LMFDB - Number field 18.10.289123420349393529839217.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 3*x^15 + 9*x^14 + 21*x^13 - x^12 - 21*x^11 + 27*x^10 - 60*x^9 - 162*x^8 - 3*x^7 + 183*x^6 + 162*x^5 - 48*x^4 - 92*x^3 + 12*x^2 + 9*x - 1)

gp: K = bnfinit(y^18 - 6*y^16 - 3*y^15 + 9*y^14 + 21*y^13 - y^12 - 21*y^11 + 27*y^10 - 60*y^9 - 162*y^8 - 3*y^7 + 183*y^6 + 162*y^5 - 48*y^4 - 92*y^3 + 12*y^2 + 9*y - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^16 - 3*x^15 + 9*x^14 + 21*x^13 - x^12 - 21*x^11 + 27*x^10 - 60*x^9 - 162*x^8 - 3*x^7 + 183*x^6 + 162*x^5 - 48*x^4 - 92*x^3 + 12*x^2 + 9*x - 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^16 - 3*x^15 + 9*x^14 + 21*x^13 - x^12 - 21*x^11 + 27*x^10 - 60*x^9 - 162*x^8 - 3*x^7 + 183*x^6 + 162*x^5 - 48*x^4 - 92*x^3 + 12*x^2 + 9*x - 1)

$ x^{18} - 6 x^{16} - 3 x^{15} + 9 x^{14} + 21 x^{13} - x^{12} - 21 x^{11} + 27 x^{10} - 60 x^{9} + \cdots - 1 $ x^18 - 6*x^16 - 3*x^15 + 9*x^14 + 21*x^13 - x^12 - 21*x^11 + 27*x^10 - 60*x^9 - 162*x^8 - 3*x^7 + 183*x^6 + 162*x^5 - 48*x^4 - 92*x^3 + 12*x^2 + 9*x - 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:	$[10, 4]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$289123420349393529839217$ 289123420349393529839217 $\medspace = 3^{24}\cdot 73^{2}\cdot 577^{3}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$20.11$	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:	$3^{4/3}73^{1/2}577^{1/2}\approx 887.9960002144848$
Ramified primes:	$3$, $73$, $577$ 3, 73, 577	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{577}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{9}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{9}a^{9}-\frac{4}{9}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{4}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{9}$, $\frac{1}{9}a^{13}+\frac{1}{3}a^{11}-\frac{1}{9}a^{10}+\frac{1}{3}a^{9}-\frac{4}{9}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{4}{9}a^{4}-\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{4}{9}a^{8}-\frac{1}{3}a^{7}-\frac{4}{9}a^{5}+\frac{1}{3}a^{3}-\frac{1}{9}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{81}a^{15}-\frac{1}{27}a^{14}+\frac{1}{27}a^{13}-\frac{2}{81}a^{12}-\frac{5}{27}a^{11}+\frac{5}{27}a^{10}+\frac{5}{27}a^{9}+\frac{1}{3}a^{8}+\frac{5}{27}a^{7}-\frac{4}{9}a^{6}-\frac{1}{3}a^{5}-\frac{5}{27}a^{4}+\frac{10}{27}a^{3}-\frac{4}{9}a^{2}-\frac{1}{9}a-\frac{8}{81}$, $\frac{1}{81}a^{16}+\frac{1}{27}a^{14}-\frac{2}{81}a^{13}-\frac{1}{27}a^{12}-\frac{4}{27}a^{11}-\frac{4}{27}a^{10}-\frac{1}{3}a^{9}-\frac{7}{27}a^{8}+\frac{2}{9}a^{7}-\frac{2}{9}a^{6}+\frac{10}{27}a^{5}-\frac{11}{27}a^{4}+\frac{1}{9}a^{3}-\frac{2}{9}a^{2}+\frac{1}{81}a+\frac{13}{27}$, $\frac{1}{1370915199}a^{17}-\frac{5101021}{1370915199}a^{16}+\frac{4718561}{1370915199}a^{15}-\frac{66728756}{1370915199}a^{14}-\frac{2973010}{1370915199}a^{13}-\frac{15985687}{1370915199}a^{12}+\frac{91185722}{456971733}a^{11}+\frac{6078875}{456971733}a^{10}-\frac{53755720}{152323911}a^{9}+\frac{111302545}{456971733}a^{8}-\frac{129050174}{456971733}a^{7}+\frac{114907360}{456971733}a^{6}-\frac{37084399}{152323911}a^{5}+\frac{93741475}{456971733}a^{4}+\frac{87550205}{456971733}a^{3}-\frac{475201700}{1370915199}a^{2}-\frac{322958338}{1370915199}a+\frac{63166922}{1370915199}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/9*a^12 + 1/3*a^11 + 1/3*a^10 - 1/9*a^9 - 4/9*a^6 + 1/3*a^5 - 1/3*a^4 - 4/9*a^3 - 1/3*a^2 - 1/9, 1/9*a^13 + 1/3*a^11 - 1/9*a^10 + 1/3*a^9 - 4/9*a^7 - 1/3*a^6 - 1/3*a^5 - 4/9*a^4 - 1/9*a + 1/3, 1/9*a^14 - 1/9*a^11 + 1/3*a^10 + 1/3*a^9 - 4/9*a^8 - 1/3*a^7 - 4/9*a^5 + 1/3*a^3 - 1/9*a^2 + 1/3*a + 1/3, 1/81*a^15 - 1/27*a^14 + 1/27*a^13 - 2/81*a^12 - 5/27*a^11 + 5/27*a^10 + 5/27*a^9 + 1/3*a^8 + 5/27*a^7 - 4/9*a^6 - 1/3*a^5 - 5/27*a^4 + 10/27*a^3 - 4/9*a^2 - 1/9*a - 8/81, 1/81*a^16 + 1/27*a^14 - 2/81*a^13 - 1/27*a^12 - 4/27*a^11 - 4/27*a^10 - 1/3*a^9 - 7/27*a^8 + 2/9*a^7 - 2/9*a^6 + 10/27*a^5 - 11/27*a^4 + 1/9*a^3 - 2/9*a^2 + 1/81*a + 13/27, 1/1370915199*a^17 - 5101021/1370915199*a^16 + 4718561/1370915199*a^15 - 66728756/1370915199*a^14 - 2973010/1370915199*a^13 - 15985687/1370915199*a^12 + 91185722/456971733*a^11 + 6078875/456971733*a^10 - 53755720/152323911*a^9 + 111302545/456971733*a^8 - 129050174/456971733*a^7 + 114907360/456971733*a^6 - 37084399/152323911*a^5 + 93741475/456971733*a^4 + 87550205/456971733*a^3 - 475201700/1370915199*a^2 - 322958338/1370915199*a + 63166922/1370915199

