14 + 3/100*a^13 - 9/100*a^12 - 29/100*a^11 - 9/25*a^10 - 3/50*a^9 + 11/50*a^8 - 11/100*a^7 + 17/50*a^6 + 9/25*a^5 - 11/25*a^4 + 13/100*a^3 + 1/100*a^2 - 19/50*a - 6/25, 1/100*a^17 - 3/100*a^15 + 1/100*a^14 + 1/20*a^13 + 9/100*a^12 + 1/50*a^11 + 3/50*a^10 + 17/50*a^9 + 1/4*a^8 + 9/25*a^7 - 3/25*a^6 + 11/25*a^5 + 1/100*a^4 - 1/20*a^3 - 1/5*a^2 + 8/25*a - 3/25

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$13$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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:	$\frac{243}{100}a^{16}-\frac{486}{25}a^{15}+\frac{9153}{100}a^{14}-\frac{30051}{100}a^{13}+\frac{69363}{100}a^{12}-\frac{113967}{100}a^{11}+\frac{57141}{50}a^{10}-\frac{2817}{25}a^{9}-\frac{45006}{25}a^{8}+\frac{344877}{100}a^{7}-\frac{161309}{50}a^{6}+\frac{30042}{25}a^{5}+\frac{17817}{25}a^{4}-\frac{117521}{100}a^{3}+\frac{30393}{100}a^{2}+\frac{8823}{50}a-\frac{1483}{25}$, $\frac{333}{50}a^{16}-\frac{1332}{25}a^{15}+\frac{6289}{25}a^{14}-\frac{20713}{25}a^{13}+\frac{96153}{50}a^{12}-\frac{79796}{25}a^{11}+\frac{82011}{25}a^{10}-\frac{27043}{50}a^{9}-\frac{236199}{50}a^{8}+\frac{236331}{25}a^{7}-\frac{459993}{50}a^{6}+\frac{98239}{25}a^{5}+\frac{34759}{25}a^{4}-\frac{147971}{50}a^{3}+\frac{43933}{50}a^{2}+\frac{19211}{50}a-\frac{3481}{25}$, $\frac{2}{5}a^{16}-\frac{16}{5}a^{15}+\frac{149}{10}a^{14}-\frac{483}{10}a^{13}+109a^{12}-\frac{1717}{10}a^{11}+\frac{771}{5}a^{10}+\frac{331}{10}a^{9}-\frac{3407}{10}a^{8}+\frac{1125}{2}a^{7}-\frac{4483}{10}a^{6}+\frac{299}{5}a^{5}+\frac{1182}{5}a^{4}-\frac{1244}{5}a^{3}+46a^{2}+\frac{447}{10}a-19$, $\frac{3}{10}a^{16}-\frac{12}{5}a^{15}+\frac{57}{5}a^{14}-\frac{189}{5}a^{13}+\frac{177}{2}a^{12}-\frac{744}{5}a^{11}+\frac{781}{5}a^{10}-\frac{319}{10}a^{9}-\frac{2147}{10}a^{8}+443a^{7}-\frac{4387}{10}a^{6}+\frac{964}{5}a^{5}+\frac{312}{5}a^{4}-\frac{1413}{10}a^{3}+\frac{85}{2}a^{2}+\frac{37}{2}a-\frac{31}{5}$, $\frac{11}{50}a^{16}-\frac{44}{25}a^{15}+\frac{213}{25}a^{14}-\frac{721}{25}a^{13}+\frac{3491}{50}a^{12}-\frac{3102}{25}a^{11}+\frac{3632}{25}a^{10}-\frac{3331}{50}a^{9}-\frac{6533}{50}a^{8}+\frac{8957}{25}a^{7}-\frac{21241}{50}a^{6}+\frac{6718}{25}a^{5}-\frac{322}{25}a^{4}-\frac{6557}{50}a^{3}+\frac{3851}{50}a^{2}-\frac{353}{50}a-\frac{507}{25}$, $\frac{253}{50}a^{17}-\frac{2017}{50}a^{16}+\frac{4686}{25}a^{15}-\frac{30269}{50}a^{14}+\frac{33977}{25}a^{13}-\frac{21203}{10}a^{12}+\frac{92679}{50}a^{11}+\frac{5289}{10}a^{10}-\frac{216641}{50}a^{9}+\frac{172018}{25}a^{8}-\frac{25886}{5}a^{7}+\frac{5678}{25}a^{6}+\frac{15458}{5}a^{5}-\frac{132179}{50}a^{4}-\frac{573}{25}a^{3}+\frac{23564}{25}a^{2}-\frac{664}{25}a-\frac{1749}{25}$, $\frac{17}{20}a^{16}-\frac{34}{5}a^{15}+\frac{643}{20}a^{14}-\frac{2121}{20}a^{13}+\frac{4933}{20}a^{12}-\frac{8213}{20}a^{11}+\frac{4241}{10}a^{10}-\frac{367}{5}a^{9}-\frac{3031}{5}a^{8}+\frac{24447}{20}a^{7}-\frac{11931}{10}a^{6}+\frac{2537}{5}a^{5}+\frac{942}{5}a^{4}-\frac{7871}{20}a^{3}+\frac{2343}{20}a^{2}+\frac{507}{10}a-\frac{91}{5}$, $\frac{1}{100}a^{16}-\frac{2}{25}a^{15}-\frac{39}{100}a^{14}+\frac{413}{100}a^{13}-\frac{1899}{100}a^{12}+\frac{5661}{100}a^{11}-\frac{5273}{50}a^{10}+\frac{5817}{50}a^{9}-\frac{719}{50}a^{8}-\frac{20371}{100}a^{7}+\frac{8461}{25}a^{6}-\frac{6026}{25}a^{5}+\frac{104}{25}a^{4}+\frac{13