LMFDB - Number field 18.0.7467330231571086237938751.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*x^2 - 9*x + 1)

gp: K = bnfinit(y^18 + 15*y^16 - 9*y^15 + 81*y^14 - 36*y^13 + 64*y^12 - 162*y^11 + 780*y^10 - 918*y^9 + 1782*y^8 - 1674*y^7 + 1792*y^6 - 1188*y^5 + 669*y^4 - 252*y^3 + 63*y^2 - 9*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*x^2 - 9*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*x^2 - 9*x + 1)

$ x^{18} + 15 x^{16} - 9 x^{15} + 81 x^{14} - 36 x^{13} + 64 x^{12} - 162 x^{11} + 780 x^{10} - 918 x^{9} + \cdots + 1 $ x^18 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*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:	$[0, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-7467330231571086237938751$ -7467330231571086237938751 $\medspace = -\,3^{24}\cdot 31^{9}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$24.09$	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:	$3^{4/3}31^{1/2}\approx 24.090317279593503$
Ramified primes:	$3$, $31$ 3, 31	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-31}) $
$\card{ \Gal(K/\Q) }$:	$18$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{28}a^{12}+\frac{1}{14}a^{11}-\frac{5}{28}a^{10}-\frac{1}{28}a^{9}-\frac{3}{28}a^{8}+\frac{1}{14}a^{7}-\frac{2}{7}a^{6}+\frac{5}{28}a^{5}+\frac{1}{7}a^{4}-\frac{1}{28}a^{3}+\frac{5}{28}a^{2}-\frac{13}{28}a+\frac{2}{7}$, $\frac{1}{140}a^{13}-\frac{1}{140}a^{12}-\frac{1}{35}a^{11}+\frac{1}{5}a^{10}-\frac{1}{20}a^{9}+\frac{8}{35}a^{8}-\frac{1}{20}a^{7}-\frac{41}{140}a^{6}-\frac{11}{140}a^{5}-\frac{17}{70}a^{4}+\frac{16}{35}a^{3}+\frac{7}{20}a^{2}-\frac{29}{70}a-\frac{31}{140}$, $\frac{1}{140}a^{14}-\frac{1}{140}a^{11}-\frac{1}{35}a^{10}-\frac{3}{28}a^{9}-\frac{5}{28}a^{8}-\frac{3}{140}a^{7}-\frac{11}{70}a^{6}-\frac{1}{7}a^{5}+\frac{3}{28}a^{4}+\frac{19}{70}a^{3}-\frac{19}{140}a^{2}+\frac{3}{20}a-\frac{13}{70}$, $\frac{1}{280}a^{15}-\frac{1}{280}a^{14}+\frac{1}{70}a^{12}-\frac{1}{10}a^{11}+\frac{17}{140}a^{10}-\frac{5}{28}a^{9}-\frac{1}{10}a^{8}-\frac{11}{70}a^{7}-\frac{27}{70}a^{6}-\frac{2}{7}a^{5}-\frac{33}{70}a^{4}+\frac{39}{140}a^{3}+\frac{3}{28}a^{2}-\frac{11}{40}a-\frac{39}{280}$, $\frac{1}{120680}a^{16}-\frac{113}{120680}a^{15}+\frac{1}{12068}a^{14}-\frac{117}{60340}a^{13}+\frac{32}{2155}a^{12}+\frac{1368}{15085}a^{11}+\frac{617}{15085}a^{10}-\frac{10631}{60340}a^{9}-\frac{24}{2155}a^{8}+\frac{3474}{15085}a^{7}+\frac{5849}{60340}a^{6}+\frac{2199}{30170}a^{5}+\frac{13371}{30170}a^{4}+\frac{1989}{30170}a^{3}+\frac{33669}{120680}a^{2}-\frac{16503}{120680}a-\frac{11771}{60340}$, $\frac{1}{9318306200}a^{17}+\frac{24121}{9318306200}a^{16}-\frac{9258169}{9318306200}a^{15}+\frac{23382927}{9318306200}a^{14}+\frac{2197987}{2329576550}a^{13}+\frac{7588801}{4659153100}a^{12}-\frac{383874917}{4659153100}a^{11}+\frac{110525857}{4659153100}a^{10}-\frac{578881193}{4659153100}a^{9}-\frac{174015448}{1164788275}a^{8}-\frac{283947063}{2329576550}a^{7}+\frac{365394488}{1164788275}a^{6}-\frac{98363227}{4659153100}a^{5}-\frac{1076187701}{4659153100}a^{4}-\frac{2382409763}{9318306200}a^{3}-\frac{386272259}{1863661240}a^{2}-\frac{2615983827}{9318306200}a-\frac{565617651}{9318306200}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/2*a^7 - 1/2, 1/2*a^8 - 1/2*a, 1/2*a^9 - 1/2*a^2, 1/2*a^10 - 1/2*a^3, 1/4*a^11 - 1/4*a^9 - 1/4*a^8 - 1/4*a^7 - 1/2*a^6 + 1/4*a^4 - 1/2*a^3 - 1/4*a^2 + 1/4*a + 1/4, 1/28*a^12 + 1/14*a^11 - 5/28*a^10 - 1/28*a^9 - 3/28*a^8 + 1/14*a^7 - 2/7*a^6 + 5/28*a^5 + 1/7*a^4 - 1/28*a^3 + 5/28*a^2 - 13/28*a + 2/7, 1/140*a^13 - 1/140*a^12 - 1/35*a^11 + 1/5*a^10 - 1/20*a^9 + 8/35*a^8 - 1/20*a^7 - 41/140*a^6 - 11/140*a^5 - 17/70*a^4 + 16/35*a^3 + 7/20*a^2 - 29/70*a - 31/140, 1/140*a^14 - 1/140*a^11 - 1/35*a^10 - 3/28*a^9 - 5/28*a^8 - 3/140*a^7 - 11/70*a^6 - 1/7*a^5 + 3/28*a^4 + 19/70*a^3 - 19/140*a^2 + 3/20*a - 13/70, 1/280*a^15 - 1/280*a^14 + 1/70*a^12 - 1/10*a^11 + 17/140*a^10 - 5/28*a^9 - 1/10*a^8 - 11/70*a^7 - 27/70*a^6 - 2/7*a^5 - 33/70*a^4 + 39/140*a^3 + 3/28*a^2 - 11/40*a - 39/280, 1/120680*a^16 - 113/120680*a^15 + 1/12068*a^14 - 117/60340*a^13 + 32/2155*a^12 + 1368/15085*a^11 + 617/15085*a^10 - 10631/60340*a^9 - 24/2155*a^8 + 3474/15085*a^7 + 5849/60340*a^6 + 2199/30170*a^5 + 13371/30170*a^4 + 1989/30170*a^3 + 33669/120680*a^2 - 16503/120680*a - 11771/60340, 1/9318306200*a^17 + 24121/9318306200*a^16 - 9258169/9318306200*a^15 + 23382927/9318306200*a^14 + 2197987/2329576550*a^13 + 7588801/4659153100*a^12 - 383874917/4659153100*a^11 + 110525857/4659153100*a^10 - 578881193/4659153100*a^9 - 174015448/1164788275*a^8 - 283947063/2329576550*a^7 + 365394488/1164788275*a^6 - 98363227/4659153100*a^5 - 1076187701/4659153100*a^4 - 2382409763/9318306200*a^3 - 386272259/1863661240*a^2 - 2615983827/9318306200*a - 565617651/9318306200