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 $1$

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:	$13$	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:	$\frac{219746270}{1370915199}a^{17}-\frac{137639242}{1370915199}a^{16}-\frac{1096927882}{1370915199}a^{15}-\frac{48902494}{1370915199}a^{14}+\frac{1339863896}{1370915199}a^{13}+\frac{3656775740}{1370915199}a^{12}-\frac{565679312}{456971733}a^{11}-\frac{354175900}{456971733}a^{10}+\frac{1880345354}{456971733}a^{9}-\frac{5835514559}{456971733}a^{8}-\frac{7279239683}{456971733}a^{7}+\frac{1028807609}{456971733}a^{6}+\frac{8220081094}{456971733}a^{5}+\frac{6419497402}{456971733}a^{4}-\frac{1577933269}{456971733}a^{3}-\frac{2542971022}{1370915199}a^{2}+\frac{216588047}{1370915199}a-\frac{719833435}{1370915199}$, $\frac{250540022}{1370915199}a^{17}-\frac{145310761}{456971733}a^{16}-\frac{1033822369}{1370915199}a^{15}+\frac{1300269023}{1370915199}a^{14}+\frac{482963615}{456971733}a^{13}+\frac{2450477636}{1370915199}a^{12}-\frac{733133410}{152323911}a^{11}+\frac{550057417}{456971733}a^{10}+\frac{2546837228}{456971733}a^{9}-\frac{976239410}{50774637}a^{8}-\frac{822387581}{456971733}a^{7}+\frac{9248360312}{456971733}a^{6}+\frac{7440135647}{456971733}a^{5}-\frac{4036976347}{456971733}a^{4}-\frac{9936636808}{456971733}a^{3}+\frac{7790227346}{1370915199}a^{2}+\frac{2095834675}{456971733}a-\frac{2352708556}{1370915199}$, $\frac{3421528}{152323911}a^{17}-\frac{298293041}{1370915199}a^{16}+\frac{21035171}{456971733}a^{15}+\frac{449723839}{456971733}a^{14}+\frac{109026949}{1370915199}a^{13}-\frac{402099368}{456971733}a^{12}-\frac{1633720918}{456971733}a^{11}+\frac{904233317}{456971733}a^{10}+\frac{222163958}{152323911}a^{9}-\frac{2950640131}{456971733}a^{8}+\frac{2152284034}{152323911}a^{7}+\frac{2739850901}{152323911}a^{6}-\frac{779945447}{456971733}a^{5}-\frac{10456473749}{456971733}a^{4}-\frac{2786234513}{152323911}a^{3}+\frac{1148133152}{152323911}a^{2}+\frac{6070915978}{1370915199}a-\frac{29106658}{152323911}$, $\frac{141834710}{1370915199}a^{17}-\frac{356323678}{1370915199}a^{16}-\frac{426475666}{1370915199}a^{15}+\frac{1135232024}{1370915199}a^{14}+\frac{511456784}{1370915199}a^{13}+\frac{1072296920}{1370915199}a^{12}-\frac{1746899849}{456971733}a^{11}+\frac{975185522}{456971733}a^{10}+\frac{971374574}{456971733}a^{9}-\frac{5583082415}{456971733}a^{8}+\frac{3004821457}{456971733}a^{7}+\frac{5796559052}{456971733}a^{6}+\frac{3679979476}{456971733}a^{5}-\frac{4979805500}{456971733}a^{4}-\frac{6465287500}{456971733}a^{3}+\frac{8476762376}{1370915199}a^{2}-\frac{1250351380}{1370915199}a+\frac{559711718}{1370915199}$, $\frac{4451557}{1370915199}a^{17}+\frac{235142386}{1370915199}a^{16}-\frac{236354480}{1370915199}a^{15}-\frac{1068299369}{1370915199}a^{14}+\frac{173380423}{1370915199}a^{13}+\frac{1267852894}{1370915199}a^{12}+\frac{1263642935}{456971733}a^{11}-\frac{916648394}{456971733}a^{10}+\frac{163939057}{456971733}a^{9}+\frac{1491407510}{456971733}a^{8}-\frac{6594351997}{456971733}a^{7}-\frac{5664670259}{456971733}a^{6}+\frac{868580966}{456971733}