LMFDB - Number field 18.0.8549394874383196572862347.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811)

gp: K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 156*y^15 + 411*y^14 - 861*y^13 + 1359*y^12 - 1407*y^11 + 264*y^10 + 2354*y^9 - 5535*y^8 + 7515*y^7 - 4059*y^6 - 4305*y^5 + 16131*y^4 - 20937*y^3 + 14379*y^2 - 5190*y + 811, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811)

$ x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 411 x^{14} - 861 x^{13} + 1359 x^{12} - 1407 x^{11} + \cdots + 811 $ x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811

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:	$-8549394874383196572862347$ -8549394874383196572862347 $\medspace = -\,3^{31}\cdot 7^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$24.27$	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:	$3^{31/18}7^{2/3}\approx 24.27210940016998$
Ramified primes:	$3$, $7$ 3, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-3}) $
$\card{ \Gal(K/\Q) }$:	$18$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{10}a^{8}+\frac{1}{10}a^{7}-\frac{1}{10}a^{5}-\frac{1}{2}a^{4}+\frac{1}{5}a^{3}-\frac{1}{2}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{10}a^{9}-\frac{1}{10}a^{7}-\frac{1}{10}a^{6}-\frac{2}{5}a^{5}-\frac{3}{10}a^{4}+\frac{3}{10}a^{3}-\frac{3}{10}a^{2}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{10}a^{10}+\frac{1}{10}a^{6}+\frac{1}{10}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{1}{10}$, $\frac{1}{10}a^{11}+\frac{1}{10}a^{7}+\frac{1}{10}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{10}a$, $\frac{1}{20}a^{12}-\frac{1}{20}a^{10}-\frac{3}{20}a^{6}+\frac{1}{5}a^{5}+\frac{9}{20}a^{4}-\frac{2}{5}a^{3}+\frac{1}{10}a^{2}-\frac{3}{20}$, $\frac{1}{20}a^{13}-\frac{1}{20}a^{11}-\frac{3}{20}a^{7}+\frac{1}{5}a^{6}+\frac{9}{20}a^{5}-\frac{2}{5}a^{4}+\frac{1}{10}a^{3}-\frac{3}{20}a$, $\frac{1}{20}a^{14}-\frac{1}{20}a^{10}-\frac{1}{20}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{3}{10}a^{5}-\frac{9}{20}a^{4}+\frac{3}{10}a^{3}+\frac{9}{20}a^{2}-\frac{3}{10}a+\frac{1}{4}$, $\frac{1}{220}a^{15}-\frac{1}{110}a^{14}+\frac{1}{55}a^{13}-\frac{3}{220}a^{12}-\frac{3}{220}a^{11}-\frac{1}{20}a^{10}-\frac{1}{44}a^{9}+\frac{3}{110}a^{8}-\frac{27}{110}a^{7}+\frac{7}{220}a^{6}+\frac{21}{44}a^{5}-\frac{63}{220}a^{4}+\frac{103}{220}a^{3}-\frac{26}{55}a^{2}-\frac{57}{220}a+\frac{93}{220}$, $\frac{1}{1210003300}a^{16}-\frac{2}{302500825}a^{15}+\frac{5686344}{302500825}a^{14}+\frac{22283003}{1210003300}a^{13}+\frac{32902}{302500825}a^{12}-\frac{48468391}{1210003300}a^{11}-\frac{705579}{121000330}a^{10}+\frac{5620591}{302500825}a^{9}+\frac{27354359}{605001650}a^{8}-\frac{189727097}{1210003300}a^{7}+\frac{2771079}{121000