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

$C_{3}$, which has order $3$ (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:	$\frac{246213}{603400}a^{17}+\frac{92133}{603400}a^{16}+\frac{461381}{75425}a^{15}-\frac{423567}{301700}a^{14}+\frac{1362861}{43100}a^{13}-\frac{112591}{43100}a^{12}+\frac{3089907}{150850}a^{11}-\frac{17458479}{301700}a^{10}+\frac{88645961}{301700}a^{9}-\frac{38587923}{150850}a^{8}+\frac{44308248}{75425}a^{7}-\frac{127988109}{301700}a^{6}+\frac{146685519}{301700}a^{5}-\frac{35763879}{150850}a^{4}+\frac{68076441}{603400}a^{3}-\frac{591795}{24136}a^{2}-\frac{37269}{150850}a+\frac{33441}{301700}$, $\frac{45291159}{2329576550}a^{17}-\frac{118430223}{1164788275}a^{16}+\frac{1235718543}{4659153100}a^{15}-\frac{7985167209}{4659153100}a^{14}+\frac{4849260027}{2329576550}a^{13}-\frac{20756200357}{2329576550}a^{12}+\frac{12621203613}{4659153100}a^{11}-\frac{22799091279}{2329576550}a^{10}+\frac{19660407521}{665593300}a^{9}-\frac{62360922591}{665593300}a^{8}+\frac{508734475239}{4659153100}a^{7}-\frac{230928488302}{1164788275}a^{6}+\frac{377552155149}{2329576550}a^{5}-\frac{834323810031}{4659153100}a^{4}+\frac{217989458213}{2329576550}a^{3}-\frac{49777350309}{931830620}a^{2}+\frac{30916349637}{2329576550}a-\frac{5122375359}{2329576550}$, $\frac{12685527}{9318306200}a^{17}+\frac{10104859}{1164788275}a^{16}+\frac{225359847}{9318306200}a^{15}+\frac{533410407}{4659153100}a^{14}+\frac{384923193}{4659153100}a^{13}+\frac{377986311}{665593300}a^{12}+\frac{119565801}{4659153100}a^{11}-\frac{78410163}{2329576550}a^{10}-\frac{1740771351}{4659153100}a^{9}+\frac{11217854733}{2329576550}a^{8}-\frac{4645846161}{2329576550}a^{7}+\frac{37487972259}{4659153100}a^{6}-\frac{31504339719}{4659153100}a^{5}+\frac{25490234463}{4659153100}a^{4}-\frac{23350966551}{9318306200}a^{3}+\frac{655158393}{931830620}a^{2}-\frac{978456039}{9318306200}a+\frac{3100484059}{4659153100}$, $\frac{2498649}{1331186600}a^{17}-\frac{184017447}{9318306200}a^{16}+\frac{84530079}{4659153100}a^{15}-\frac{1402816197}{4659153100}a^{14}+\frac{127001871}{665593300}a^{13}-\frac{6527125197}{4659153100}a^{12}+\frac{12116583}{166398325}a^{11}-\frac{1864237869}{4659153100}a^{10}+\frac{20625216021}{4659153100}a^{9}-\frac{17988600344}{1164788275}a^{8}+\frac{28000140561}{2329576550}a^{7}-\frac{101260009899}{4659153100}a^{6}+\frac{77626772499}{4659153100}a^{5}-\frac{13892236182}{1164788275}a^{4}+\frac{48217615371}{9318306200}a^{3}-\frac{2622191379}{1863661240}a^{2}+\frac{969570297}{4659153100}a-\frac{8871376259}{4659153100}$, $\frac{203415207}{1331186600}a^{17}-\frac{37413489}{2329576550}a^{16}+\frac{20994481279}{9318306200}a^{15}-\frac{7675926981}{4659153100}a^{14}+\frac{7945334523}{665593300}a^{13}-\frac{16078963183}{2329576550}a^{12}+\frac{34881594957}{4659153100}a^{11}-\frac{124247605847}{4659153100}a^{10}+\frac{140068072987}{1164788275}a^{9}-\frac{685642194403}{4659153100}a^{8}+\frac{303673166099}{1164788275}a^{7}-\frac{181539007471}{665593300}a^{6}+\frac{603461325941}{2329576550}a^{5}-\frac{871484466659}{4659153100}a^{4}+\frac{785977468973}{9318306200}a^{3}-\frac{7393171297}{232957655}a^{2}+\frac{40940057427}{9318306200}a-\frac{1071301237}{4659153100}$, $\frac{4320392611}{4659153100}a^{17}+\frac{3139532117}{9318306