a^{5}+\frac{8396261825}{456971733}a^{4}+\frac{5353012516}{456971733}a^{3}-\frac{7114255898}{1370915199}a^{2}-\frac{1137751910}{1370915199}a-\frac{986142716}{1370915199}$, $\frac{219746270}{1370915199}a^{17}-\frac{137639242}{1370915199}a^{16}-\frac{1096927882}{1370915199}a^{15}-\frac{48902494}{1370915199}a^{14}+\frac{1339863896}{1370915199}a^{13}+\frac{3656775740}{1370915199}a^{12}-\frac{565679312}{456971733}a^{11}-\frac{354175900}{456971733}a^{10}+\frac{1880345354}{456971733}a^{9}-\frac{5835514559}{456971733}a^{8}-\frac{7279239683}{456971733}a^{7}+\frac{1028807609}{456971733}a^{6}+\frac{8220081094}{456971733}a^{5}+\frac{6419497402}{456971733}a^{4}-\frac{1577933269}{456971733}a^{3}-\frac{2542971022}{1370915199}a^{2}+\frac{216588047}{1370915199}a-\frac{2090748634}{1370915199}$, $\frac{41391283}{456971733}a^{17}+\frac{66701018}{1370915199}a^{16}-\frac{91840268}{152323911}a^{15}-\frac{7445367}{16924879}a^{14}+\frac{1205015765}{1370915199}a^{13}+\frac{910503919}{456971733}a^{12}+\frac{288918676}{456971733}a^{11}-\frac{1177414622}{456971733}a^{10}+\frac{55474560}{16924879}a^{9}-\frac{2233397378}{456971733}a^{8}-\frac{2831620247}{152323911}a^{7}-\frac{314534867}{152323911}a^{6}+\frac{7051022000}{456971733}a^{5}+\frac{7829233925}{456971733}a^{4}-\frac{156979312}{50774637}a^{3}-\frac{3600043799}{456971733}a^{2}+\frac{3873193346}{1370915199}a+\frac{236352386}{456971733}$, $\frac{19782335}{152323911}a^{17}+\frac{183298439}{1370915199}a^{16}-\frac{1136655685}{1370915199}a^{15}-\frac{168305935}{152323911}a^{14}+\frac{1356641435}{1370915199}a^{13}+\frac{5016381800}{1370915199}a^{12}+\frac{39239075}{16924879}a^{11}-\frac{1524938029}{456971733}a^{10}+\frac{948751993}{456971733}a^{9}-\frac{1973024588}{456971733}a^{8}-\frac{14029270943}{456971733}a^{7}-\frac{2540726797}{152323911}a^{6}+\frac{11088055625}{456971733}a^{5}+\frac{17645605666}{456971733}a^{4}+\frac{4521304658}{456971733}a^{3}-\frac{802343362}{50774637}a^{2}-\frac{5669554768}{1370915199}a+\frac{1984820657}{1370915199}$, $\frac{63261886}{1370915199}a^{17}+\frac{6189506}{16924879}a^{16}-\frac{277939669}{456971733}a^{15}-\frac{2573753417}{1370915199}a^{14}+\frac{351759836}{456971733}a^{13}+\frac{53892930}{16924879}a^{12}+\frac{2597408885}{456971733}a^{11}-\frac{844593185}{152323911}a^{10}+\frac{138264503}{456971733}a^{9}+\frac{951004252}{152323911}a^{8}-\frac{5822616157}{152323911}a^{7}-\frac{12749804339}{456971733}a^{6}+\frac{9098795959}{456971733}a^{5}+\frac{7706997925}{152323911}a^{4}+\frac{3222975758}{152323911}a^{3}-\frac{31016137019}{1370915199}a^{2}-\frac{1630303151}{456971733}a+\frac{696873961}{456971733}$, $\frac{507908975}{1370915199}a^{17}-\frac{89282579}{456971733}a^{16}-\frac{2668059184}{1370915199}a^{15}-\frac{295639576}{1370915199}a^{14}+\frac{1195700725}{456971733}a^{13}+\frac{8805069023}{1370915199}a^{12}-\frac{413820578}{152323911}a^{11}-\frac{1599177488}{456971733}a^{10}+\frac{4572167123}{456971733}a^{9}-\frac{4247554916}{152323911}a^{8}-\frac{18814329842}{456971733}a^{7}+\frac{3303729212}{456971733}a^{6}+\frac{22087036850}{456971733}a^{5}+\frac{16839