330}a^{6}+\frac{13066167}{110000300}a^{5}+\frac{63315067}{302500825}a^{4}+\frac{56874853}{605001650}a^{3}-\frac{380730861}{1210003300}a^{2}-\frac{95317689}{1210003300}a+\frac{419206351}{1210003300}$, $\frac{1}{219010597300}a^{17}+\frac{41}{109505298650}a^{16}+\frac{88186389}{54752649325}a^{15}-\frac{293570201}{109505298650}a^{14}-\frac{2663910857}{219010597300}a^{13}+\frac{2119382209}{219010597300}a^{12}+\frac{319661059}{43802119460}a^{11}-\frac{5392051771}{219010597300}a^{10}-\frac{3536951741}{109505298650}a^{9}-\frac{545229181}{109505298650}a^{8}+\frac{8115185843}{43802119460}a^{7}-\frac{38612413563}{219010597300}a^{6}+\frac{21279287663}{219010597300}a^{5}+\frac{88208851921}{219010597300}a^{4}+\frac{64658892139}{219010597300}a^{3}-\frac{24146316587}{109505298650}a^{2}-\frac{26299037576}{54752649325}a+\frac{12074426669}{43802119460}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/2, 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^2 - 1/2*a - 1/2, 1/10*a^8 + 1/10*a^7 - 1/10*a^5 - 1/2*a^4 + 1/5*a^3 - 1/2*a^2 + 1/5*a + 2/5, 1/10*a^9 - 1/10*a^7 - 1/10*a^6 - 2/5*a^5 - 3/10*a^4 + 3/10*a^3 - 3/10*a^2 + 1/5*a - 2/5, 1/10*a^10 + 1/10*a^6 + 1/10*a^5 - 1/5*a^4 + 2/5*a^3 + 1/5*a^2 - 1/5*a - 1/10, 1/10*a^11 + 1/10*a^7 + 1/10*a^6 - 1/5*a^5 + 2/5*a^4 + 1/5*a^3 - 1/5*a^2 - 1/10*a, 1/20*a^12 - 1/20*a^10 - 3/20*a^6 + 1/5*a^5 + 9/20*a^4 - 2/5*a^3 + 1/10*a^2 - 3/20, 1/20*a^13 - 1/20*a^11 - 3/20*a^7 + 1/5*a^6 + 9/20*a^5 - 2/5*a^4 + 1/10*a^3 - 3/20*a, 1/20*a^14 - 1/20*a^10 - 1/20*a^8 - 1/5*a^7 - 1/5*a^6 - 3/10*a^5 - 9/20*a^4 + 3/10*a^3 + 9/20*a^2 - 3/10*a + 1/4, 1/220*a^15 - 1/110*a^14 + 1/55*a^13 - 3/220*a^12 - 3/220*a^11 - 1/20*a^10 - 1/44*a^9 + 3/110*a^8 - 27/110*a^7 + 7/220*a^6 + 21/44*a^5 - 63/220*a^4 + 103/220*a^3 - 26/55*a^2 - 57/220*a + 93/220, 1/1210003300*a^16 - 2/302500825*a^15 + 5686344/302500825*a^14 + 22283003/1210003300*a^13 + 32902/302500825*a^12 - 48468391/1210003300*a^11 - 705579/121000330*a^10 + 5620591/302500825*a^9 + 27354359/605001650*a^8 - 189727097/1210003300*a^7 + 2771079/121000330*a^6 + 13066167/110000300*a^5 + 63315067/302500825*a^4 + 56874853/605001650*a^3 - 380730861/1210003300*a^2 - 95317689/1210003300*a + 419206351/1210003300, 1/219010597300*a^17 + 41/109505298650*a^16 + 88186389/54752649325*a^15 - 293570201/109505298650*a^14 - 2663910857/219010597300*a^13 + 2119382209/219010597300*a^12 + 319661059/43802119460*a^11 - 5392051771/219010597300*a^10 - 3536951741/109505298650*a^9 - 545229181/109505298650*a^8 + 8115185843/43802119460*a^7 - 38612413563/219010597300*a^6 + 21279287663/219010597300*a^5 + 88208851921/219010597300*a^4 + 64658892139/219010597300*a^3 - 24146316587/109505298650*a^2 - 26299037576/54752649325*a + 12074426669/43802119460