200}a^{16}+\frac{65389370621}{4659153100}a^{15}-\frac{30886859441}{9318306200}a^{14}+\frac{172121617069}{2329576550}a^{13}-\frac{35242509943}{4659153100}a^{12}+\frac{131743146893}{2329576550}a^{11}-\frac{626133741551}{4659153100}a^{10}+\frac{3133753407209}{4659153100}a^{9}-\frac{708220347511}{1164788275}a^{8}+\frac{6727446278233}{4659153100}a^{7}-\frac{1250399616724}{1164788275}a^{6}+\frac{5995641347251}{4659153100}a^{5}-\frac{1615880268371}{2329576550}a^{4}+\frac{927668442331}{2329576550}a^{3}-\frac{206547665979}{1863661240}a^{2}+\frac{82490257293}{4659153100}a-\frac{19573116237}{9318306200}$, $\frac{1045364583}{9318306200}a^{17}+\frac{554069513}{9318306200}a^{16}+\frac{3908962747}{2329576550}a^{15}-\frac{127523168}{1164788275}a^{14}+\frac{5651243781}{665593300}a^{13}+\frac{2183821419}{2329576550}a^{12}+\frac{22005631399}{4659153100}a^{11}-\frac{15995476836}{1164788275}a^{10}+\frac{363466665281}{4659153100}a^{9}-\frac{130854795773}{2329576550}a^{8}+\frac{653876608997}{4659153100}a^{7}-\frac{339510570159}{4659153100}a^{6}+\frac{221251238627}{2329576550}a^{5}-\frac{82666286903}{4659153100}a^{4}-\frac{18846965919}{9318306200}a^{3}+\frac{3019797463}{266237320}a^{2}-\frac{6690779497}{1164788275}a+\frac{3758518391}{4659153100}$, $\frac{4451912589}{4659153100}a^{17}+\frac{560507599}{4659153100}a^{16}+\frac{9570309137}{665593300}a^{15}-\frac{31684381297}{4659153100}a^{14}+\frac{358623581607}{4659153100}a^{13}-\frac{29301886108}{1164788275}a^{12}+\frac{280207508989}{4659153100}a^{11}-\frac{694475551059}{4659153100}a^{10}+\frac{3385846491401}{4659153100}a^{9}-\frac{921224412939}{1164788275}a^{8}+\frac{3791889340171}{2329576550}a^{7}-\frac{6650863457109}{4659153100}a^{6}+\frac{524236059221}{332796650}a^{5}-\frac{4603387092183}{4659153100}a^{4}+\frac{1276445317029}{2329576550}a^{3}-\frac{9097354717}{46591531}a^{2}+\frac{179925372227}{4659153100}a-\frac{8936419729}{4659153100}$ 246213/603400a^17 + 92133/603400a^16 + 461381/75425a^15 - 423567/301700a^14 + 1362861/43100a^13 - 112591/43100a^12 + 3089907/150850a^11 - 17458479/301700a^10 + 88645961/301700a^9 - 38587923/150850a^8 + 44308248/75425a^7 - 127988109/301700a^6 + 146685519/301700a^5 - 35763879/150850a^4 + 68076441/603400a^3 - 591795/24136a^2 - 37269/150850a + 33441/301700, 45291159/2329576550a^17 - 118430223/1164788275a^16 + 1235718543/4659153100a^15 - 7985167209/4659153100a^14 + 4849260027/2329576550a^13 - 20756200357/2329576550a^12 + 12621203613/4659153100a^11 - 22799091279/2329576550a^10 + 19660407521/665593300a^9 - 62360922591/665593300a^8 + 508734475239/4659153100a^7 - 230928488302/1164788275a^6 + 377552155149/2329576550a^5 - 834323810031/4659153100a^4 + 217989458213/2329576550a^3 - 49777350309/931830620a^2 + 30916349637/2329576550a - 5122375359/2329576550, 12685527/9318306200a^17 + 10104859/1164788275a^16 + 225359847/9318306200a^15 + 533410407/4659153100a^14 + 384923193/4659153100a^13 + 377986311/665593300a^12 + 119565801/4659153100a^11 - 78410163/2329576550a^10 - 1740771351/4659153100a^9 + 11217854733/2329576550a^8 - 4645846161/2329576550a^7 + 37487972259/4659153100a^6 - 31504339719/4659153100a^5 + 25490234463/4659153100a^4 - 23350966551/9318306200a^3 + 655158393/931830620a^2 - 978456039/9318306200a + 3100484059/4659153100, 