LMFDB - Number field 18.0.315164892649262158461997559808.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484)

gp: K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 138*y^15 + 279*y^14 - 369*y^13 + 318*y^12 - 315*y^11 + 1413*y^10 - 5320*y^9 + 12681*y^8 - 22095*y^7 + 29967*y^6 - 25938*y^5 + 5652*y^4 + 8688*y^3 - 2448*y^2 - 396*y + 484, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484)

$ x^{18} - 9 x^{17} + 45 x^{16} - 138 x^{15} + 279 x^{14} - 369 x^{13} + 318 x^{12} - 315 x^{11} + \cdots + 484 $ x^18 - 9*x^17 + 45*x^16 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484

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:	$-315164892649262158461997559808$ -315164892649262158461997559808 $\medspace = -\,2^{12}\cdot 3^{33}\cdot 7^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$43.53$	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:	$2^{2/3}3^{11/6}7^{2/3}\approx 43.531903466499294$
Ramified primes:	$2$, $3$, $7$ 2, 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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{2}{9}a^{5}-\frac{2}{9}a^{4}+\frac{2}{9}a^{3}+\frac{4}{9}a^{2}-\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{45}a^{9}-\frac{1}{45}a^{8}+\frac{1}{45}a^{7}+\frac{2}{45}a^{6}+\frac{7}{45}a^{5}-\frac{16}{45}a^{4}+\frac{13}{45}a^{3}+\frac{14}{45}a^{2}-\frac{1}{9}a+\frac{1}{5}$, $\frac{1}{45}a^{10}+\frac{1}{15}a^{7}-\frac{2}{15}a^{6}-\frac{1}{5}a^{5}-\frac{1}{15}a^{4}-\frac{1}{15}a^{3}+\frac{1}{5}a^{2}+\frac{4}{45}a-\frac{2}{15}$, $\frac{1}{45}a^{11}-\frac{2}{45}a^{8}-\frac{1}{45}a^{7}+\frac{1}{45}a^{6}-\frac{13}{45}a^{5}+\frac{7}{45}a^{4}-\frac{16}{45}a^{3}-\frac{16}{45}a^{2}+\frac{14}{45}a-\frac{1}{9}$, $\frac{1}{270}a^{12}+\frac{1}{270}a^{9}+\frac{1}{45}a^{8}-\frac{1}{45}a^{7}+\frac{11}{90}a^{6}+\frac{8}{45}a^{5}-\frac{14}{45}a^{4}-\frac{77}{270}a^{3}+\frac{1}{45}a^{2}+\frac{4}{9}a-\frac{19}{135}$, $\frac{1}{540}a^{13}+\frac{1}{540}a^{10}+\frac{1}{20}a^{7}+\frac{1}{15}a^{6}+\frac{4}{15}a^{5}+\frac{19}{540}a^{4}-\frac{2}{15}a^{3}+\frac{1}{15}a^{2}+\frac{131}{270}a+\frac{2}{5}$, $\frac{1}{2700}a^{14}+\frac{1}{1350}a^{13}+\frac{1}{1350}a^{12}+\frac{1}{108}a^{11}-\frac{1}{270}a^{10}+\frac{13}{1350}a^{9}+\frac{1}{100}a^{8}-\frac{1}{150}a^{7}-\frac{1}{50}a^{6}-\frac{593}{2700}a^{5}+\frac{289}{1350}a^{4}+\frac{451}{1350}a^{3}-\frac{11}{270}a^{2}-\frac{313}{675}a+\frac{56}{675}$, $\frac{1}{2700}a^{15}-\frac{1}{1350}a^{13}+\frac{1}{2700}a^{12}-\frac{7}{1350}a^{10}+\frac{1}{180}a^{9}-\frac{2}{75}a^{8}-\frac{7}{50}a^{7}-\frac{61}{540}a^{6}-\frac{31}{75}a^{5}+\frac{233}{1350}a^{4}+\frac{113}{1350}a^{3}-\frac{17}{75}a^{2}-\frac{143}{675}a-\frac{49}{225}$, $\frac{1}{16200}a^{16}+\frac{1}{16200}a^{15}-\frac{7}{16200}a^{13}-\