LMFDB - Number field 18.14.4627653845488381463962761.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 15*x^16 - 15*x^15 + 54*x^14 + 114*x^13 - 32*x^12 - 327*x^11 - 102*x^10 + 457*x^9 + 102*x^8 - 327*x^7 + 32*x^6 + 114*x^5 - 54*x^4 - 15*x^3 + 15*x^2 - 1)

gp: K = bnfinit(y^18 - 15*y^16 - 15*y^15 + 54*y^14 + 114*y^13 - 32*y^12 - 327*y^11 - 102*y^10 + 457*y^9 + 102*y^8 - 327*y^7 + 32*y^6 + 114*y^5 - 54*y^4 - 15*y^3 + 15*y^2 - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 15*x^16 - 15*x^15 + 54*x^14 + 114*x^13 - 32*x^12 - 327*x^11 - 102*x^10 + 457*x^9 + 102*x^8 - 327*x^7 + 32*x^6 + 114*x^5 - 54*x^4 - 15*x^3 + 15*x^2 - 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 15*x^16 - 15*x^15 + 54*x^14 + 114*x^13 - 32*x^12 - 327*x^11 - 102*x^10 + 457*x^9 + 102*x^8 - 327*x^7 + 32*x^6 + 114*x^5 - 54*x^4 - 15*x^3 + 15*x^2 - 1)

$ x^{18} - 15 x^{16} - 15 x^{15} + 54 x^{14} + 114 x^{13} - 32 x^{12} - 327 x^{11} - 102 x^{10} + 457 x^{9} + \cdots - 1 $ x^18 - 15*x^16 - 15*x^15 + 54*x^14 + 114*x^13 - 32*x^12 - 327*x^11 - 102*x^10 + 457*x^9 + 102*x^8 - 327*x^7 + 32*x^6 + 114*x^5 - 54*x^4 - 15*x^3 + 15*x^2 - 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$18$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[14, 2]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$4627653845488381463962761$ 4627653845488381463962761 $\medspace = 3^{24}\cdot 199^{2}\cdot 20341^{2}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$23.46$	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:	$3^{4/3}199^{1/2}20341^{1/2}\approx 8705.111958789244$
Ramified primes:	$3$, $199$, $20341$ 3, 199, 20341	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{201909}a^{16}+\frac{89}{5457}a^{15}+\frac{9960}{67303}a^{14}+\frac{1132}{201909}a^{13}-\frac{24091}{201909}a^{12}+\frac{28915}{201909}a^{11}+\frac{6521}{201909}a^{10}+\frac{88973}{201909}a^{9}+\frac{172}{1819}a^{8}-\frac{32083}{67303}a^{7}-\frac{66239}{201909}a^{6}-\frac{85304}{201909}a^{5}+\frac{23203}{201909}a^{4}+\frac{57076}{201909}a^{3}-\frac{50156}{201909}a^{2}-\frac{89}{5457}a-\frac{15158}{67303}$, $\frac{1}{201909}a^{17}+\frac{21814}{201909}a^{15}+\frac{3278}{201909}a^{14}+\frac{17201}{201909}a^{13}+\frac{3447}{67303}a^{12}+\frac{1363}{11877}a^{11}-\frac{16543}{67303}a^{10}+\frac{615}{1819}a^{9}+\frac{96706}{201909}a^{8}+\frac{6398}{67303}a^{7}-\frac{89300}{201909}a^{6}-\frac{20250}{67303}a^{5}-\frac{28801}{201909}a^{4}+\frac{14386}{67303}a^{3}+\frac{596}{1819}a^{2}-\frac{37408}{201909}a-\frac{636}{1819}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/3*a^12 - 1/3*a^10 - 1/3*a^9 - 1/3*a^8 - 1/3*a^7 + 1/3*a^5 - 1/3*a^4 + 1/3*a^3 - 1/3*a^2 + 1/3, 1/3*a^13 - 1/3*a^11 - 1/3*a^10 - 1/3*a^9 - 1/3*a^8 + 1/3*a^6 - 1/3*a^5 + 1/3*a^4 - 1/3*a^3 + 1/3*a, 1/3*a^14 - 1/3*a^11 + 1/3*a^10 + 1/3*a^9 - 1/3*a^8 - 1/3*a^6 - 1/3*a^5 + 1/3*a^4 + 1/3*a^3 + 1/3, 1/3*a^15 + 1/3*a^11 + 1/3*a^9 - 1/3*a^8 + 1/3*a^7 - 1/3*a^6 - 1/3*a^5 + 1/3*a^3 - 1/3*a^2 + 1/3*a + 1/3, 1/201909*a^16 + 89/5457*a^15 + 9960/67303*a^14 + 1132/201909*a^13 - 24091/201909*a^12 + 28915/201909*a^11 + 6521/201909*a^10 + 88973/201909*a^9 + 172/1819*a^8 - 32083/67303*a^7 - 66239/201909*a^6 - 85304/201909*a^5 + 23203/201909*a^4 + 57076/201909*a^3 - 50156/201909*a^2 - 89/5457*a - 15158/67303, 1/201909*a^17 + 21814/201909*a^15 + 3278/201909*a^14 + 17201/201909*a^13 + 3447/67303*a^12 + 1363/11877*a^11 - 16543/67303*a^10 + 615/1819*a^9 + 96706/201909*a^8 + 6398/67303*a^7 - 89300/201909*a^6 - 20250/67303*a^5 - 28801/201909*a^4 + 14386/67303*a^3 + 596/1819*a^2 - 37408/201909*a - 636/1819