573}{100}a^{3}-\frac{4889}{100}a^{2}-\frac{562}{25}a+\frac{89}{25}$, $\frac{347}{100}a^{17}-\frac{2583}{100}a^{16}+\frac{11563}{100}a^{15}-\frac{8974}{25}a^{14}+\frac{19074}{25}a^{13}-\frac{5552}{5}a^{12}+\frac{79771}{100}a^{11}+\frac{3209}{5}a^{10}-\frac{64461}{25}a^{9}+\frac{352689}{100}a^{8}-\frac{41227}{20}a^{7}-\frac{13964}{25}a^{6}+\frac{8881}{5}a^{5}-\frac{109161}{100}a^{4}-\frac{20077}{50}a^{3}+\frac{42067}{100}a^{2}+\frac{1232}{25}a-\frac{823}{25}$, $\frac{1}{100}a^{16}-\frac{2}{25}a^{15}-\frac{49}{100}a^{14}+\frac{483}{100}a^{13}-\frac{2199}{100}a^{12}+\frac{6551}{100}a^{11}-\frac{6163}{50}a^{10}+\frac{3511}{25}a^{9}-\frac{727}{25}a^{8}-\frac{22021}{100}a^{7}+\frac{19527}{50}a^{6}-\frac{7656}{25}a^{5}+\frac{1064}{25}a^{4}+\frac{13393}{100}a^{3}-\frac{6289}{100}a^{2}-\frac{679}{50}a+\frac{169}{25}$, $\frac{9039}{100}a^{17}-\frac{3066}{5}a^{16}+\frac{258223}{100}a^{15}-\frac{746451}{100}a^{14}+\frac{282929}{20}a^{13}-\frac{1713929}{100}a^{12}+\frac{109357}{25}a^{11}+\frac{1342397}{50}a^{10}-\frac{1358446}{25}a^{9}+\frac{205565}{4}a^{8}-\frac{159233}{50}a^{7}-\frac{1920109}{50}a^{6}+\frac{812114}{25}a^{5}-\frac{279001}{100}a^{4}-\frac{448939}{20}a^{3}+\frac{7877}{5}a^{2}+\frac{284939}{50}a+\frac{27183}{25}$, $\frac{90239}{20}a^{17}-\frac{4289259}{100}a^{16}+\frac{22477347}{100}a^{15}-\frac{40890237}{50}a^{14}+\frac{107363019}{50}a^{13}-\frac{103028701}{25}a^{12}+\frac{547042581}{100}a^{11}-\frac{92432539}{25}a^{10}-\frac{133447653}{50}a^{9}+\frac{1123621187}{100}a^{8}-\frac{1586535641}{100}a^{7}+\frac{600317437}{50}a^{6}-\frac{75322511}{25}a^{5}-\frac{346096169}{100}a^{4}+\frac{181176449}{50}a^{3}-\frac{62741949}{100}a^{2}-\frac{24631219}{50}a+\frac{3549769}{25}$, $\frac{46479037}{50}a^{17}-\frac{702600803}{100}a^{16}+\frac{1585143291}{50}a^{15}-\frac{9931709249}{100}a^{14}+\frac{21381165071}{100}a^{13}-\frac{31654204097}{100}a^{12}+\frac{4802612413}{20}a^{11}+\frac{7812385501}{50}a^{10}-\frac{35579710073}{50}a^{9}+\frac{25278023781}{25}a^{8}-\frac{63743885163}{100}a^{7}-\frac{518643216}{5}a^{6}+\frac{12148904487}{25}a^{5}-\frac{16440158857}{50}a^{4}-\frac{8870686149}{100}a^{3}+\frac{12198934827}{100}a^{2}+\frac{207410403}{25}a-\frac{258315739}{25}$ 243/100a^16 - 486/25a^15 + 9153/100a^14 - 30051/100a^13 + 69363/100a^12 - 113967/100a^11 + 57141/50a^10 - 2817/25a^9 - 45006/25a^8 + 344877/100a^7 - 161309/50a^6 + 30042/25a^5 + 17817/25a^4 - 117521/100a^3 + 30393/100a^2 + 8823/50a - 1483/25, 333/50a^16 - 1332/25a^15 + 6289/25a^14 - 20713/25a^13 + 96153/50a^12 - 79796/25a^11 + 82011/25a^10 - 27043/50a^9 - 236199/50a^8 + 236331/25a^7 - 459993/50a^6 + 98239/25a^5 + 34759/25a^4 - 147971/50a^3 + 43933/50a^2 + 19211/50a - 3481/25, 2/5a^16 - 16/5a^15 + 149/10a^14 - 483/10a^13 + 109a^12 - 1717/10a^11 + 771/5a^10 + 331/10a^9 - 3407/10a^8 + 1125/2a^7 - 4483/10a^6 + 299/5a^5 + 1182/5a^4 - 1244/5a^3 + 46a^2 + 447/10a - 19, 3/10a^16 - 12/5a^15 + 57/5a^14 - 189/5a^13 + 177/2a^12 - 744/5a^11 + 781/5a^10 - 319/10a^9 - 2147/10a^8 + 443a^7 - 4387/10a^6 + 964/5a^5 + 312/5a^4 - 1413/10a^3 + 85/2a^2 + 37/2a - 31/5, 11/50a^16 - 44/25a^15 + 213/25a^14 - 721/25a^13 + 3491/50a^12 - 3102/25a^11 + 3632/25a^10 - 3331/50a^9 - 6533/50a^8 + 8957/25a^7 - 21241/50a^6 + 6718/25a^5 - 322/25a^4 - 6557/50a^3 + 3851/50a^2 - 353/50a - 