LMFDB - Number field 18.0.258151783382020583032356864.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301)

gp: K = bnfinit(y^18 + 18*y^16 - 18*y^15 + 135*y^14 - 270*y^13 + 699*y^12 - 1620*y^11 + 3123*y^10 - 5760*y^9 + 10044*y^8 - 15390*y^7 + 21528*y^6 - 28674*y^5 + 34641*y^4 - 33822*y^3 + 32643*y^2 - 28566*y + 12301, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301)

$ x^{18} + 18 x^{16} - 18 x^{15} + 135 x^{14} - 270 x^{13} + 699 x^{12} - 1620 x^{11} + 3123 x^{10} + \cdots + 12301 $ x^18 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301

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:	$[0, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-258151783382020583032356864$ -258151783382020583032356864 $\medspace = -\,2^{18}\cdot 3^{44}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$29.33$	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:	$2\cdot 3^{70/27}\approx 34.51525023937336$
Ramified primes:	$2$, $3$ 2, 3	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-1}) $
$\card{ \Aut(K/\Q) }$:	$9$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{16}a^{12}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{3}{16}a^{4}+\frac{1}{4}a^{3}-\frac{3}{8}a^{2}+\frac{3}{8}a-\frac{3}{16}$, $\frac{1}{16}a^{13}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{7}{16}a-\frac{1}{4}$, $\frac{1}{16}a^{14}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{3}{16}a^{6}-\frac{3}{8}a^{4}-\frac{3}{8}a^{3}+\frac{5}{16}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{112784}a^{15}+\frac{15}{112784}a^{13}+\frac{48}{7049}a^{12}+\frac{45}{56392}a^{11}+\frac{576}{7049}a^{10}+\frac{307}{7049}a^{9}-\frac{1865}{14098}a^{8}-\frac{14209}{112784}a^{7}+\frac{11441}{56392}a^{6}+\frac{12793}{112784}a^{5}-\frac{4051}{28196}a^{4}-\frac{34745}{112784}a^{3}-\frac{16515}{56392}a^{2}-\frac{29101}{112784}a-\frac{3483}{56392}$, $\frac{1}{112784}a^{16}+\frac{15}{112784}a^{14}+\frac{48}{7049}a^{13}+\frac{45}{56392}a^{12}+\frac{576}{7049}a^{11}+\frac{307}{7049}a^{10}+\frac{3319}{28196}a^{9}-\frac{14209}{112784}a^{8}-\frac{2657}{56392}a^{7}-\frac{15403}{112784}a^{6}-\frac{2775}{7049}a^{5}+\frac{21647}{112784}a^{4}+\frac{11681}{56392}a^{3}+\frac{55487}{112784}a^{2}+\frac{10615}{56392}a-\frac{1}{4}$, $\frac{1}{467038544}a^{17}-\frac{13}{66719792}a^{16}+\frac{17}{467038544}a^{15}-\frac{11405879}{467038544}a^{14}-\frac{342785}{12290488}a^{13}-\frac{1960263}{66719792}a^{12}+\frac{5106555}{116759636}a^{11}-\frac{5313989}{58379818}a^{10}+\frac{14732337}{467038544}a^{9}-\frac{11604219}{66719792}a^{8}+\frac{12155191}{467038544}a^{7}-\frac{88147459}{467038544}a^{6}-\frac{195032519}{467038544}a^{5}-\frac{189097}{5695592}a^{4}+\frac{204339363}{467038544}a^{3}+\frac{1524559}{24580976}a^{2}+\frac{16665309}{58379818}a-\frac{89952123}{467038544}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^2 - 1/2, 1/2*a^7 - 1/2*a^3 - 1/2*a, 1/2*a^8 - 1/2*a^4 - 1/2*a^2, 1/4*a^9 - 1/4*a^7 - 1/4*a^6 - 1/4*a^5 - 1/2*a^4 - 1/2*a^3 - 1/4*a^2 + 1/4*a - 1/4, 1/4*a^10 - 1/4*a^8 - 1/4*a^7 - 1/4*a^6 - 1/2*a^5 - 1/2*a^4 - 1/4*a^3 + 1/4*a^2 - 1/4*a, 1/4*a^11 - 1/4*a^8 - 1/4*a^6 + 1/4*a^5 + 1/4*a^4 + 1/4*a^3 - 1/4*a + 1/4, 1/16*a^12 - 1/8*a^9 + 1/8*a^8 + 1/8*a^7 - 1/8*a^6 + 1/8*a^5 - 3/16*a^4 + 1/4*a^3 - 3/8*a^2 + 3/8*a - 3/16, 1/16*a^13 - 1/8*a^10 - 1/8*a^9 + 1/8*a^8 + 1/8*a^7 - 1/8*a^6 + 1/16*a^5 - 1/4*a^4 + 1/8*a^3 + 1/8*a^2 - 7/16*a - 1/4, 1/16*a^14 - 1/8*a^11 - 1/8*a^10 - 1/8*a^9 + 1/8*a^8 + 1/8*a^7 - 3/16*a^6 - 3/8*a^4 - 3/8*a^3 + 5/16*a^2 - 1/2*a - 1/4, 1/112784*a^15 + 15/112784*a^13 + 48/7049*a^12 + 45/56392*a^11 + 576/7049*a^10 + 307/7049*a^9 - 1865/14098*a^8 - 14209/112784*a^7 + 11441/56392*a^6 + 12793/112784*a^5 - 4051/28196*a^4 - 34745/112784*a^3 - 16515/56392*a^2 - 29101/112784*a - 3483/56392, 1/112784*a^16 + 15/112784*a^14 + 48/7049*a^13 + 45/56392*a^12 + 576/7049*a^11 + 307/7049*a^10 + 3319/28196*a^9 - 14209/112784*a^8 - 2657/56392*a^7 - 15403/112784*a^6 - 2775/7049*a^5 + 21647/112784*a^4 + 11681/56392*a^3 + 55487/112784*a^2 + 10615/56392*a - 1/4, 1/467038544*a^17 - 13/66719792*a^16 + 17/467038544*a^15 - 11405879/467038544*a^14 - 342785/12290488*a^13 - 1960263/66719792*a^12 + 5106555/116759636*a^11 - 5313989/58379818*a^10 + 14732337/467038544*a^9 - 11604219/66719792*a^8 + 12155191/467038544*a^7 - 88147459/467038544*a^6 - 195032519/467038544*a^5 - 189097/5695592*a^4 + 204339363/467038544*a^3 + 1524559/24580976*a^2 + 16665309/58379818*a - 89952123/467038544