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

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

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$15$	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:	$\frac{10615}{5457}a^{17}+\frac{7768}{5457}a^{16}-\frac{52153}{1819}a^{15}-\frac{272729}{5457}a^{14}+\frac{416095}{5457}a^{13}+\frac{1544504}{5457}a^{12}+\frac{212291}{1819}a^{11}-\frac{1092851}{1819}a^{10}-\frac{1116178}{1819}a^{9}+\frac{3228403}{5457}a^{8}+\frac{1174287}{1819}a^{7}-\frac{1987133}{5457}a^{6}-\frac{1064180}{5457}a^{5}+\frac{337183}{1819}a^{4}-\frac{44242}{5457}a^{3}-\frac{275705}{5457}a^{2}+\frac{40552}{5457}a+\frac{1237}{321}$, $\frac{10615}{5457}a^{17}+\frac{7768}{5457}a^{16}-\frac{52153}{1819}a^{15}-\frac{272729}{5457}a^{14}+\frac{416095}{5457}a^{13}+\frac{1544504}{5457}a^{12}+\frac{212291}{1819}a^{11}-\frac{1092851}{1819}a^{10}-\frac{1116178}{1819}a^{9}+\frac{3228403}{5457}a^{8}+\frac{1174287}{1819}a^{7}-\frac{1987133}{5457}a^{6}-\frac{1064180}{5457}a^{5}+\frac{337183}{1819}a^{4}-\frac{44242}{5457}a^{3}-\frac{275705}{5457}a^{2}+\frac{40552}{5457}a+\frac{1558}{321}$, $\frac{464837}{201909}a^{17}+\frac{42978}{67303}a^{16}-\frac{6882836}{201909}a^{15}-\frac{2966212}{67303}a^{14}+\frac{21844090}{201909}a^{13}+\frac{58469920}{201909}a^{12}+\frac{4221445}{201909}a^{11}-\frac{145560317}{201909}a^{10}-\frac{30063640}{67303}a^{9}+\frac{57178111}{67303}a^{8}+\frac{92630503}{201909}a^{7}-\frac{105452753}{201909}a^{6}-\frac{4527267}{67303}a^{5}+\frac{37801952}{201909}a^{4}-\frac{11153722}{201909}a^{3}-\frac{2646792}{67303}a^{2}+\frac{925627}{67303}a+\frac{660350}{201909}$, $a^{17}-15a^{15}-15a^{14}+54a^{13}+114a^{12}-32a^{11}-327a^{10}-102a^{9}+457a^{8}+102a^{7}-327a^{6}+32a^{5}+114a^{4}-54a^{3}-15a^{2}+15a$, $\frac{42055}{67303}a^{17}-\frac{52198}{67303}a^{16}-\frac{621302}{67303}a^{15}+\frac{406199}{201909}a^{14}+\frac{2913338}{67303}a^{13}+\frac{2077928}{67303}a^{12}-\frac{19695251}{201909}a^{11}-\frac{35539825}{201909}a^{10}+\frac{32354141}{201909}a^{9}+\frac{65818990}{201909}a^{8}-\frac{15806456}{67303}a^{7}-\frac{1112852}{5457}a^{6}+\frac{39580402}{201909}a^{5}+\frac{316496}{201909}a^{4}-\frac{12693853}{201909}a^{3}+\frac{1590626}{67303}a^{2}+\frac{279549}{67303}a-\frac{576709}{201909}$, $\frac{1638854}{201909}a^{17}+\frac{876131}{201909}a^{16}-\frac{8017291}{67303}a^{15}-\frac{37436146}{201909}a^{14}+\frac{67574297}{201909}a^{13}+\frac{221942593}{201909}a^{12}+\frac{23037730}{67303}a^{11}-\frac{163990318}{67303}a^{10}-\frac{143448107}{67303}a^{9}+\frac{501240731}{201909}a^{8}+\frac