frac{1}{648}a^{12}+\frac{1}{450}a^{11}-\frac{149}{16200}a^{10}+\frac{17}{3240}a^{9}+\frac{7}{225}a^{8}-\frac{449}{16200}a^{7}-\frac{2519}{16200}a^{6}+\frac{139}{450}a^{5}-\frac{791}{8100}a^{4}+\frac{646}{2025}a^{3}+\frac{16}{75}a^{2}-\frac{733}{2025}a-\frac{223}{810}$, $\frac{1}{412843959000}a^{17}-\frac{1391279}{206421979500}a^{16}-\frac{55112903}{412843959000}a^{15}-\frac{52259851}{412843959000}a^{14}+\frac{166493879}{206421979500}a^{13}-\frac{34446781}{412843959000}a^{12}-\frac{3031134173}{412843959000}a^{11}-\frac{1115574689}{206421979500}a^{10}-\frac{22019455}{3302751672}a^{9}+\frac{2511859727}{82568791800}a^{8}+\frac{29002923553}{206421979500}a^{7}+\frac{65562917941}{412843959000}a^{6}-\frac{40536095351}{206421979500}a^{5}+\frac{81986719}{1876563450}a^{4}+\frac{16552546579}{51605494875}a^{3}-\frac{4988304029}{10321098975}a^{2}-\frac{1820351201}{51605494875}a+\frac{1475727119}{9382817250}$ 1, a, a^2, a^3, a^4, a^5, 1/3*a^6 - 1/3*a^3 + 1/3, 1/3*a^7 - 1/3*a^4 + 1/3*a, 1/9*a^8 - 1/9*a^7 + 1/9*a^6 + 2/9*a^5 - 2/9*a^4 + 2/9*a^3 + 4/9*a^2 - 4/9*a + 4/9, 1/45*a^9 - 1/45*a^8 + 1/45*a^7 + 2/45*a^6 + 7/45*a^5 - 16/45*a^4 + 13/45*a^3 + 14/45*a^2 - 1/9*a + 1/5, 1/45*a^10 + 1/15*a^7 - 2/15*a^6 - 1/5*a^5 - 1/15*a^4 - 1/15*a^3 + 1/5*a^2 + 4/45*a - 2/15, 1/45*a^11 - 2/45*a^8 - 1/45*a^7 + 1/45*a^6 - 13/45*a^5 + 7/45*a^4 - 16/45*a^3 - 16/45*a^2 + 14/45*a - 1/9, 1/270*a^12 + 1/270*a^9 + 1/45*a^8 - 1/45*a^7 + 11/90*a^6 + 8/45*a^5 - 14/45*a^4 - 77/270*a^3 + 1/45*a^2 + 4/9*a - 19/135, 1/540*a^13 + 1/540*a^10 + 1/20*a^7 + 1/15*a^6 + 4/15*a^5 + 19/540*a^4 - 2/15*a^3 + 1/15*a^2 + 131/270*a + 2/5, 1/2700*a^14 + 1/1350*a^13 + 1/1350*a^12 + 1/108*a^11 - 1/270*a^10 + 13/1350*a^9 + 1/100*a^8 - 1/150*a^7 - 1/50*a^6 - 593/2700*a^5 + 289/1350*a^4 + 451/1350*a^3 - 11/270*a^2 - 313/675*a + 56/675, 1/2700*a^15 - 1/1350*a^13 + 1/2700*a^12 - 7/1350*a^10 + 1/180*a^9 - 2/75*a^8 - 7/50*a^7 - 61/540*a^6 - 31/75*a^5 + 233/1350*a^4 + 113/1350*a^3 - 17/75*a^2 - 143/675*a - 49/225, 1/16200*a^16 + 1/16200*a^15 - 7/16200*a^13 - 1/648*a^12 + 1/450*a^11 - 149/16200*a^10 + 17/3240*a^9 + 7/225*a^8 - 449/16200*a^7 - 2519/16200*a^6 + 139/450*a^5 - 791/8100*a^4 + 646/2025*a^3 + 16/75*a^2 - 733/2025*a - 223/810, 1/412843959000*a^17 - 1391279/206421979500*a^16 - 55112903/412843959000*a^15 - 52259851/412843959000*a^14 + 166493879/206421979500*a^13 - 34446781/412843959000*a^12 - 3031134173/412843959000*a^11 - 1115574689/206421979500*a^10 - 22019455/3302751672*a^9 + 2511859727/82568791800*a^8 + 29002923553/206421979500*a^7 + 65562917941/412843959000*a^6 - 40536095351/206421979500*a^5 + 81986719/1876563450*a^4 + 16552546579/51605494875*a^3 - 4988304029/10321098975*a^2 - 1820351201/51605494875*a + 1475727119/9382817250