046637}{456971733}a^{4}-\frac{7386260884}{456971733}a^{3}-\frac{20573429323}{1370915199}a^{2}+\frac{764545624}{152323911}a+\frac{225281870}{1370915199}$, $\frac{459789136}{1370915199}a^{17}-\frac{378611983}{1370915199}a^{16}-\frac{2113950196}{1370915199}a^{15}+\frac{181713307}{1370915199}a^{14}+\frac{2358656888}{1370915199}a^{13}+\frac{7409097881}{1370915199}a^{12}-\frac{1566079894}{456971733}a^{11}+\frac{50953703}{456971733}a^{10}+\frac{1142208557}{152323911}a^{9}-\frac{12473707430}{456971733}a^{8}-\frac{12121957046}{456971733}a^{7}+\frac{1306048717}{456971733}a^{6}+\frac{5194402475}{152323911}a^{5}+\frac{10905504112}{456971733}a^{4}-\frac{3799918900}{456971733}a^{3}+\frac{1697453944}{1370915199}a^{2}-\frac{309898393}{1370915199}a-\frac{1727492839}{1370915199}$, $\frac{476362594}{1370915199}a^{17}-\frac{151247000}{1370915199}a^{16}-\frac{2760425633}{1370915199}a^{15}-\frac{689044358}{1370915199}a^{14}+\frac{4293459634}{1370915199}a^{13}+\frac{9152633965}{1370915199}a^{12}-\frac{967064473}{456971733}a^{11}-\frac{2930853875}{456971733}a^{10}+\frac{4456069093}{456971733}a^{9}-\frac{10929770605}{456971733}a^{8}-\frac{21684366724}{456971733}a^{7}+\frac{4223476897}{456971733}a^{6}+\frac{28600744007}{456971733}a^{5}+\frac{20854331621}{456971733}a^{4}-\frac{11437716089}{456971733}a^{3}-\frac{38697273323}{1370915199}a^{2}+\frac{5474656693}{1370915199}a+\frac{3398324245}{1370915199}$, $\frac{795734944}{1370915199}a^{17}-\frac{32016037}{152323911}a^{16}-\frac{4291681499}{1370915199}a^{15}-\frac{1239023051}{1370915199}a^{14}+\frac{655335289}{152323911}a^{13}+\frac{15170485276}{1370915199}a^{12}-\frac{456417508}{152323911}a^{11}-\frac{3179238001}{456971733}a^{10}+\frac{6403295341}{456971733}a^{9}-\frac{6055039697}{152323911}a^{8}-\frac{33554253856}{456971733}a^{7}+\frac{579380923}{456971733}a^{6}+\frac{40395987460}{456971733}a^{5}+\frac{33238603726}{456971733}a^{4}-\frac{10057985696}{456971733}a^{3}-\frac{45121261385}{1370915199}a^{2}+\frac{1183808752}{456971733}a+\frac{3558790078}{1370915199}$ 219746270/1370915199a^17 - 137639242/1370915199a^16 - 1096927882/1370915199a^15 - 48902494/1370915199a^14 + 1339863896/1370915199a^13 + 3656775740/1370915199a^12 - 565679312/456971733a^11 - 354175900/456971733a^10 + 1880345354/456971733a^9 - 5835514559/456971733a^8 - 7279239683/456971733a^7 + 1028807609/456971733a^6 + 8220081094/456971733a^5 + 6419497402/456971733a^4 - 1577933269/456971733a^3 - 2542971022/1370915199a^2 + 216588047/1370915199a - 719833435/1370915199, 250540022/1370915199a^17 - 145310761/456971733a^16 - 1033822369/1370915199a^15 + 1300269023/1370915199a^14 + 482963615/456971733a^13 + 2450477636/1370915199a^12 - 733133410/152323911a^11 + 550057417/456971733a^10 + 2546837228/456971733a^9 - 976239410/50774637a^8 - 822387581/456971733a^7 + 9248360312/456971733a^6 + 7440135647/456971733a^5 - 4036976347/456971733a^4 - 9936636808/456971733a^3 + 7790227346/1370915199a^2 + 2095834675/456971733a - 2352708556/1370915199, 3421528/152323911a^17 - 298293041/1370915199a^16 + 21035171/456971733a^15 + 449723839/456971733a^14 + 