{142315323}{67303}a^{7}-\frac{287154100}{201909}a^{6}-\frac{95619940}{201909}a^{5}+\frac{41424825}{67303}a^{4}-\frac{19463318}{201909}a^{3}-\frac{31915258}{201909}a^{2}+\frac{5549432}{201909}a+\frac{161498}{11877}$, $\frac{103516}{67303}a^{17}+\frac{300170}{201909}a^{16}-\frac{4618045}{201909}a^{15}-\frac{3018276}{67303}a^{14}+\frac{685568}{11877}a^{13}+\frac{49457293}{201909}a^{12}+\frac{24997546}{201909}a^{11}-\frac{101257195}{201909}a^{10}-\frac{21983}{37}a^{9}+\frac{31892529}{67303}a^{8}+\frac{133930264}{201909}a^{7}-\frac{20676561}{67303}a^{6}-\frac{45455410}{201909}a^{5}+\frac{37553260}{201909}a^{4}-\frac{2054777}{201909}a^{3}-\frac{11185313}{201909}a^{2}+\frac{1758566}{201909}a+\frac{834916}{201909}$, $\frac{318754}{201909}a^{17}+\frac{99473}{67303}a^{16}-\frac{4683629}{201909}a^{15}-\frac{3056504}{67303}a^{14}+\frac{11304811}{201909}a^{13}+\frac{447748}{1819}a^{12}+\frac{27277790}{201909}a^{11}-\frac{98883472}{201909}a^{10}-\frac{122013181}{201909}a^{9}+\frac{85736005}{201909}a^{8}+\frac{44404054}{67303}a^{7}-\frac{48279187}{201909}a^{6}-\frac{983252}{3959}a^{5}+\frac{31411073}{201909}a^{4}+\frac{4013534}{201909}a^{3}-\frac{3929387}{67303}a^{2}+\frac{927850}{201909}a+\frac{1277707}{201909}$, $\frac{113086}{67303}a^{17}+\frac{294724}{201909}a^{16}-\frac{4892812}{201909}a^{15}-\frac{9357383}{201909}a^{14}+\frac{649837}{11877}a^{13}+\frac{49432606}{201909}a^{12}+\frac{29743244}{201909}a^{11}-\frac{829959}{1819}a^{10}-\frac{2286352}{3959}a^{9}+\frac{68945000}{201909}a^{8}+\frac{105729373}{201909}a^{7}-\frac{33058397}{201909}a^{6}-\frac{25132184}{201909}a^{5}+\frac{22651438}{201909}a^{4}+\frac{208217}{67303}a^{3}-\frac{6257395}{201909}a^{2}+\frac{144665}{201909}a+\frac{142049}{67303}$, $\frac{420290}{201909}a^{17}+\frac{3797}{3959}a^{16}-\frac{6151436}{201909}a^{15}-\frac{9136154}{201909}a^{14}+\frac{17557858}{201909}a^{13}+\frac{18334827}{67303}a^{12}+\frac{292182}{3959}a^{11}-\frac{41164818}{67303}a^{10}-\frac{100128565}{201909}a^{9}+\frac{42569249}{67303}a^{8}+\frac{93482234}{201909}a^{7}-\frac{24391657}{67303}a^{6}-\frac{14161994}{201909}a^{5}+\frac{30472792}{201909}a^{4}-\frac{7665346}{201909}a^{3}-\frac{7514876}{201909}a^{2}+\frac{714813}{67303}a+\frac{630491}{201909}$, $\frac{1005}{3959}a^{17}-\frac{213643}{201909}a^{16}-\frac{709339}{201909}a^{15}+\frac{783538}{67303}a^{14}+\frac{5126089}{201909}a^{13}-\frac{5356216}{201909}a^{12}-\frac{598811}{5457}a^{11}-\frac{7751258}{201909}a^{10}+\frac{55696049}{201909}a^{9}+\frac{31122134}{201909}a^{8}-\frac{25173108}{67303}a^{7}-\frac{4654007}{67303}a^{6}+\frac{15364