507/25, 253/50a^17 - 2017/50a^16 + 4686/25a^15 - 30269/50a^14 + 33977/25a^13 - 21203/10a^12 + 92679/50a^11 + 5289/10a^10 - 216641/50a^9 + 172018/25a^8 - 25886/5a^7 + 5678/25a^6 + 15458/5a^5 - 132179/50a^4 - 573/25a^3 + 23564/25a^2 - 664/25a - 1749/25, 17/20a^16 - 34/5a^15 + 643/20a^14 - 2121/20a^13 + 4933/20a^12 - 8213/20a^11 + 4241/10a^10 - 367/5a^9 - 3031/5a^8 + 24447/20a^7 - 11931/10a^6 + 2537/5a^5 + 942/5a^4 - 7871/20a^3 + 2343/20a^2 + 507/10a - 91/5, 1/100a^16 - 2/25a^15 - 39/100a^14 + 413/100a^13 - 1899/100a^12 + 5661/100a^11 - 5273/50a^10 + 5817/50a^9 - 719/50a^8 - 20371/100a^7 + 8461/25a^6 - 6026/25a^5 + 104/25a^4 + 13573/100a^3 - 4889/100a^2 - 562/25a + 89/25, 347/100a^17 - 2583/100a^16 + 11563/100a^15 - 8974/25a^14 + 19074/25a^13 - 5552/5a^12 + 79771/100a^11 + 3209/5a^10 - 64461/25a^9 + 352689/100a^8 - 41227/20a^7 - 13964/25a^6 + 8881/5a^5 - 109161/100a^4 - 20077/50a^3 + 42067/100a^2 + 1232/25a - 823/25, 1/100a^16 - 2/25a^15 - 49/100a^14 + 483/100a^13 - 2199/100a^12 + 6551/100a^11 - 6163/50a^10 + 3511/25a^9 - 727/25a^8 - 22021/100a^7 + 19527/50a^6 - 7656/25a^5 + 1064/25a^4 + 13393/100a^3 - 6289/100a^2 - 679/50a + 169/25, 9039/100a^17 - 3066/5a^16 + 258223/100a^15 - 746451/100a^14 + 282929/20a^13 - 1713929/100a^12 + 109357/25a^11 + 1342397/50a^10 - 1358446/25a^9 + 205565/4a^8 - 159233/50a^7 - 1920109/50a^6 + 812114/25a^5 - 279001/100a^4 - 448939/20a^3 + 7877/5a^2 + 284939/50a + 27183/25, 90239/20a^17 - 4289259/100a^16 + 22477347/100a^15 - 40890237/50a^14 + 107363019/50a^13 - 103028701/25a^12 + 547042581/100a^11 - 92432539/25a^10 - 133447653/50a^9 + 1123621187/100a^8 - 1586535641/100a^7 + 600317437/50a^6 - 75322511/25a^5 - 346096169/100a^4 + 181176449/50a^3 - 62741949/100a^2 - 24631219/50a + 3549769/25, 46479037/50a^17 - 702600803/100a^16 + 1585143291/50a^15 - 9931709249/100a^14 + 21381165071/100a^13 - 31654204097/100a^12 + 4802612413/20a^11 + 7812385501/50a^10 - 35579710073/50a^9 + 25278023781/25a^8 - 63743885163/100a^7 - 518643216/5a^6 + 12148904487/25a^5 - 16440158857/50a^4 - 8870686149/100a^3 + 12198934827/100a^2 + 207410403/25*a - 258315739/25 (assuming GRH)	comment:Fundamental units 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:	$ 5673649067000 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 5673649067000 \cdot 1}{2\cdot\sqrt{36163754150128780430177325000000000000}}\cr\approx \mathstrut & 0.752861606595519 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 384*x^14 - 672*x^13 + 749*x^12 - 204*x^11 - 1008*x^10 + 2191*x^9 - 2253*x^8 + 876*x^7 + 684*x^6 - 1107*x^5 + 414*x^4 + 269*x^3 - 180*x^2 - 24*x + 16) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 384*x^14 - 672*x^13 + 749*x^12 - 204*x^11 - 1008*x^10 + 2191*x^9 - 2253*x^8 + 876*x^7 + 684*x^6 - 1107*x^5 + 414*x^4 + 269*x^3 - 180*x^2 - 24*x + 16, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 384*x^14 - 672*x^13 + 749*x^12 - 204*x^11 - 1008*x^10 + 2191*x^9 - 2253*x^8 + 876*x^7 + 684*x^6 - 1107*x^5 + 414*x^4 + 269*x^3 - 180*x^2 - 24*x + 16); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 384*x^14 - 672*x^13 + 749*x^12 - 204*x^11 - 1008*x^10 + 2191*x^9 - 2253*x^8 + 876*x^7 + 684*x^6 - 1107*x^5 + 414*x^4 + 269*x^3 - 180*x^2 - 24*x + 16); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^5.S_4^2$ (as 18T623):