2498649/1331186600a^17 - 184017447/9318306200a^16 + 84530079/4659153100a^15 - 1402816197/4659153100a^14 + 127001871/665593300a^13 - 6527125197/4659153100a^12 + 12116583/166398325a^11 - 1864237869/4659153100a^10 + 20625216021/4659153100a^9 - 17988600344/1164788275a^8 + 28000140561/2329576550a^7 - 101260009899/4659153100a^6 + 77626772499/4659153100a^5 - 13892236182/1164788275a^4 + 48217615371/9318306200a^3 - 2622191379/1863661240a^2 + 969570297/4659153100a - 8871376259/4659153100, 203415207/1331186600a^17 - 37413489/2329576550a^16 + 20994481279/9318306200a^15 - 7675926981/4659153100a^14 + 7945334523/665593300a^13 - 16078963183/2329576550a^12 + 34881594957/4659153100a^11 - 124247605847/4659153100a^10 + 140068072987/1164788275a^9 - 685642194403/4659153100a^8 + 303673166099/1164788275a^7 - 181539007471/665593300a^6 + 603461325941/2329576550a^5 - 871484466659/4659153100a^4 + 785977468973/9318306200a^3 - 7393171297/232957655a^2 + 40940057427/9318306200a - 1071301237/4659153100, 4320392611/4659153100a^17 + 3139532117/9318306200a^16 + 65389370621/4659153100a^15 - 30886859441/9318306200a^14 + 172121617069/2329576550a^13 - 35242509943/4659153100a^12 + 131743146893/2329576550a^11 - 626133741551/4659153100a^10 + 3133753407209/4659153100a^9 - 708220347511/1164788275a^8 + 6727446278233/4659153100a^7 - 1250399616724/1164788275a^6 + 5995641347251/4659153100a^5 - 1615880268371/2329576550a^4 + 927668442331/2329576550a^3 - 206547665979/1863661240a^2 + 82490257293/4659153100a - 19573116237/9318306200, 1045364583/9318306200a^17 + 554069513/9318306200a^16 + 3908962747/2329576550a^15 - 127523168/1164788275a^14 + 5651243781/665593300a^13 + 2183821419/2329576550a^12 + 22005631399/4659153100a^11 - 15995476836/1164788275a^10 + 363466665281/4659153100a^9 - 130854795773/2329576550a^8 + 653876608997/4659153100a^7 - 339510570159/4659153100a^6 + 221251238627/2329576550a^5 - 82666286903/4659153100a^4 - 18846965919/9318306200a^3 + 3019797463/266237320a^2 - 6690779497/1164788275a + 3758518391/4659153100, 4451912589/4659153100a^17 + 560507599/4659153100a^16 + 9570309137/665593300a^15 - 31684381297/4659153100a^14 + 358623581607/4659153100a^13 - 29301886108/1164788275a^12 + 280207508989/4659153100a^11 - 694475551059/4659153100a^10 + 3385846491401/4659153100a^9 - 921224412939/1164788275a^8 + 3791889340171/2329576550a^7 - 6650863457109/4659153100a^6 + 524236059221/332796650a^5 - 4603387092183/4659153100a^4 + 1276445317029/2329576550a^3 - 9097354717/46591531a^2 + 179925372227/4659153100a - 8936419729/4659153100 (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:	$ 42855.4917555 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 42855.4917555 \cdot 3}{2\cdot\sqrt{7467330231571086237938751}}\cr\approx \mathstrut & 0.359032595169 \end{aligned}\] (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 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*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 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*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 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*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 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*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