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

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:	$8$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{81}{56392} a^{15} + \frac{1215}{56392} a^{13} - \frac{1233}{56392} a^{12} + \frac{3645}{28196} a^{11} - \frac{3699}{14098} a^{10} + \frac{7831}{14098} a^{9} - \frac{33291}{28196} a^{8} + \frac{117891}{56392} a^{7} - \frac{88335}{28196} a^{6} + \frac{274941}{56392} a^{5} - \frac{360909}{56392} a^{4} + \frac{399999}{56392} a^{3} - \frac{195777}{28196} a^{2} + \frac{434223}{56392} a - \frac{247041}{56392} $ (81)/(56392)a^(15) + (1215)/(56392)a^(13) - (1233)/(56392)a^(12) + (3645)/(28196)a^(11) - (3699)/(14098)a^(10) + (7831)/(14098)a^(9) - (33291)/(28196)a^(8) + (117891)/(56392)a^(7) - (88335)/(28196)a^(6) + (274941)/(56392)a^(5) - (360909)/(56392)a^(4) + (399999)/(56392)a^(3) - (195777)/(28196)a^(2) + (434223)/(56392)a - (247041)/(56392) (order $4$)	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:	$\frac{203383}{116759636}a^{17}+\frac{1396903}{467038544}a^{16}+\frac{2074529}{58379818}a^{15}+\frac{2219005}{116759636}a^{14}+\frac{14491431}{58379818}a^{13}-\frac{5641783}{33359896}a^{12}+\frac{11053383}{12290488}a^{11}-\frac{58811633}{33359896}a^{10}+\frac{704756809}{233519272}a^{9}-\frac{2846428617}{467038544}a^{8}+\frac{557654421}{58379818}a^{7}-\frac{3391095163}{233519272}a^{6}+\frac{2118433007}{116759636}a^{5}-\frac{296450313}{11391184}a^{4}+\frac{3245672945}{116759636}a^{3}-\frac{1301766177}{58379818}a^{2}+\frac{323542547}{12290488}a-\frac{514955306}{29189909}$, $\frac{322681}{233519272}a^{17}-\frac{45691}{233519272}a^{16}+\frac{353079}{16679948}a^{15}-\frac{10109811}{467038544}a^{14}+\frac{8735233}{66719792}a^{13}-\frac{16757087}{66719792}a^{12}+\frac{125861733}{233519272}a^{11}-\frac{135404323}{116759636}a^{10}+\frac{223298571}{116759636}a^{9}-\frac{177236587}{58379818}a^{8}+\frac{1101370159}{233519272}a^{7}-\frac{2804792187}{467038544}a^{6}+\frac{3375474955}{467038544}a^{5}-\frac{91617417}{11391184}a^{4}+\frac{452402681}{58379818}a^{3}-\frac{2743267199}{467038544}a^{2}+\frac{1646739925}{467038544}a-\frac{544859325}{467038544}$, $\frac{138751}{467038544}a^{17}-\frac{29419}{467038544}a^{16}+\frac{434087}{58379818}a^{15}-\frac{3196409}{467038544}a^{14}+\frac{33942375}{467038544}a^{13}-\frac{15354811}{116759636}a^{12}+\frac{93107585}{233519272}a^{11}-\frac{118173473}{116759636}a^{10}+\frac{5979965}{3511568}a^{9}-\frac{1834899275}{467038544}a^{8}+\frac{1413648379}{233519272}a^{7}-\frac{4470390317}{467038544}a^{6}+\frac{404776923}{29189909}a^{5}-\frac{201786245}{11391184}a^{4}+\frac{30390069}{1536311}a^{3}-\frac{9148130877}{467038544}a^{2}+\frac{1304744665}{66719792}a-\frac{687014299}{58379818}$, $\frac{21363}{233519272}a^{17}+\frac{190819}{467038544}a^{16}-\frac{387587}{467038544}a^{15}+\frac{374029}{233519272}a^{14}-\frac{890819}{29189909}a^{13}+\frac{3887165}{233519272}a^{12}-\frac{23768951}{116759636}a^{11}+\frac{135497}{438946}a^{10}-\frac{9388319}{12290488}a^{9}+\frac{109080831}{66719792}a^{8}-\frac{1297922187}{467038544}a^{7}+\frac{254658113}{58379818}a^{6}-