LMFDB - Number field 18.0.60436082803655481715851264.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256)

gp: K = bnfinit(y^18 + 18*y^16 - 24*y^15 + 162*y^14 - 288*y^13 + 1068*y^12 - 1872*y^11 + 4473*y^10 - 7616*y^9 + 13050*y^8 - 18360*y^7 + 23988*y^6 - 26784*y^5 + 24408*y^4 - 16992*y^3 + 8352*y^2 - 2304*y + 256, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 18*x^16 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 18*x^16 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256)

$ x^{18} + 18 x^{16} - 24 x^{15} + 162 x^{14} - 288 x^{13} + 1068 x^{12} - 1872 x^{11} + 4473 x^{10} + \cdots + 256 $ x^18 + 18*x^16 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256

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:	$-60436082803655481715851264$ -60436082803655481715851264 $\medspace = -\,2^{27}\cdot 3^{37}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$27.06$	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:	$2^{3/2}3^{37/18}\approx 27.05790946495165$
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{-6}) $
$\card{ \Gal(K/\Q) }$:	$18$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}-\frac{1}{16}a^{8}+\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{13}+\frac{1}{32}a^{11}-\frac{1}{32}a^{9}+\frac{3}{32}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}+\frac{1}{32}a^{10}+\frac{1}{32}a^{8}-\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{1088}a^{15}-\frac{1}{1088}a^{14}+\frac{11}{1088}a^{13}+\frac{31}{1088}a^{12}+\frac{53}{1088}a^{11}-\frac{19}{1088}a^{10}-\frac{55}{1088}a^{9}-\frac{19}{1088}a^{8}+\frac{45}{544}a^{7}-\frac{1}{16}a^{6}-\frac{15}{272}a^{5}+\frac{11}{272}a^{4}-\frac{41}{136}a^{3}+\frac{7}{17}a^{2}-\frac{8}{17}a-\frac{3}{17}$, $\frac{1}{2176}a^{16}-\frac{3}{272}a^{14}+\frac{1}{272}a^{13}-\frac{9}{1088}a^{12}+\frac{7}{544}a^{10}+\frac{3}{68}a^{9}+\frac{37}{2176}a^{8}-\frac{5}{136}a^{7}-\frac{49}{544}a^{6}+\frac{15}{272}a^{5}+\frac{31}{544}a^{4}+\frac{67}{136}a^{3}-\frac{21}{136}a^{2}-\frac{11}{34}a+\frac{7}{17}$, $\frac{1}{10569569224448}a^{17}-\frac{758859609}{5284784612224}a^{16}+\frac{514208499}{2642392306112}a^{15}-\frac{14106836161}{1321196153056}a^{14}+\frac{21416486097}{5284784612224}a^{13}-\frac{5081856671}{203260946624}a^{12}-\frac{23642997}{685623328}a^{11}+\frac{81277697559}{1321196153056}a^{10}-\frac{439851939999}{10569569224448}a^{9}-\frac{88589790557}{5284784612224}a^{8}-\frac{193988823677}{2642392306112}a^{7}+\frac{12464565213}{660598076528}a^{6}-\frac{33790372685}{155434841536}a^{5}-\frac{221685412901}{1321196153056}a^{4}+\frac{113746523867}{660598076528}a^{3}-\frac{1780590869}{25407618328}a^{2}+\frac{11987875502}{41287379783}a-\frac{16729895994}{41287379783}$ 1, a, a^2, a^3, 1/2*a^4 - 1/2*a^2, 1/2*a^5 - 1/2*a^3, 1/4*a^6 - 1/4*a^5 - 1/4*a^4 + 1/4*a^3 - 1/2*a^2 - 1/2*a, 1/4*a^7 + 1/4*a^3 - 1/2*a, 1/8*a^8 - 1/4*a^5 + 1/8*a^4 - 1/4*a^3 - 1/4*a^2 - 1/2*a, 1/8*a^9 - 1/8*a^5 - 1/2*a^2 - 1/2*a, 1/8*a^10 - 1/8*a^6 - 1/2*a^3 - 1/2*a^2, 1/8*a^11 - 1/8*a^7 - 1/2*a^3 - 1/2*a^2, 1/16*a^12 - 1/16*a^10 - 1/16*a^8 + 1/16*a^6 - 1/4*a^5 - 1/4*a^4 - 1/4*a^3 + 1/4*a^2 - 1/2*a, 1/32*a^13 + 1/32*a^11 - 1/32*a^9 + 3/32*a^7 - 1/4*a^5 + 1/8*a^3 - 1/2*a^2 - 1/2*a, 1/32*a^14 - 1/32*a^12 + 1/32*a^10 + 1/32*a^8 - 1/16*a^6 - 1/4*a^5 - 1/4*a^3 - 1/2*a, 1/1088*a^15 - 1/1088*a^14 + 11/1088*a^13 + 31/1088*a^12 + 53/1088*a^11 - 19/1088*a^10 - 55/1088*a^9 - 19/1088*a^8 + 45/544*a^7 - 1/16*a^6 - 15/272*a^5 + 11/272*a^4 - 41/136*a^3 + 7/17*a^2 - 8/17*a - 3/17, 1/2176*a^16 - 3/272*a^14 + 1/272*a^13 - 9/1088*a^12 + 7/544*a^10 + 3/68*a^9 + 37/2176*a^8 - 5/136*a^7 - 49/544*a^6 + 15/272*a^5 + 31/544*a^4 + 67/136*a^3 - 21/136*a^2 - 11/34*a + 7/17, 1/10569569224448*a^17 - 758859609/5284784612224*a^16 + 514208499/2642392306112*a^15 - 14106836161/1321196153056*a^14 + 21416486097/5284784612224*a^13 - 5081856671/203260946624*a^12 - 23642997/685623328*a^11 + 81277697559/1321196153056*a^10 - 439851939999/10569569224448*a^9 - 88589790557/5284784612224*a^8 - 193988823677/2642392306112*a^7 + 12464565213/660598076528*a^6 - 33790372685/155434841536*a^5 - 221685412901/1321196153056*a^4 + 113746523867/660598076528*a^3 - 1780590869/25407618328*a^2 + 11987875502/41287379783*a - 16729895994/41287379783