109026949/1370915199a^13 - 402099368/456971733a^12 - 1633720918/456971733a^11 + 904233317/456971733a^10 + 222163958/152323911a^9 - 2950640131/456971733a^8 + 2152284034/152323911a^7 + 2739850901/152323911a^6 - 779945447/456971733a^5 - 10456473749/456971733a^4 - 2786234513/152323911a^3 + 1148133152/152323911a^2 + 6070915978/1370915199a - 29106658/152323911, 141834710/1370915199a^17 - 356323678/1370915199a^16 - 426475666/1370915199a^15 + 1135232024/1370915199a^14 + 511456784/1370915199a^13 + 1072296920/1370915199a^12 - 1746899849/456971733a^11 + 975185522/456971733a^10 + 971374574/456971733a^9 - 5583082415/456971733a^8 + 3004821457/456971733a^7 + 5796559052/456971733a^6 + 3679979476/456971733a^5 - 4979805500/456971733a^4 - 6465287500/456971733a^3 + 8476762376/1370915199a^2 - 1250351380/1370915199a + 559711718/1370915199, 4451557/1370915199a^17 + 235142386/1370915199a^16 - 236354480/1370915199a^15 - 1068299369/1370915199a^14 + 173380423/1370915199a^13 + 1267852894/1370915199a^12 + 1263642935/456971733a^11 - 916648394/456971733a^10 + 163939057/456971733a^9 + 1491407510/456971733a^8 - 6594351997/456971733a^7 - 5664670259/456971733a^6 + 868580966/456971733a^5 + 8396261825/456971733a^4 + 5353012516/456971733a^3 - 7114255898/1370915199a^2 - 1137751910/1370915199a - 986142716/1370915199, 219746270/1370915199a^17 - 137639242/1370915199a^16 - 1096927882/1370915199a^15 - 48902494/1370915199a^14 + 1339863896/1370915199a^13 + 3656775740/1370915199a^12 - 565679312/456971733a^11 - 354175900/456971733a^10 + 1880345354/456971733a^9 - 5835514559/456971733a^8 - 7279239683/456971733a^7 + 1028807609/456971733a^6 + 8220081094/456971733a^5 + 6419497402/456971733a^4 - 1577933269/456971733a^3 - 2542971022/1370915199a^2 + 216588047/1370915199a - 2090748634/1370915199, 41391283/456971733a^17 + 66701018/1370915199a^16 - 91840268/152323911a^15 - 7445367/16924879a^14 + 1205015765/1370915199a^13 + 910503919/456971733a^12 + 288918676/456971733a^11 - 1177414622/456971733a^10 + 55474560/16924879a^9 - 2233397378/456971733a^8 - 2831620247/152323911a^7 - 314534867/152323911a^6 + 7051022000/456971733a^5 + 7829233925/456971733a^4 - 156979312/50774637a^3 - 3600043799/456971733a^2 + 3873193346/1370915199a + 236352386/456971733, 19782335/152323911a^17 + 183298439/1370915199a^16 - 1136655685/1370915199a^15 - 168305935/152323911a^14 + 1356641435/1370915199a^13 + 5016381800/1370915199a^12 + 39239075/16924879a^11 - 1524938029/456971733a^10 + 948751993/456971733a^9 - 1973024588/456971733a^8 - 14029270943/456971733a^7 - 2540726797/152323911a^6 + 11088055625/456971733a^5 + 17645605666/456971733a^4 + 4521304658/456971733a^3 - 802343362/50774637a^2 - 5669554768/1370915199a + 1984820657/1370915199, 63261886/1370915199a^17 + 6189506/16924879a^16 - 277939669/456971733a^15 - 2573753417/1370915199a^14 + 351759836/456971733a^13 + 53892930/16924879a^12 + 2597408885/456971733a^11 - 844593185/152323911a^10 + 138264503/456971733a^9 + 951004252/152323911a^8 - 5822616157/152323911a^7 - 12749804339/456971733a^6 + 9098795959/456971733a^5 + 7706997925/152323911a^4 + 