$C_3\times S_3$ (as 18T3):

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 9 conjugacy class representatives for $S_3 \times C_3$

Character table for $S_3 \times C_3$

Intermediate fields

$\Q(\sqrt{-31}) $, 3.1.31.1 x3, $\Q(\zeta_{9})^+$, 6.0.29791.1, 6.0.195458751.2 x2, 6.0.195458751.1, 9.3.15832158831.4 x3

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 6 sibling:	6.0.195458751.2
Degree 9 sibling:	9.3.15832158831.4
Minimal sibling:	6.0.195458751.2

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.3.0.1}{3} }^{6}$	R	${\href{/padicField/5.3.0.1}{3} }^{6}$	${\href{/padicField/7.3.0.1}{3} }^{6}$	${\href{/padicField/11.6.0.1}{6} }^{3}$	${\href{/padicField/13.6.0.1}{6} }^{3}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.6.0.1}{6} }^{3}$	${\href{/padicField/29.6.0.1}{6} }^{3}$	R	${\href{/padicField/37.2.0.1}{2} }^{9}$	${\href{/padicField/41.3.0.1}{3} }^{6}$	${\href{/padicField/43.6.0.1}{6} }^{3}$	${\href{/padicField/47.3.0.1}{3} }^{6}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.3.0.1}{3} }^{6}$

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
$3$ 3	3.6.8.3	$x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$	$3$	$2$	$8$	$C_6$	$[2]^{2}$
	3.6.8.3	$x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$	$3$	$2$	$8$	$C_6$	$[2]^{2}$
	3.6.8.3	$x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$	$3$	$2$	$8$	$C_6$	$[2]^{2}$
$31$ 31	31.6.3.2	$x^{6} + 95 x^{4} + 56 x^{3} + 2884 x^{2} - 5152 x + 28684$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	31.6.3.2	$x^{6} + 95 x^{4} + 56 x^{3} + 2884 x^{2} - 5152 x + 28684$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	31.6.3.2	$x^{6} + 95 x^{4} + 56 x^{3} + 2884 x^{2} - 5152 x + 28684$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

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