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

$C_{2}$, which has order $2$ (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:	$ -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:	$\frac{32372462639}{5284784612224}a^{17}+\frac{25896992153}{5284784612224}a^{16}+\frac{17296272443}{155434841536}a^{15}-\frac{155591228561}{2642392306112}a^{14}+\frac{73710806387}{82574759566}a^{13}-\frac{101917723439}{101630473312}a^{12}+\frac{7252559365}{1371246656}a^{11}-\frac{17536243625297}{2642392306112}a^{10}+\frac{102450940149245}{5284784612224}a^{9}-\frac{144986693718689}{5284784612224}a^{8}+\frac{63372737458105}{1321196153056}a^{7}-\frac{79762030424379}{1321196153056}a^{6}+\frac{98778459944797}{1321196153056}a^{5}-\frac{100981130466491}{1321196153056}a^{4}+\frac{9048071859447}{165149519132}a^{3}-\frac{790782759761}{25407618328}a^{2}+\frac{587744366461}{82574759566}a-\frac{17874229220}{41287379783}$, $\frac{12014002345}{5284784612224}a^{17}-\frac{8022814697}{5284784612224}a^{16}+\frac{26030646307}{660598076528}a^{15}-\frac{13319515531}{165149519132}a^{14}+\frac{1012929070481}{2642392306112}a^{13}-\frac{170910721855}{203260946624}a^{12}+\frac{920899193}{342811664}a^{11}-\frac{7072730357453}{1321196153056}a^{10}+\frac{62341753215649}{5284784612224}a^{9}-\frac{111659922122341}{5284784612224}a^{8}+\frac{12075351716381}{330299038264}a^{7}-\frac{67581630513379}{1321196153056}a^{6}+\frac{89788951225433}{1321196153056}a^{5}-\frac{99928248064831}{1321196153056}a^{4}+\frac{23348524063801}{330299038264}a^{3}-\frac{1129244489815}{25407618328}a^{2}+\frac{1401946175861}{82574759566}a-\frac{112924202982}{41287379783}$, $\frac{5256138709}{101630473312}a^{17}+\frac{1077697595}{50815236656}a^{16}+\frac{89154585867}{101630473312}a^{15}-\frac{93384898125}{101630473312}a^{14}+\frac{43910415103}{6351904582}a^{13}-\frac{1136120875375}{101630473312}a^{12}+\frac{554711871}{13185064}a^{11}-\frac{6861519496773}{101630473312}a^{10}+\frac{15462340099975}{101630473312}a^{9}-\frac{26077316670827}{101630473312}a^{8}+\frac{38952260869877}{101630473312}a^{7}-\frac{25318588898055}{50815236656}a^{6}+\frac{14570514317089}{25407618328}a^{5}-\frac{835098882353}{1494565784}a^{4}+\frac{8408642560611}{25407618328}a^{3}-\frac{43012378315}{747282892}a^{2}-\frac{74315646147}{3175952291}a+\frac{22658926947}{3175952291}$, $\frac{25218021013}{2642392306112}a^{17}-\frac{138447729}{330299038264}a^{16}+\frac{223062593917}{1321196153056}a^{15}-\frac{158250046109}{660598076528}a^{14}+\frac{495477096229}{330299038264}a^{13}-\frac{71212555945}{25407618328}a^{12}+\frac{6752511143}{685623328}a^{11}-\frac{5926348952649}{330299038264}a^{10}+\frac{107626488133607}{2642392306112}a^{9}-\frac{5927165978105}{82574759566}a^{8}+\frac{77823917199733}{660598076528}a^{7}-\frac{6532317197405}{38858710384}a^{6}+\frac{140346134182063}{660598076528}a^{5}-\frac{39088572471799}{165149519132}a^{4}+\frac{34070833534289}{165149519132}a^{3}-\frac{1709457761533}{12703809164}a^{2}+\frac{286112227483}{4857338798}a-\frac{390986381169}{41287379783}$, $\frac{48017533}{50815236656}a^{17}+\frac{44348785}{25407618328}a^{16}+\frac{3616742243}{203260946624}a^{15}+\frac{1373922627}{203260946624}a^{14}+\frac{1527864219}{11956526272}a^{13}-\frac{9081499413}{203260946624}a^{12}+\frac{75296887}{105480512}a^{11}-\frac{99079400007}{203260946624}a^{10}+\frac{517790877757}{203260946624}a^{9}-\frac{609921374343}{203260946624}a^{8}+\frac{322160633489}{50815236656}a^{7}-\frac{394845178529}{50815236656}a^{6}+\frac{575651247363}{50815236656}a^