739}{67303}a^{5}-\frac{3991518}{67303}a^{4}-\frac{9337159}{201909}a^{3}+\frac{2288004}{67303}a^{2}+\frac{258176}{201909}a-\frac{915301}{201909}$, $\frac{178886}{201909}a^{17}+\frac{10420}{11877}a^{16}-\frac{2559331}{201909}a^{15}-\frac{5238994}{201909}a^{14}+\frac{1733801}{67303}a^{13}+\frac{8874479}{67303}a^{12}+\frac{1105769}{11877}a^{11}-\frac{15362856}{67303}a^{10}-\frac{66503608}{201909}a^{9}+\frac{9434666}{67303}a^{8}+\frac{19223774}{67303}a^{7}-\frac{10825367}{201909}a^{6}-\frac{12241145}{201909}a^{5}+\frac{10164230}{201909}a^{4}-\frac{250283}{201909}a^{3}-\frac{2632325}{201909}a^{2}+\frac{28277}{201909}a-\frac{5941}{201909}$, $\frac{438316}{201909}a^{17}+\frac{127770}{67303}a^{16}-\frac{6464494}{201909}a^{15}-\frac{4072989}{67303}a^{14}+\frac{16281352}{201909}a^{13}+\frac{67462429}{201909}a^{12}+\frac{11266482}{67303}a^{11}-\frac{138020285}{201909}a^{10}-\frac{161056033}{201909}a^{9}+\frac{125234879}{201909}a^{8}+\frac{58925828}{67303}a^{7}-\frac{71030186}{201909}a^{6}-\frac{20952944}{67303}a^{5}+\frac{42201358}{201909}a^{4}+\frac{2631703}{201909}a^{3}-\frac{15130718}{201909}a^{2}+\frac{1205927}{201909}a+\frac{527981}{67303}$, $\frac{1045544}{201909}a^{17}+\frac{481972}{201909}a^{16}-\frac{15359852}{201909}a^{15}-\frac{22763825}{201909}a^{14}+\frac{14828455}{67303}a^{13}+\frac{138179236}{201909}a^{12}+\frac{35168995}{201909}a^{11}-\frac{314653843}{201909}a^{10}-\frac{253104799}{201909}a^{9}+\frac{331951370}{201909}a^{8}+\frac{248524939}{201909}a^{7}-\frac{191759804}{201909}a^{6}-\frac{48065455}{201909}a^{5}+\frac{77119984}{201909}a^{4}-\frac{15582436}{201909}a^{3}-\frac{18909652}{201909}a^{2}+\frac{1151046}{67303}a+\frac{1820279}{201909}$, $\frac{128938}{67303}a^{17}+\frac{155434}{67303}a^{16}-\frac{5522728}{201909}a^{15}-\frac{4187063}{67303}a^{14}+\frac{192857}{3959}a^{13}+\frac{61670777}{201909}a^{12}+\frac{50377850}{201909}a^{11}-\frac{100565681}{201909}a^{10}-\frac{3321144}{3959}a^{9}+\frac{50232266}{201909}a^{8}+\frac{52681291}{67303}a^{7}-\frac{13909892}{201909}a^{6}-\frac{16796713}{67303}a^{5}+\frac{24508894}{201909}a^{4}+\frac{8848090}{201909}a^{3}-\frac{10362253}{201909}a^{2}-\frac{1221016}{201909}a+\frac{1008259}{201909}$ 10615/5457a^17 + 7768/5457a^16 - 52153/1819a^15 - 272729/5457a^14 + 416095/5457a^13 + 1544504/5457a^12 + 212291/1819a^11 - 1092851/1819a^10 - 1116178/1819a^9 + 3228403/5457a^8 + 1174287/1819a^7 - 1987133/5457a^6 - 1064180/5457a^5 + 337183/1819a^4 - 44242/5457a^3 - 275705/5457a^2 + 40552/5457a + 1237/321, 10615/5457a^17 + 7768/5457a^16 - 52153/1819a^15 - 272729/5457a^14 + 