3222975758/152323911a^3 - 31016137019/1370915199a^2 - 1630303151/456971733a + 696873961/456971733, 507908975/1370915199a^17 - 89282579/456971733a^16 - 2668059184/1370915199a^15 - 295639576/1370915199a^14 + 1195700725/456971733a^13 + 8805069023/1370915199a^12 - 413820578/152323911a^11 - 1599177488/456971733a^10 + 4572167123/456971733a^9 - 4247554916/152323911a^8 - 18814329842/456971733a^7 + 3303729212/456971733a^6 + 22087036850/456971733a^5 + 16839046637/456971733a^4 - 7386260884/456971733a^3 - 20573429323/1370915199a^2 + 764545624/152323911a + 225281870/1370915199, 459789136/1370915199a^17 - 378611983/1370915199a^16 - 2113950196/1370915199a^15 + 181713307/1370915199a^14 + 2358656888/1370915199a^13 + 7409097881/1370915199a^12 - 1566079894/456971733a^11 + 50953703/456971733a^10 + 1142208557/152323911a^9 - 12473707430/456971733a^8 - 12121957046/456971733a^7 + 1306048717/456971733a^6 + 5194402475/152323911a^5 + 10905504112/456971733a^4 - 3799918900/456971733a^3 + 1697453944/1370915199a^2 - 309898393/1370915199a - 1727492839/1370915199, 476362594/1370915199a^17 - 151247000/1370915199a^16 - 2760425633/1370915199a^15 - 689044358/1370915199a^14 + 4293459634/1370915199a^13 + 9152633965/1370915199a^12 - 967064473/456971733a^11 - 2930853875/456971733a^10 + 4456069093/456971733a^9 - 10929770605/456971733a^8 - 21684366724/456971733a^7 + 4223476897/456971733a^6 + 28600744007/456971733a^5 + 20854331621/456971733a^4 - 11437716089/456971733a^3 - 38697273323/1370915199a^2 + 5474656693/1370915199a + 3398324245/1370915199, 795734944/1370915199a^17 - 32016037/152323911a^16 - 4291681499/1370915199a^15 - 1239023051/1370915199a^14 + 655335289/152323911a^13 + 15170485276/1370915199a^12 - 456417508/152323911a^11 - 3179238001/456971733a^10 + 6403295341/456971733a^9 - 6055039697/152323911a^8 - 33554253856/456971733a^7 + 579380923/456971733a^6 + 40395987460/456971733a^5 + 33238603726/456971733a^4 - 10057985696/456971733a^3 - 45121261385/1370915199a^2 + 1183808752/456971733*a + 3558790078/1370915199	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:	$ 113389.185023 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 113389.185023 \cdot 1}{2\cdot\sqrt{289123420349393529839217}}\cr\approx \mathstrut & 0.168274935112 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 3*x^15 + 9*x^14 + 21*x^13 - x^12 - 21*x^11 + 27*x^10 - 60*x^9 - 162*x^8 - 3*x^7 + 183*x^6 + 162*x^5 - 48*x^4 - 92*x^3 + 12*x^2 + 9*x - 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 - 6*x^16 - 3*x^15 + 9*x^14 + 21*x^13 - x^12 - 21*x^11 + 27*x^10 - 60*x^9 - 162*x^8 - 3*x^7 + 183*x^6 + 162*x^5 - 48*x^4 - 92*x^3 + 12*x^2 + 9*x - 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 - 6*x^16 - 3*x^15 + 9*x^14 + 21*x^13 - x^12 - 21*x^11 + 27*x^10 - 60*x^9 - 162*x^8 - 3*x^7 + 183*x^6 + 162*x^5 - 48*x^4 - 92*x^3 + 12*x^2 + 9*x - 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 - 6*x^16 - 3*x^15 + 9*x^14 + 21*x^13 - x^12 - 21*x^11 + 27*x^10 - 60*x^9 - 162*x^8 - 3*x^7 + 183*x^6 + 162*x^5 - 48*x^4 - 92*x^3 + 12*x^2 + 9*x - 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