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

$C_{2}\times C_{6}$, which has order $12$ (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{1408897}{5096839000} a^{17} + \frac{31378883}{15290517000} a^{16} - \frac{66212771}{7645258500} a^{15} + \frac{299763671}{15290517000} a^{14} - \frac{980199139}{45871551000} a^{13} - \frac{210458471}{22935775500} a^{12} + \frac{2735295419}{45871551000} a^{11} - \frac{217190789}{5096839000} a^{10} - \frac{381713957}{1529051700} a^{9} + \frac{2662205437}{3058103400} a^{8} - \frac{6797093357}{5096839000} a^{7} + \frac{2620921159}{2548419500} a^{6} + \frac{400444193}{637104875} a^{5} - \frac{2024948501}{417014100} a^{4} + \frac{102313447901}{11467887750} a^{3} - \frac{12581378303}{2293577550} a^{2} - \frac{2619885139}{1911314625} a + \frac{208098308}{173755875} $ -(1408897)/(5096839000)a^(17) + (31378883)/(15290517000)a^(16) - (66212771)/(7645258500)a^(15) + (299763671)/(15290517000)a^(14) - (980199139)/(45871551000)a^(13) - (210458471)/(22935775500)a^(12) + (2735295419)/(45871551000)a^(11) - (217190789)/(5096839000)a^(10) - (381713957)/(1529051700)a^(9) + (2662205437)/(3058103400)a^(8) - (6797093357)/(5096839000)a^(7) + (2620921159)/(2548419500)a^(6) + (400444193)/(637104875)a^(5) - (2024948501)/(417014100)a^(4) + (102313447901)/(11467887750)a^(3) - (12581378303)/(2293577550)a^(2) - (2619885139)/(1911314625)*a + (208098308)/(173755875) (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:	$\frac{3077363}{2752293060}a^{17}-\frac{277633219}{27522930600}a^{16}+\frac{1390767089}{27522930600}a^{15}-\frac{2127621641}{13761465300}a^{14}+\frac{2849807671}{9174310200}a^{13}-\frac{3682135187}{9174310200}a^{12}+\frac{1513262141}{4587155100}a^{11}-\frac{8829227441}{27522930600}a^{10}+\frac{43229095213}{27522930600}a^{9}-\frac{82307033243}{13761465300}a^{8}+\frac{390487528283}{27522930600}a^{7}-\frac{669264999151}{27522930600}a^{6}+\frac{111897273799}{3440366325}a^{5}-\frac{457549451}{16680564}a^{4}+\frac{4109256803}{1146788775}a^{3}+\frac{546755573}{45871551}a^{2}-\frac{3939650017}{3440366325}a-\frac{1037510023}{625521150}$, $\frac{287223}{463349000}a^{17}-\frac{1250147}{231674500}a^{16}+\frac{36244363}{1390047000}a^{15}-\frac{35695283}{463349000}a^{14}+\frac{34332467}{231674500}a^{13}-\frac{784027597}{4170141000}a^{12}+\frac{218363333}{1390047000}a^{11}-\frac{126527531}{695023500}a^{10}+\frac{710427937}{834028200}a^{9}-\frac{280511921}{92669800}a^{8}+\frac{1584032729}{231674500}a^{7}-\frac{5405125027}{463349000}a^{6}+\frac{1804191011}{115837250}a^{5}-\frac{290178341}{23167450}a^{4}+\frac{1262505853}{521267625}a^{3}+\frac{175453039}{69502350}a^{2}-\frac{13720784}{173755875}a+\frac{18261623}{1042535250}$, $\frac{5845139}{11467887750}a^{17}-\frac{196578743}{45871551000}a^{16}+\frac{2828240011}{137614653000}a^{15}-\frac{4116245879}{68807326500}a^{14}+\frac{15788190859}{137614653000}a^{13}-\frac{19784702263