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

$C_{3}$, which has order $3$

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{142478274}{54752649325} a^{17} - \frac{1211065329}{54752649325} a^{16} + \frac{1164716012}{10950529865} a^{15} - \frac{3891108774}{10950529865} a^{14} + \frac{49458012006}{54752649325} a^{13} - \frac{19974525194}{10950529865} a^{12} + \frac{148166499222}{54752649325} a^{11} - \frac{12233974344}{4977513575} a^{10} - \frac{19546183936}{54752649325} a^{9} + \frac{320886743151}{54752649325} a^{8} - \frac{640182526626}{54752649325} a^{7} + \frac{785562146878}{54752649325} a^{6} - \frac{237498764142}{54752649325} a^{5} - \frac{678971147157}{54752649325} a^{4} + \frac{1973970226214}{54752649325} a^{3} - \frac{189472903824}{4977513575} a^{2} + \frac{1171992574812}{54752649325} a - \frac{242868047207}{54752649325} $ (142478274)/(54752649325)a^(17) - (1211065329)/(54752649325)a^(16) + (1164716012)/(10950529865)a^(15) - (3891108774)/(10950529865)a^(14) + (49458012006)/(54752649325)a^(13) - (19974525194)/(10950529865)a^(12) + (148166499222)/(54752649325)a^(11) - (12233974344)/(4977513575)a^(10) - (19546183936)/(54752649325)a^(9) + (320886743151)/(54752649325)a^(8) - (640182526626)/(54752649325)a^(7) + (785562146878)/(54752649325)a^(6) - (237498764142)/(54752649325)a^(5) - (678971147157)/(54752649325)a^(4) + (1973970226214)/(54752649325)a^(3) - (189472903824)/(4977513575)a^(2) + (1171992574812)/(54752649325)*a - (242868047207)/(54752649325) (order $6$)	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:	$\frac{19389452329}{219010597300}a^{17}-\frac{77686286239}{109505298650}a^{16}+\frac{64821606027}{19910054300}a^{15}-\frac{2271825330299}{219010597300}a^{14}+\frac{250063449837}{9955027150}a^{13}-\frac{2636861844653}{54752649325}a^{12}+\frac{14206624397271}{219010597300}a^{11}-\frac{9941311346449}{219010597300}a^{10}-\frac{9145381361857}{219010597300}a^{9}+\frac{39696491321319}{219010597300}a^{8}-\frac{16298401019952}{54752649325}a^{7}+\frac{17209938523762}{54752649325}a^{6}+\frac{1938020684659}{43802119460}a^{5}-\frac{95420354464979}{219010597300}a^{4}+\frac{391153675969}{398201086}a^{3}-\frac{6537823774157}{8760423892}a^{2}+\frac{24961290645469}{109505298650}a-\frac{669994429421}{219010597300}$, $\frac{5607270454}{54752649325}a^{17}-\frac{197214557263}{219010597300}a^{16}+\frac{964731717161}{219010597300}a^{15}-\frac{817334577723}{54752649325}a^{14}+\frac{8404812138543}{219010597300}a^{13}-\frac{17136806598861}{219010597300}a^{12}+\frac{235300455901}{1991005430}a^{11}-\frac{24190207618931}{219010597300}a^{10}-\frac{1781250149607}{219010597300}a^{9}+\frac{27145068846029}{109505298650}a^{8}-\frac{22222065314521}{43802119460}a^{7}+\frac{137943743060957}{219010597300}a^{6}-\frac{25039072233051}{109505298650}a^{5}-\frac{119920452189669}{219010597300}a^{4}+\frac{334862308880889}{219010597300}a^{3}-\frac{381916849354499}{219010597300}a^{2}+\frac{101162094287253}{109505298650}a-\frac{2187425304703}{10950529865}$, $\frac{614367601}{19910054300}a^{17}-\frac{2228883959}{8760423892}a^{16}+\frac{261022249857}{219010597300}a^{15}-\frac{212229120106}{54752649325}a^{14}+\frac{1049210734871}{109505298650}a^{13}-\frac{4112062811597}{219010597300}a^{12}+\frac{5782338201987}{219010597300}a^{11}-\frac{4571608292231}{219010597300}a^{10}-\frac{97565232821}{8760423892}a^{9}+\frac{7303890786151}{109505298650}a^{8}-\frac{6468220132354}{54752649325}a^{7}+\frac{29078563023087}{219010597300}a^{6}-\frac{1931011746171}{219010597300}a^{5}-\frac{622365149581}{3982010860}a^{4}+\frac{41249401891121}{109505298650}a^{3}-\frac{73819279554607}{219010597300}a^{2}+\frac{14816678164797}{109505298650}a-\frac{973889329718}{54752649325}$, $\frac{1333426362}{54752649325}a^{17}-\frac{2019895483}{9955027150}a^{16}+\frac{52427746111}{54752649325}a^{15}-\frac{343547270889}{109505298650}a^{14}+\frac{427759636427}{54752649325}a^{13}-\frac{844858560461}{54752649325}a^{12}+\frac{1205732262648}{54752649325}a^{11}-\frac{1981380792379}{109505298650}