{5}-\frac{600273699665}{50815236656}a^{4}+\frac{122563386267}{12703809164}a^{3}-\frac{37691284385}{6351904582}a^{2}+\frac{17158449935}{6351904582}a-\frac{800193243}{3175952291}$, $\frac{996938485}{406521893248}a^{17}-\frac{1189441041}{50815236656}a^{16}+\frac{678157603}{101630473312}a^{15}-\frac{50718152037}{101630473312}a^{14}+\frac{64939083833}{203260946624}a^{13}-\frac{391375342869}{101630473312}a^{12}+\frac{1713553}{387796}a^{11}-\frac{2154295415243}{101630473312}a^{10}+\frac{9957471677805}{406521893248}a^{9}-\frac{7312468630761}{101630473312}a^{8}+\frac{9907900512117}{101630473312}a^{7}-\frac{465765143437}{3175952291}a^{6}+\frac{19508046090095}{101630473312}a^{5}-\frac{2644647816827}{12703809164}a^{4}+\frac{4477077179647}{25407618328}a^{3}-\frac{1325225395343}{12703809164}a^{2}+\frac{200748149173}{6351904582}a-\frac{10703075375}{3175952291}$, $\frac{241351327}{5284784612224}a^{17}-\frac{2480109273}{5284784612224}a^{16}-\frac{781281617}{330299038264}a^{15}-\frac{10986360759}{1321196153056}a^{14}-\frac{101961768803}{2642392306112}a^{13}+\frac{612547135}{203260946624}a^{12}-\frac{226596297}{685623328}a^{11}+\frac{136960255615}{330299038264}a^{10}-\frac{11214895204133}{5284784612224}a^{9}+\frac{17665425891983}{5284784612224}a^{8}-\frac{627394032617}{77717420768}a^{7}+\frac{18113871970947}{1321196153056}a^{6}-\frac{26779947951641}{1321196153056}a^{5}+\frac{36037145239413}{1321196153056}a^{4}-\frac{283941556445}{9714677596}a^{3}+\frac{687165025283}{25407618328}a^{2}-\frac{1053576479919}{82574759566}a+\frac{79318038900}{41287379783}$, $\frac{3180922631}{2642392306112}a^{17}+\frac{14395787875}{5284784612224}a^{16}+\frac{30594610807}{1321196153056}a^{15}+\frac{1440244397}{77717420768}a^{14}+\frac{100719322245}{660598076528}a^{13}+\frac{5743984743}{203260946624}a^{12}+\frac{490448609}{685623328}a^{11}+\frac{22307479191}{660598076528}a^{10}+\frac{4822439650861}{2642392306112}a^{9}-\frac{3523846894781}{5284784612224}a^{8}+\frac{1020523097727}{660598076528}a^{7}-\frac{73228704307}{1321196153056}a^{6}-\frac{14651713901}{19429355192}a^{5}+\frac{5325927808725}{1321196153056}a^{4}-\frac{2324268162815}{330299038264}a^{3}+\frac{167878926759}{25407618328}a^{2}-\frac{220616187191}{82574759566}a+\frac{15229355134}{41287379783}$ 32372462639/5284784612224a^17 + 25896992153/5284784612224a^16 + 17296272443/155434841536a^15 - 155591228561/2642392306112a^14 + 73710806387/82574759566a^13 - 101917723439/101630473312a^12 + 7252559365/1371246656a^11 - 17536243625297/2642392306112a^10 + 102450940149245/5284784612224a^9 - 144986693718689/5284784612224a^8 + 63372737458105/1321196153056a^7 - 79762030424379/1321196153056a^6 + 98778459944797/1321196153056a^5 - 100981130466491/1321196153056a^4 + 9048071859447/165149519132a^3 - 790782759761/25407618328a^2 + 587744366461/82574759566a - 17874229220/41287379783, 12014002345/5284784612224a^17 - 8022814697/5284784612224a^16 + 26030646307/660598076528a^15 - 13319515531/165149519132a^14 + 1012929070481/2642392306112a^13 - 170910721855/203260946624a^12 + 920899193/342811664a^11 - 7072730357453/1321196153056a^10 + 62341753215649/5284784612224a^9 - 111659922122341/5284784612224a^8 + 12075351716381/330299038264a^7 - 67581630513379/1321196153056a^6 + 89788951225433/1321196153056a^5 - 99928248064831/1321196153056a^4 + 23348524063801/330299038264a^3 - 1129244489815/25407618328a^2 + 1401946175861/82574759566a - 112924202982/41287379783, 5256138709/101630473312a^17 + 1077697595/50815236656a^16 + 89154585867/101630473312a^15 - 93384898125/101630473312a^14 + 43910415103/6351904582a^13 - 1136120875375/101630473312a^12 + 554711871/13185064a^11 - 6861519496773/101630473312a^10 + 15462340099975/101630473312a^9 - 26077316670827/101630473312a^8 + 38952260869877/101630473312a^7 - 25318588898055/50815236656a^6 + 14570514317089/25407618328a^5 - 835098882353/1494565784a^4 + 8408642560611/25407618328a^3 - 43012378315/747282892a^2 - 74315646147/3175952291a + 22658926947/3175952291, 25218021013/2642392306112a^17 - 138447729/330299038264a^16 + 223062593917/1321196153056a^15 - 158250046109/660598076528a^14 + 495477096229/330299038264a^13 - 71212555945/25407618328a^12 + 6752511143/685623328a^11 - 5926348952649/330299038264a^10 + 107626488133607/2642392306112a^9 - 5927165978105/82574759566a^8 + 77823917199733/660598076528a^7 - 6532317197405/38858710384a^6 + 140346134182063/660598076528a^5 - 39088572471799/165149519132a^4 + 34070833534289/165149519132a^3 - 1709457761533/12703809164a^2 + 286112227483/4857338798a - 390986381169/41287379783, 48017533/50815236656a^17 + 44348785/25407618328a^16 + 3616742243/203260946624a^15 + 1373922627/203260946624a^14 + 1527864219/11956526272a^13 - 9081499413/203260946624a^12 + 75296887/105480512a^11 - 99079400007/203260946624a^10 + 517790877757/203260946624a^9 - 609921374343/203260946624a^8 + 322160633489/50815236656a^7 - 394845178529/50815236656a^6 + 575651247363/50815236656a^5 - 600273699665/50815236656a^4 + 122563386267/12703809164a^3 - 37691284385/6351904582a^2 + 17158449935/6351904582a - 800193243/3175952291, 996938485/406521893248a^17 - 1189441041/50815236656a^16 + 678157603/101630473312a^15 - 50718152037/101630473312a^14 + 64939083833/203260946624a^13 - 391375342869/101630473312a^12 + 1713553/387796a^11 - 2154295415243/101630473312a^10 + 9957471677805/406521893248a^9 - 7312468630761/101630473312a^8 + 9907900512117/101630473312a^7 - 465765143437/3175952291a^6 + 19508046090095/101630473312a^5 - 2644647816827/12703809164a^4 + 4477077179647/25407618328a^3 - 1325225395343/12703809164a^2 + 200748149173/6351904582a - 10703075375/3175952291, 241351327/5284784612224a^17 - 2480109273/5284784612224a^16 - 781281617/330299038264a^15 - 10986360759/1321196153056a^14 - 101961768803/2642392306112a^13 + 612547135/203260946624a^12 - 226596297/685623328a^11 + 136960255615/330299038264a^10 - 11214895204133/5284784612224a^9 + 17665425891983/5284784612224a^8 - 627394032617/77717420768a^7 + 18113871970947/1321196153056a^6 - 26779947951641/1321196153056a^5 + 36037145239413/1321196153056a^4 - 283941556445/9714677596a^3 + 687165025283/25407618328a^2 - 1053576479919/82574759566a + 79318038900/41287379783, 3180922631/2642392306112a^17 + 14395787875/5284784612224a^16 + 30594610807/1321196153056a^15 + 1440244397/77717420768a^14 + 100719322245/660598076528a^13 + 5743984743/203260946624a^12 + 490448609/685623328a^11 + 22307479191/660598076528a^10 + 4822439650861/2642392306112a^9 - 3523846894781/5284784612224a^8 + 1020523097727/660598076528a^7 - 73228704307/1321196153056a^6 - 14651713901/19429355192a^5 + 5325927808725/1321196153056a^4 - 2324268162815/330299038264a^3 + 167878926759/25407618328a^2 - 220616187191/82574759566a + 15229355134/41287379783 (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:	$ 46631737.4813 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 46631737.4813 \cdot 2}{2\cdot\sqrt{60436082803655481715851264}}\cr\approx \mathstrut & 91.5487162723 \end{aligned}\] (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 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256)