comment:Galois group

sage:K.galois_group(type='pari')

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 18432

The 120 conjugacy class representatives for $C_2^5.S_4^2$

Character table for $C_2^5.S_4^2$

Intermediate fields

3.3.1620.1, 3.3.1300.1, 9.9.5837879385000000.1

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

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(b)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 18 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$	${\href{/padicField/11.6.0.1}{6} }^{3}$	R	${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$	${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }$	${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$	${\href{/padicField/29.3.0.1}{3} }^{6}$	${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$	${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$	${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$	${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }$	${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }$	${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$	${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.1.3.2a1.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
	2.1.3.2a1.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
	2.4.3.8a1.2	$x^{12} + 3 x^{9} + 3 x^{8} + 3 x^{6} + 6 x^{5} + 3 x^{4} + x^{3} + 3 x^{2} + 3 x + 3$	$3$	$4$	$8$	$C_3 : C_4$	$$[\ ]_{3}^{4}$$
$3$ 3	3.6.3.24a1.1	$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 29 x^{12} + 24 x^{11} + 51 x^{10} + 36 x^{9} + 72 x^{8} + 60 x^{7} + 85 x^{6} + 78 x^{5} + 69 x^{4} + 44 x^{3} + 60 x^{2} + 48 x + 23$	$3$	$6$	$24$	$S_3 \times C_3$	not computed
$5$ 5	5.2.3.4a1.2	$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$	$3$	$2$	$4$	$S_3$	$$[\ ]_{3}^{2}$$
$5$ 5	5.2.6.10a1.2	$x^{12} + 24 x^{11} + 252 x^{10} + 1520 x^{9} + 5820 x^{8} + 14784 x^{7} + 25376 x^{6} + 29568 x^{5} + 23280 x^{4} + 12160 x^{3} + 4032 x^{2} + 768 x + 69$	$6$	$2$	$10$	$D_6$	$$[\ ]_{6}^{2}$$
$13$ 13	13.2.1.0a1.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	13.2.1.0a1.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	13.2.1.0a1.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	13.2.2.2a1.2	$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	13.4.2.4a1.2	$x^{8} + 6 x^{6} + 24 x^{5} + 13 x^{4} + 72 x^{3} + 156 x^{2} + 48 x + 17$	$2$	$4$	$4$	$C_4\times C_2$	$$[\ ]_{2}^{4}$$
$1061117$ 1061117		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 $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$

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