416095/5457a^13 + 1544504/5457a^12 + 212291/1819a^11 - 1092851/1819a^10 - 1116178/1819a^9 + 3228403/5457a^8 + 1174287/1819a^7 - 1987133/5457a^6 - 1064180/5457a^5 + 337183/1819a^4 - 44242/5457a^3 - 275705/5457a^2 + 40552/5457a + 1558/321, 464837/201909a^17 + 42978/67303a^16 - 6882836/201909a^15 - 2966212/67303a^14 + 21844090/201909a^13 + 58469920/201909a^12 + 4221445/201909a^11 - 145560317/201909a^10 - 30063640/67303a^9 + 57178111/67303a^8 + 92630503/201909a^7 - 105452753/201909a^6 - 4527267/67303a^5 + 37801952/201909a^4 - 11153722/201909a^3 - 2646792/67303a^2 + 925627/67303a + 660350/201909, a^17 - 15a^15 - 15a^14 + 54a^13 + 114a^12 - 32a^11 - 327a^10 - 102a^9 + 457a^8 + 102a^7 - 327a^6 + 32a^5 + 114a^4 - 54a^3 - 15a^2 + 15a, 42055/67303a^17 - 52198/67303a^16 - 621302/67303a^15 + 406199/201909a^14 + 2913338/67303a^13 + 2077928/67303a^12 - 19695251/201909a^11 - 35539825/201909a^10 + 32354141/201909a^9 + 65818990/201909a^8 - 15806456/67303a^7 - 1112852/5457a^6 + 39580402/201909a^5 + 316496/201909a^4 - 12693853/201909a^3 + 1590626/67303a^2 + 279549/67303a - 576709/201909, 1638854/201909a^17 + 876131/201909a^16 - 8017291/67303a^15 - 37436146/201909a^14 + 67574297/201909a^13 + 221942593/201909a^12 + 23037730/67303a^11 - 163990318/67303a^10 - 143448107/67303a^9 + 501240731/201909a^8 + 142315323/67303a^7 - 287154100/201909a^6 - 95619940/201909a^5 + 41424825/67303a^4 - 19463318/201909a^3 - 31915258/201909a^2 + 5549432/201909a + 161498/11877, 103516/67303a^17 + 300170/201909a^16 - 4618045/201909a^15 - 3018276/67303a^14 + 685568/11877a^13 + 49457293/201909a^12 + 24997546/201909a^11 - 101257195/201909a^10 - 21983/37a^9 + 31892529/67303a^8 + 133930264/201909a^7 - 20676561/67303a^6 - 45455410/201909a^5 + 37553260/201909a^4 - 2054777/201909a^3 - 11185313/201909a^2 + 1758566/201909a + 834916/201909, 318754/201909a^17 + 99473/67303a^16 - 4683629/201909a^15 - 3056504/67303a^14 + 11304811/201909a^13 + 447748/1819a^12 + 27277790/201909a^11 - 98883472/201909a^10 - 122013181/201909a^9 + 85736005/201909a^8 + 44404054/67303a^7 - 48279187/201909a^6 - 983252/3959a^5 + 31411073/201909a^4 + 4013534/201909a^3 - 3929387/67303a^2 + 927850/201909a + 1277707/201909, 113086/67303a^17 + 294724/201909a^16 - 4892812/201909a^15 - 9357383/201909a^14 + 649837/11877a^13 + 49432606/201909a^12 + 29743244/201909a^11 - 829959/1819a^10 - 2286352/3959a^9 + 68945000/201909a^8 + 105729373/201909a^7 - 33058397/201909a^6 - 25132184/201909a^5 + 22651438/201909a^4 + 208217/67303a^3 - 6257395/201909a^2 + 144665/201909a + 142049/67303, 420290/201909a^17 + 3797/3959a^16 - 