}{137614653000}a^{12}+\frac{8053482463}{68807326500}a^{11}-\frac{18255312239}{137614653000}a^{10}+\frac{18553545167}{27522930600}a^{9}-\frac{10866057503}{4587155100}a^{8}+\frac{244982742431}{45871551000}a^{7}-\frac{1238592339137}{137614653000}a^{6}+\frac{815864051947}{68807326500}a^{5}-\frac{11667307111}{1251042300}a^{4}+\frac{44407743329}{34403663250}a^{3}+\frac{19849415743}{6880732650}a^{2}+\frac{18032487347}{17201831625}a+\frac{703521977}{3127605750}$, $\frac{50765402}{10321098975}a^{17}-\frac{1179284041}{27522930600}a^{16}+\frac{17117170877}{82568791800}a^{15}-\frac{12552977773}{20642197950}a^{14}+\frac{31700942117}{27522930600}a^{13}-\frac{114032866433}{82568791800}a^{12}+\frac{10482871184}{10321098975}a^{11}-\frac{10344559607}{9174310200}a^{10}+\frac{541278443429}{82568791800}a^{9}-\frac{495460018643}{20642197950}a^{8}+\frac{1481454347363}{27522930600}a^{7}-\frac{1466093177339}{16513758360}a^{6}+\frac{1175610830467}{10321098975}a^{5}-\frac{25849421572}{312760575}a^{4}-\frac{6327793337}{825687918}a^{3}+\frac{87360850546}{2064219795}a^{2}+\frac{6976696939}{2293577550}a-\frac{7405335737}{1876563450}$, $\frac{17692547}{103210989750}a^{17}-\frac{553755929}{412843959000}a^{16}+\frac{2199784691}{412843959000}a^{15}-\frac{1175540357}{103210989750}a^{14}+\frac{2948693339}{412843959000}a^{13}+\frac{6682707397}{412843959000}a^{12}-\frac{2498808578}{51605494875}a^{11}+\frac{1777478041}{412843959000}a^{10}+\frac{16292524627}{82568791800}a^{9}-\frac{12423028463}{20642197950}a^{8}+\frac{221481070933}{412843959000}a^{7}-\frac{53074011157}{412843959000}a^{6}-\frac{118542120509}{103210989750}a^{5}+\frac{440933359}{150125076}a^{4}-\frac{237929463523}{51605494875}a^{3}-\frac{7925984146}{10321098975}a^{2}+\frac{75823010702}{51605494875}a+\frac{4087174747}{9382817250}$, $\frac{425939323}{412843959000}a^{17}-\frac{423420053}{51605494875}a^{16}+\frac{15772537711}{412843959000}a^{15}-\frac{43658629243}{412843959000}a^{14}+\frac{10018034948}{51605494875}a^{13}-\frac{95468407483}{412843959000}a^{12}+\frac{82598888431}{412843959000}a^{11}-\frac{29517313291}{103210989750}a^{10}+\frac{109812525071}{82568791800}a^{9}-\frac{351367329457}{82568791800}a^{8}+\frac{469310023381}{51605494875}a^{7}-\frac{6302724650117}{412843959000}a^{6}+\frac{1035246083623}{51605494875}a^{5}-\frac{29531126021}{1876563450}a^{4}+\frac{159061548262}{51605494875}a^{3}-\frac{412244023}{20642197950}a^{2}-\frac{46810178}{51605494875}a-\frac{1881389473}{9382817250}$, $\frac{63061657}{137614653000}a^{17}-\frac{150842557}{45871551000}a^{16}+\frac{332449159}{22935775500}a^{15}-\frac{5039657947}{137614653000}a^{14}+\frac{311331963}{5096839000}a^{13}-\frac{482135749}{7645258500}a^{12}+\frac{5502459799}{137614653000}a^{11}-\frac{3851100907}{45871551000}a^{10}+\frac{2296759397}{4587155100}a^{9}-\frac{8602066709}{5504586120}a^{8}+\frac{139742961509}{45871551000}a^{7}-\frac{111009091553}{22935775500}a