a^{10}-\frac{832890151667}{109505298650}a^{9}+\frac{2933447587977}{54752649325}a^{8}-\frac{5348695136074}{54752649325}a^{7}+\frac{558874949859}{4977513575}a^{6}-\frac{162987917671}{10950529865}a^{5}-\frac{13609149955969}{109505298650}a^{4}+\frac{3378157670952}{10950529865}a^{3}-\frac{6361157692793}{21901059730}a^{2}+\frac{6881256178884}{54752649325}a-\frac{2118388860361}{109505298650}$, $\frac{5293225387}{109505298650}a^{17}-\frac{9170497793}{21901059730}a^{16}+\frac{222156649579}{109505298650}a^{15}-\frac{746280123773}{109505298650}a^{14}+\frac{1903114164509}{109505298650}a^{13}-\frac{3849878794749}{109505298650}a^{12}+\frac{2868094962287}{54752649325}a^{11}-\frac{473521979337}{9955027150}a^{10}-\frac{76807972731}{10950529865}a^{9}+\frac{6243238199102}{54752649325}a^{8}-\frac{12410393409176}{54752649325}a^{7}+\frac{30258437993999}{109505298650}a^{6}-\frac{12014180087}{140211650}a^{5}-\frac{5595134590163}{21901059730}a^{4}+\frac{75693154657829}{109505298650}a^{3}-\frac{82568515263889}{109505298650}a^{2}+\frac{21035784992239}{54752649325}a-\frac{8627128160399}{109505298650}$, $\frac{299573531}{43802119460}a^{17}-\frac{2849541721}{54752649325}a^{16}+\frac{50173931407}{219010597300}a^{15}-\frac{13882917519}{19910054300}a^{14}+\frac{351550530843}{219010597300}a^{13}-\frac{634901599117}{219010597300}a^{12}+\frac{34015170367}{9955027150}a^{11}-\frac{2601318697}{1991005430}a^{10}-\frac{1070098525631}{219010597300}a^{9}+\frac{2819866847883}{219010597300}a^{8}-\frac{3770212694327}{219010597300}a^{7}+\frac{57284878317}{3982010860}a^{6}+\frac{1560370527241}{109505298650}a^{5}-\frac{1836009750293}{54752649325}a^{4}+\frac{625158278243}{9955027150}a^{3}-\frac{5739703255951}{219010597300}a^{2}-\frac{1608187837579}{219010597300}a+\frac{408224677504}{54752649325}$, $\frac{1710867203}{19910054300}a^{17}-\frac{84703053267}{109505298650}a^{16}+\frac{840765709033}{219010597300}a^{15}-\frac{2886123107521}{219010597300}a^{14}+\frac{1876378041306}{54752649325}a^{13}-\frac{15471641406233}{219010597300}a^{12}+\frac{4760450997191}{43802119460}a^{11}-\frac{5768995272042}{54752649325}a^{10}+\frac{387844366989}{219010597300}a^{9}+\frac{47491601655899}{219010597300}a^{8}-\frac{5058875340297}{10950529865}a^{7}+\frac{11689045845711}{19910054300}a^{6}-\frac{55353778046891}{219010597300}a^{5}-\frac{51625807452111}{109505298650}a^{4}+\frac{75040849503198}{54752649325}a^{3}-\frac{363003559696197}{219010597300}a^{2}+\frac{100520481943129}{109505298650}a-\frac{4532926327689}{21901059730}$, $\frac{8928075671}{219010597300}a^{17}-\frac{37494361077}{109505298650}a^{16}+\frac{355927262909}{219010597300}a^{15}-\frac{1172697848163}{219010597300}a^{14}+\frac{587263274177}{43802119460}a^{13}-\frac{1458067420621}{54752649325}a^{12}+\frac{4205278455313}{109505298650}a^{11}-\frac{644704544971}{19910054300}a^{10}-\frac{2458708230601}{219010597300}a^{9}+\frac{3982249681633}{43802119460}a^{8}-\frac{37046395801183}{219010597300}a^{7}+\frac{21602900425001}{109505298650}a^{6}-\frac{3822479237867}{109505298650}a^{5}-\frac{45777033099387}{219010597300}a^{4}+\frac{58027664429829}{109505298650}a^{3}-\frac{113385698469613}{219010597300}a^{2}+\frac{2065568490099}{8760423892}a-\frac{8994043798151}{219010597300}$ 19389452329/219010597300a^17 - 77686286239/109505298650a^16 + 64821606027/19910054300a^15 - 2271825330299/219010597300a^14 + 250063449837/9955027150a^13 - 2636861844653/54752649325a^12 + 14206624397271/219010597300a^11 - 9941311346449/219010597300a^10 - 9145381361857/219010597300a^9 + 39696491321319/219010597300a^8 - 16298401019952/54752649325a^7 + 17209938523762/54752649325a^6 + 1938020684659/43802119460a^5 - 95420354464979/219010597300a^4 + 391153675969/398201086a^3 - 6537823774157/8760423892a^2 + 24961290645469/109505298650a - 669994429421/219010597300, 5607270454/54752649325a^17 - 197214557263/219010597300a^16 + 964731717161/219010597300a^15 - 817334577723/54752649325a^14 + 8404812138543/219010597300a^13 - 17136806598861/219010597300a^12 + 235300455901/1991005430a^11 - 