$C_2\times S_4^3.A_4$ (as 18T879):

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 331776

The 360 conjugacy class representatives for $C_2\times S_4^3.A_4$

Character table for $C_2\times S_4^3.A_4$

Intermediate fields

$\Q(\zeta_{9})^+$, 9.9.22384826361.1

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 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	${\href{/padicField/2.9.0.1}{9} }^{2}$	R	$18$	$18$	${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$	$18$	${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$	${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$	${\href{/padicField/23.9.0.1}{9} }^{2}$	$18$	${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$	${\href{/padicField/41.6.0.1}{6} }^{3}$	$18$	$18$	${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$

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	3.9.12.1	$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$	$3$	$3$	$12$	$C_3^2$	$[2]^{3}$
$3$ 3	3.9.12.1	$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$	$3$	$3$	$12$	$C_3^2$	$[2]^{3}$
$73$ 73	73.2.0.1	$x^{2} + 70 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	73.2.0.1	$x^{2} + 70 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	73.2.0.1	$x^{2} + 70 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	73.2.1.1	$x^{2} + 73$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	73.2.1.1	$x^{2} + 73$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	73.2.0.1	$x^{2} + 70 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	73.6.0.1	$x^{6} + 45 x^{3} + 23 x^{2} + 48 x + 5$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$577$ 577	$\Q_{577}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{577}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{577}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{577}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		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 $4$	$2$	$2$	$2$

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