6151436/201909a^15 - 9136154/201909a^14 + 17557858/201909a^13 + 18334827/67303a^12 + 292182/3959a^11 - 41164818/67303a^10 - 100128565/201909a^9 + 42569249/67303a^8 + 93482234/201909a^7 - 24391657/67303a^6 - 14161994/201909a^5 + 30472792/201909a^4 - 7665346/201909a^3 - 7514876/201909a^2 + 714813/67303a + 630491/201909, 1005/3959a^17 - 213643/201909a^16 - 709339/201909a^15 + 783538/67303a^14 + 5126089/201909a^13 - 5356216/201909a^12 - 598811/5457a^11 - 7751258/201909a^10 + 55696049/201909a^9 + 31122134/201909a^8 - 25173108/67303a^7 - 4654007/67303a^6 + 15364739/67303a^5 - 3991518/67303a^4 - 9337159/201909a^3 + 2288004/67303a^2 + 258176/201909a - 915301/201909, 178886/201909a^17 + 10420/11877a^16 - 2559331/201909a^15 - 5238994/201909a^14 + 1733801/67303a^13 + 8874479/67303a^12 + 1105769/11877a^11 - 15362856/67303a^10 - 66503608/201909a^9 + 9434666/67303a^8 + 19223774/67303a^7 - 10825367/201909a^6 - 12241145/201909a^5 + 10164230/201909a^4 - 250283/201909a^3 - 2632325/201909a^2 + 28277/201909a - 5941/201909, 438316/201909a^17 + 127770/67303a^16 - 6464494/201909a^15 - 4072989/67303a^14 + 16281352/201909a^13 + 67462429/201909a^12 + 11266482/67303a^11 - 138020285/201909a^10 - 161056033/201909a^9 + 125234879/201909a^8 + 58925828/67303a^7 - 71030186/201909a^6 - 20952944/67303a^5 + 42201358/201909a^4 + 2631703/201909a^3 - 15130718/201909a^2 + 1205927/201909a + 527981/67303, 1045544/201909a^17 + 481972/201909a^16 - 15359852/201909a^15 - 22763825/201909a^14 + 14828455/67303a^13 + 138179236/201909a^12 + 35168995/201909a^11 - 314653843/201909a^10 - 253104799/201909a^9 + 331951370/201909a^8 + 248524939/201909a^7 - 191759804/201909a^6 - 48065455/201909a^5 + 77119984/201909a^4 - 15582436/201909a^3 - 18909652/201909a^2 + 1151046/67303a + 1820279/201909, 128938/67303a^17 + 155434/67303a^16 - 5522728/201909a^15 - 4187063/67303a^14 + 192857/3959a^13 + 61670777/201909a^12 + 50377850/201909a^11 - 100565681/201909a^10 - 3321144/3959a^9 + 50232266/201909a^8 + 52681291/67303a^7 - 13909892/201909a^6 - 16796713/67303a^5 + 24508894/201909a^4 + 8848090/201909a^3 - 10362253/201909a^2 - 1221016/201909*a + 1008259/201909 (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:	$ 1134548.71484 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 1134548.71484 \cdot 1}{2\cdot\sqrt{4627653845488381463962761}}\cr\approx \mathstrut & 0.170565972863 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^18 - 15*x^16 - 15*x^15 + 54*x^14 + 114*x^13 - 32*x^12 - 327*x^11 - 102*x^10 + 457*x^9 + 102*x^8 - 327*x^7 + 32*x^6 + 114*x^5 - 54*x^4 - 15*x^3 + 15*x^2 - 1)