\frac{784190465}{116759636}a^{5}+\frac{107350023}{11391184}a^{4}-\frac{4463952547}{467038544}a^{3}+\frac{2454564503}{233519272}a^{2}-\frac{2866863819}{233519272}a+\frac{1439937899}{233519272}$, $\frac{1037819}{467038544}a^{17}+\frac{9976}{29189909}a^{16}+\frac{3662245}{116759636}a^{15}-\frac{5689521}{233519272}a^{14}+\frac{77827569}{467038544}a^{13}-\frac{5093429}{16679948}a^{12}+\frac{135377637}{233519272}a^{11}-\frac{145773423}{116759636}a^{10}+\frac{880998033}{467038544}a^{9}-\frac{626973523}{233519272}a^{8}+\frac{433381777}{116759636}a^{7}-\frac{540108843}{116759636}a^{6}+\frac{902681049}{233519272}a^{5}-\frac{2479433}{813656}a^{4}+\frac{843840175}{233519272}a^{3}+\frac{10003157}{12290488}a^{2}-\frac{119153379}{24580976}a+\frac{156164307}{233519272}$, $\frac{241967}{233519272}a^{17}+\frac{806195}{233519272}a^{16}+\frac{5506241}{467038544}a^{15}+\frac{15772131}{467038544}a^{14}-\frac{73747}{12290488}a^{13}+\frac{68737175}{467038544}a^{12}-\frac{12668812}{29189909}a^{11}+\frac{101969225}{116759636}a^{10}-\frac{59687515}{29189909}a^{9}+\frac{480916159}{116759636}a^{8}-\frac{3200675627}{467038544}a^{7}+\frac{4766059093}{467038544}a^{6}-\frac{1856020685}{116759636}a^{5}+\frac{33050197}{1627312}a^{4}-\frac{9275212949}{467038544}a^{3}+\frac{1536071223}{66719792}a^{2}-\frac{676309607}{29189909}a+\frac{231936451}{24580976}$, $\frac{96025}{467038544}a^{17}-\frac{110119}{233519272}a^{16}+\frac{332533}{116759636}a^{15}-\frac{3944467}{467038544}a^{14}+\frac{5121607}{233519272}a^{13}-\frac{1959695}{33359896}a^{12}+\frac{956739}{8339974}a^{11}-\frac{28942143}{116759636}a^{10}+\frac{166434083}{467038544}a^{9}-\frac{85885379}{116759636}a^{8}+\frac{6747469}{8339974}a^{7}-\frac{651809147}{467038544}a^{6}+\frac{118631051}{66719792}a^{5}-\frac{7415903}{2847796}a^{4}+\frac{716431165}{233519272}a^{3}-\frac{1943269585}{467038544}a^{2}+\frac{160110778}{29189909}a-\frac{12286475}{4169987}$, $\frac{89771}{233519272}a^{17}+\frac{4657}{116759636}a^{16}+\frac{7527}{12290488}a^{15}-\frac{4426545}{467038544}a^{14}-\frac{9538435}{233519272}a^{13}-\frac{376133}{6145244}a^{12}-\frac{60315931}{233519272}a^{11}+\frac{26980671}{233519272}a^{10}-\frac{88223955}{116759636}a^{9}+\frac{308182253}{233519272}a^{8}-\frac{297791413}{116759636}a^{7}+\frac{1628225371}{467038544}a^{6}-\frac{680144973}{116759636}a^{5}+\frac{6476667}{813656}a^{4}-\frac{865991179}{116759636}a^{3}+\frac{3577548351}{467038544}a^{2}-\frac{2519586759}{233519272}a+\frac{27558827}{16679948}$ 203383/116759636a^17 + 1396903/467038544a^16 + 2074529/58379818a^15 + 2219005/116759636a^14 + 14491431/58379818a^13 - 5641783/33359896a^12 + 11053383/12290488a^11 - 58811633/33359896a^10 + 704756809/233519272a^9 - 2846428617/467038544a^8 + 557654421/58379818a^7 - 3391095163/233519272a^6 + 2118433007/116759636a^5 - 296450313/11391184a^4 + 3245672945/116759636a^3 - 1301766177/58379818a^2 + 323542547/12290488a - 514955306/29189909, 322681/233519272a^17 - 45691/233519272a^16 + 353079/16679948a^15 - 10109811/467038544a^14 + 8735233/66719792a^13 - 16757087/66719792a^12 + 125861733/233519272a^11 - 135404323/116759636a^10 + 223298571/116759636a^9 - 177236587/58379818a^8 + 1101370159/233519272a^7 - 2804792187/467038544a^6 + 3375474955/467038544a^5 - 91617417/11391184a^4 + 452402681/58379818a^3 - 2743267199/467038544a^2 + 1646739925/467038544a - 544859325/467038544, 138751/467038544a^17 - 29419/467038544a^16 + 434087/58379818a^15 - 3196409/467038544a^14 + 33942375/467038544a^13 - 15354811/116759636a^12 + 93107585/233519272a^11 - 118173473/116759636a^10 + 5979965/3511568a^9 - 1834899275/467038544a^8 + 1413648379/233519272a^7 - 4470390317/467038544a^6 + 404776923/29189909a^5 - 201786245/11391184a^4 + 30390069/1536311a^3 - 9148130877/467038544a^2 + 1304744665/66719792a - 687014299/58379818, 21363/233519272a^17 + 190819/467038544a^16 - 387587/467038544a^15 + 374029/233519272a^14 - 890819/29189909a^13 + 3887165/233519272a^12 - 23768951/116759636a^11 + 135497/438946a^10 - 9388319/12290488a^9 + 109080831/66719792a^8 - 1297922187/467038544a^7 + 254658113/58379818a^6 - 784190465/116759636a^5 + 107350023/11391184a^4 - 4463952547/467038544a^3 + 2454564503/233519272a^2 - 2866863819/233519272a + 1439937899/233519272, 1037819/467038544a^17 + 9976/29189909a^16 + 3662245/116759636a^15 - 5689521/233519272a^14 + 77827569/467038544a^13 - 5093429/16679948a^12 + 135377637/233519272a^11 - 145773423/116759636a^10 + 880998033/467038544a^9 - 626973523/233519272a^8 + 433381777/116759636a^7 - 540108843/116759636a^6 + 902681049/233519272a^5 - 2479433/813656a^4 + 843840175/233519272a^3 + 10003157/12290488a^2 - 119153379/24580976a + 156164307/233519272, 241967/233519272a^17 + 806195/233519272a^16 + 5506241/467038544a^15 + 15772131/467038544a^14 - 73747/12290488a^13 + 68737175/467038544a^12 - 12668812/29189909a^11 + 101969225/116759636a^10 - 59687515/29189909a^9 + 480916159/116759636a^8 - 3200675627/467038544a^7 + 4766059093/467038544a^6 - 1856020685/116759636a^5 + 33050197/1627312a^4 - 9275212949/467038544a^3 + 1536071223/66719792a^2 - 676309607/29189909a + 231936451/24580976, 96025/467038544a^17 - 110119/233519272a^16 + 332533/116759636a^15 - 3944467/467038544a^14 + 5121607/233519272a^13 - 1959695/33359896a^12 + 956739/8339974a^11 - 28942143/116759636a^10 + 166434083/467038544a^9 - 85885379/116759636a^8 + 6747469/8339974a^7 - 651809147/467038544a^6 + 118631051/66719792a^5 - 7415903/2847796a^4 + 716431165/233519272a^3 - 1943269585/467038544a^2 + 160110778/29189909a - 12286475/4169987, 89771/233519272a^17 + 4657/116759636a^16 + 7527/12290488a^15 - 4426545/467038544a^14 - 9538435/233519272a^13 - 376133/6145244a^12 - 60315931/233519272a^11 + 26980671/233519272a^10 - 88223955/116759636a^9 + 308182253/233519272a^8 - 297791413/116759636a^7 + 1628225371/467038544a^6 - 680144973/116759636a^5 + 6476667/813656a^4 - 865991179/116759636a^3 + 3577548351/467038544a^2 - 2519586759/233519272a + 27558827/16679948 (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:	$ 1082911.90412 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1082911.90412 \cdot 1}{4\cdot\sqrt{258151783382020583032356864}}\cr\approx \mathstrut & 0.257166495662 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301)