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 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256, 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 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256);

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 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256);

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

$D_9$ (as 18T5):

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 18

The 6 conjugacy class representatives for $D_9$

Character table for $D_9$

Intermediate fields

$\Q(\sqrt{-6}) $, 3.1.216.1 x3, 6.0.1119744.1, 9.1.1586874322944.5 x9

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 9 sibling:	9.1.1586874322944.5
Minimal sibling:	9.1.1586874322944.5

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	${\href{/padicField/5.9.0.1}{9} }^{2}$	${\href{/padicField/7.9.0.1}{9} }^{2}$	${\href{/padicField/11.9.0.1}{9} }^{2}$	${\href{/padicField/13.2.0.1}{2} }^{9}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	${\href{/padicField/19.2.0.1}{2} }^{9}$	${\href{/padicField/23.2.0.1}{2} }^{9}$	${\href{/padicField/29.3.0.1}{3} }^{6}$	${\href{/padicField/31.9.0.1}{9} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{9}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.2.0.1}{2} }^{9}$	${\href{/padicField/47.2.0.1}{2} }^{9}$	${\href{/padicField/53.9.0.1}{9} }^{2}$	${\href{/padicField/59.3.0.1}{3} }^{6}$

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
$2$ 2	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.2	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
$3$ 3		Deg $18$	$18$	$1$	$37$

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