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))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^18 - 15*x^16 - 15*x^15 + 54*x^14 + 114*x^13 - 32*x^12 - 327*x^11 - 102*x^10 + 457*x^9 + 102*x^8 - 327*x^7 + 32*x^6 + 114*x^5 - 54*x^4 - 15*x^3 + 15*x^2 - 1, 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)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 15*x^16 - 15*x^15 + 54*x^14 + 114*x^13 - 32*x^12 - 327*x^11 - 102*x^10 + 457*x^9 + 102*x^8 - 327*x^7 + 32*x^6 + 114*x^5 - 54*x^4 - 15*x^3 + 15*x^2 - 1);

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)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 15*x^16 - 15*x^15 + 54*x^14 + 114*x^13 - 32*x^12 - 327*x^11 - 102*x^10 + 457*x^9 + 102*x^8 - 327*x^7 + 32*x^6 + 114*x^5 - 54*x^4 - 15*x^3 + 15*x^2 - 1);

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

$S_4\wr C_3$ (as 18T703):

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 41472

The 55 conjugacy class representatives for $S_4\wr C_3$

Character table for $S_4\wr C_3$

Intermediate fields

$\Q(\zeta_{9})^+$, 9.7.2151198234819.1

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

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 12 sibling:	data not computed
Degree 18 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	12.10.174247057020339.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$	R	${\href{/padicField/5.9.0.1}{9} }^{2}$	${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$	${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$	${\href{/padicField/13.9.0.1}{9} }^{2}$	${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$	${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$	${\href{/padicField/23.9.0.1}{9} }^{2}$	${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$	${\href{/padicField/31.9.0.1}{9} }^{2}$	${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$	${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$	${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
$3$ 3	3.6.8.3	$x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$	$3$	$2$	$8$	$C_6$	$[2]^{2}$
$3$ 3	3.12.16.14	$x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$	$3$	$4$	$16$	$C_{12}$	$[2]^{4}$
$199$ 199	$\Q_{199}$	$x + 196$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{199}$	$x + 196$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{199}$	$x + 196$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{199}$	$x + 196$	$1$	$1$	$0$	Trivial	$[\ ]$
	199.2.0.1	$x^{2} + 193 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	199.2.0.1	$x^{2} + 193 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	199.2.1.2	$x^{2} + 199$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	199.2.1.2	$x^{2} + 199$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	199.3.0.1	$x^{3} + x + 196$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	199.3.0.1	$x^{3} + x + 196$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
$20341$ 20341	$\Q_{20341}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{20341}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{20341}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{20341}$	$x$	$1$	$1$	$0$	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 $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$

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