24190207618931/219010597300a^10 - 1781250149607/219010597300a^9 + 27145068846029/109505298650a^8 - 22222065314521/43802119460a^7 + 137943743060957/219010597300a^6 - 25039072233051/109505298650a^5 - 119920452189669/219010597300a^4 + 334862308880889/219010597300a^3 - 381916849354499/219010597300a^2 + 101162094287253/109505298650a - 2187425304703/10950529865, 614367601/19910054300a^17 - 2228883959/8760423892a^16 + 261022249857/219010597300a^15 - 212229120106/54752649325a^14 + 1049210734871/109505298650a^13 - 4112062811597/219010597300a^12 + 5782338201987/219010597300a^11 - 4571608292231/219010597300a^10 - 97565232821/8760423892a^9 + 7303890786151/109505298650a^8 - 6468220132354/54752649325a^7 + 29078563023087/219010597300a^6 - 1931011746171/219010597300a^5 - 622365149581/3982010860a^4 + 41249401891121/109505298650a^3 - 73819279554607/219010597300a^2 + 14816678164797/109505298650a - 973889329718/54752649325, 1333426362/54752649325a^17 - 2019895483/9955027150a^16 + 52427746111/54752649325a^15 - 343547270889/109505298650a^14 + 427759636427/54752649325a^13 - 844858560461/54752649325a^12 + 1205732262648/54752649325a^11 - 1981380792379/109505298650a^10 - 832890151667/109505298650a^9 + 2933447587977/54752649325a^8 - 5348695136074/54752649325a^7 + 558874949859/4977513575a^6 - 162987917671/10950529865a^5 - 13609149955969/109505298650a^4 + 3378157670952/10950529865a^3 - 6361157692793/21901059730a^2 + 6881256178884/54752649325a - 2118388860361/109505298650, 5293225387/109505298650a^17 - 9170497793/21901059730a^16 + 222156649579/109505298650a^15 - 746280123773/109505298650a^14 + 1903114164509/109505298650a^13 - 3849878794749/109505298650a^12 + 2868094962287/54752649325a^11 - 473521979337/9955027150a^10 - 76807972731/10950529865a^9 + 6243238199102/54752649325a^8 - 12410393409176/54752649325a^7 + 30258437993999/109505298650a^6 - 12014180087/140211650a^5 - 5595134590163/21901059730a^4 + 75693154657829/109505298650a^3 - 82568515263889/109505298650a^2 + 21035784992239/54752649325a - 8627128160399/109505298650, 299573531/43802119460a^17 - 2849541721/54752649325a^16 + 50173931407/219010597300a^15 - 13882917519/19910054300a^14 + 351550530843/219010597300a^13 - 634901599117/219010597300a^12 + 34015170367/9955027150a^11 - 2601318697/1991005430a^10 - 1070098525631/219010597300a^9 + 2819866847883/219010597300a^8 - 3770212694327/219010597300a^7 + 57284878317/3982010860a^6 + 1560370527241/109505298650a^5 - 1836009750293/54752649325a^4 + 625158278243/9955027150a^3 - 5739703255951/219010597300a^2 - 1608187837579/219010597300a + 408224677504/54752649325, 1710867203/19910054300a^17 - 84703053267/109505298650a^16 + 840765709033/219010597300a^15 - 2886123107521/219010597300a^14 + 1876378041306/54752649325a^13 - 15471641406233/219010597300a^12 + 4760450997191/43802119460a^11 - 5768995272042/54752649325a^10 + 387844366989/219010597300a^9 + 47491601655899/219010597300a^8 - 5058875340297/10950529865a^7 + 11689045845711/19910054300a^6 - 55353778046891/219010597300a^5 - 51625807452111/109505298650a^4 + 75040849503198/54752649325a^3 - 363003559696197/219010597300a^2 + 100520481943129/109505298650a - 4532926327689/21901059730, 8928075671/219010597300a^17 - 37494361077/109505298650a^16 + 355927262909/219010597300a^15 - 1172697848163/219010597300a^14 + 587263274177/43802119460a^13 - 1458067420621/54752649325a^12 + 4205278455313/109505298650a^11 - 644704544971/19910054300a^10 - 2458708230601/219010597300a^9 + 3982249681633/43802119460a^8 - 37046395801183/219010597300a^7 + 21602900425001/109505298650a^6 - 3822479237867/109505298650a^5 - 45777033099387/219010597300a^4 + 58027664429829/109505298650a^3 - 113385698469613/219010597300a^2 + 2065568490099/8760423892a - 8994043798151/219010597300	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:	$ 102362.355108 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 102362.355108 \cdot 3}{6\cdot\sqrt{8549394874383196572862347}}\cr\approx \mathstrut & 0.267153827516 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811)