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 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301, 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 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301);

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 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301);

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

$C_3\times D_9$ (as 18T19):

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 54

The 18 conjugacy class representatives for $C_3\times D_9$

Character table for $C_3\times D_9$

Intermediate fields

$\Q(\sqrt{-1}) $, 3.1.324.1 x3, 6.0.419904.2

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 27 sibling:	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	${\href{/padicField/5.9.0.1}{9} }^{2}$	${\href{/padicField/7.6.0.1}{6} }^{3}$	${\href{/padicField/11.6.0.1}{6} }^{3}$	${\href{/padicField/13.9.0.1}{9} }^{2}$	${\href{/padicField/17.9.0.1}{9} }^{2}$	${\href{/padicField/19.2.0.1}{2} }^{9}$	${\href{/padicField/23.6.0.1}{6} }^{3}$	${\href{/padicField/29.9.0.1}{9} }^{2}$	${\href{/padicField/31.6.0.1}{6} }^{3}$	${\href{/padicField/37.9.0.1}{9} }^{2}$	${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{9}$	${\href{/padicField/43.6.0.1}{6} }^{3}$	${\href{/padicField/47.6.0.1}{6} }^{3}$	${\href{/padicField/53.3.0.1}{3} }^{6}$	${\href{/padicField/59.6.0.1}{6} }^{3}$

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
$2$ 2	2.6.6.3	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
	2.6.6.3	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
	2.6.6.3	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$3$ 3		Deg $18$	$9$	$2$	$44$

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