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 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811, 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 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811);

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 - 9*x^17 + 45*x^16 - 156*x^15 + 411*x^14 - 861*x^13 + 1359*x^12 - 1407*x^11 + 264*x^10 + 2354*x^9 - 5535*x^8 + 7515*x^7 - 4059*x^6 - 4305*x^5 + 16131*x^4 - 20937*x^3 + 14379*x^2 - 5190*x + 811);

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{-3}) $, 3.1.1323.1 x3, 3.3.3969.2, 6.0.5250987.1, 6.0.964467.2 x2, 6.0.47258883.1, 9.3.1688134559643.1 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:	6.0.964467.2
Degree 9 sibling:	9.3.1688134559643.1
Minimal sibling:	6.0.964467.2

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }^{3}$	R	${\href{/padicField/5.2.0.1}{2} }^{9}$	R	${\href{/padicField/11.2.0.1}{2} }^{9}$	${\href{/padicField/13.3.0.1}{3} }^{6}$	${\href{/padicField/17.6.0.1}{6} }^{3}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.2.0.1}{2} }^{9}$	${\href{/padicField/29.6.0.1}{6} }^{3}$	${\href{/padicField/31.3.0.1}{3} }^{6}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	${\href{/padicField/41.6.0.1}{6} }^{3}$	${\href{/padicField/43.3.0.1}{3} }^{6}$	${\href{/padicField/47.6.0.1}{6} }^{3}$	${\href{/padicField/53.6.0.1}{6} }^{3}$	${\href{/padicField/59.6.0.1}{6} }^{3}$

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
$3$ 3		Deg $18$	$18$	$1$	$31$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$

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