^{6}+\frac{386174923303}{68807326500}a^{5}-\frac{365007911}{139004700}a^{4}-\frac{1175033817}{1274209750}a^{3}+\frac{2711709709}{3440366325}a^{2}-\frac{536098604}{5733943875}a-\frac{50081827}{521267625}$, $\frac{9856481}{733293000}a^{17}-\frac{8881156}{91661625}a^{16}+\frac{313831757}{733293000}a^{15}-\frac{784633601}{733293000}a^{14}+\frac{645265879}{366646500}a^{13}-\frac{1226503601}{733293000}a^{12}+\frac{797795837}{733293000}a^{11}-\frac{809668459}{366646500}a^{10}+\frac{2186101981}{146658600}a^{9}-\frac{6512968691}{146658600}a^{8}+\frac{32215049573}{366646500}a^{7}-\frac{98171541799}{733293000}a^{6}+\frac{13930942556}{91661625}a^{5}-\frac{424985123}{6666300}a^{4}-\frac{4627660006}{91661625}a^{3}+\frac{648822127}{36664650}a^{2}+\frac{203821573}{183323250}a-\frac{54892931}{16665750}$ 3077363/2752293060a^17 - 277633219/27522930600a^16 + 1390767089/27522930600a^15 - 2127621641/13761465300a^14 + 2849807671/9174310200a^13 - 3682135187/9174310200a^12 + 1513262141/4587155100a^11 - 8829227441/27522930600a^10 + 43229095213/27522930600a^9 - 82307033243/13761465300a^8 + 390487528283/27522930600a^7 - 669264999151/27522930600a^6 + 111897273799/3440366325a^5 - 457549451/16680564a^4 + 4109256803/1146788775a^3 + 546755573/45871551a^2 - 3939650017/3440366325a - 1037510023/625521150, 287223/463349000a^17 - 1250147/231674500a^16 + 36244363/1390047000a^15 - 35695283/463349000a^14 + 34332467/231674500a^13 - 784027597/4170141000a^12 + 218363333/1390047000a^11 - 126527531/695023500a^10 + 710427937/834028200a^9 - 280511921/92669800a^8 + 1584032729/231674500a^7 - 5405125027/463349000a^6 + 1804191011/115837250a^5 - 290178341/23167450a^4 + 1262505853/521267625a^3 + 175453039/69502350a^2 - 13720784/173755875a + 18261623/1042535250, 5845139/11467887750a^17 - 196578743/45871551000a^16 + 2828240011/137614653000a^15 - 4116245879/68807326500a^14 + 15788190859/137614653000a^13 - 19784702263/137614653000a^12 + 8053482463/68807326500a^11 - 18255312239/137614653000a^10 + 18553545167/27522930600a^9 - 10866057503/4587155100a^8 + 244982742431/45871551000a^7 - 1238592339137/137614653000a^6 + 815864051947/68807326500a^5 - 11667307111/1251042300a^4 + 44407743329/34403663250a^3 + 19849415743/6880732650a^2 + 18032487347/17201831625a + 703521977/3127605750, 50765402/10321098975a^17 - 1179284041/27522930600a^16 + 17117170877/82568791800a^15 - 12552977773/20642197950a^14 + 31700942117/27522930600a^13 - 114032866433/82568791800a^12 + 10482871184/10321098975a^11 - 10344559607/9174310200a^10 + 541278443429/82568791800a^9 - 495460018643/20642197950a^8 + 1481454347363/27522930600a^7 - 1466093177339/16513758360a^6 + 1175610830467/10321098975a^5 - 25849421572/312760575a^4 - 6327793337/825687918a^3 + 87360850546/2064219795a^2 + 6976696939/2293577550a - 7405335737/1876563450, 17692547/103210989750a^17 - 553755929/412843959000a^16 + 2199784691/412843959000a^15 - 1175540357/103210989750a^14 + 2948693339/412843959000a^13 + 6682707397/412843959000a^12 - 2498808578/51605494875a^11 + 1777478041/412843959000a^10 + 16292524627/82568791800a^9 - 12423028463/20642197950a^8 + 221481070933/412843959000a^7 - 53074011157/412843959000a^6 - 118542120509/103210989750a^5 + 440933359/150125076a^4 - 237929463523/51605494875a^3 - 7925984146/10321098975a^2 + 75823010702/51605494875a + 4087174747/9382817250, 425939323/412843959000a^17 - 423420053/51605494875a^16 + 15772537711/412843959000a^15 - 43658629243/412843959000a^14 + 10018034948/51605494875a^13 - 95468407483/412843959000a^12 + 82598888431/412843959000a^11 - 29517313291/103210989750a^10 + 109812525071/82568791800a^9 - 351367329457/82568791800a^8 + 469310023381/51605494875a^7 - 6302724650117/412843959000a^6 + 1035246083623/51605494875a^5 - 29531126021/1876563450a^4 + 159061548262/51605494875a^3 - 412244023/20642197950a^2 - 46810178/51605494875a - 1881389473/9382817250, 63061657/137614653000a^17 - 150842557/45871551000a^16 + 332449159/22935775500a^15 - 5039657947/137614653000a^14 + 311331963/5096839000a^13 - 482135749/7645258500a^12 + 5502459799/137614653000a^11 - 3851100907/45871551000a^10 + 2296759397/4587155100a^9 - 8602066709/5504586120a^8 + 139742961509/45871551000a^7 - 111009091553/22935775500a^6 + 386174923303/68807326500a^5 - 365007911/139004700a^4 - 1175033817/1274209750a^3 + 2711709709/3440366325a^2 - 536098604/5733943875a - 50081827/521267625, 9856481/733293000a^17 - 8881156/91661625a^16 + 313831757/733293000a^15 - 784633601/733293000a^14 + 645265879/366646500a^13 - 1226503601/733293000a^12 + 797795837/733293000a^11 - 809668459/366646500a^10 + 2186101981/146658600a^9 - 6512968691/146658600a^8 + 32215049573/366646500a^7 - 98171541799/733293000a^6 + 13930942556/91661625a^5 - 424985123/6666300a^4 - 4627660006/91661625a^3 + 648822127/36664650a^2 + 203821573/183323250a - 54892931/16665750 (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:	$ 31677601.476 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 31677601.476 \cdot 12}{6\cdot\sqrt{315164892649262158461997559808}}\cr\approx \mathstrut & 1.7223927341 \end{aligned}\] (assuming GRH)

# 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 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484)

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 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484, 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 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484);

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 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484);

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

$C_3:S_3$ (as 18T4):

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 $C_3^2 : C_2$

Character table for $C_3^2 : C_2$

Intermediate fields

$\Q(\sqrt{-3}) $, 3.1.11907.1 x3, 3.1.47628.2 x3, 3.1.972.1 x3, 3.1.588.1 x3, 6.0.425329947.3, 6.0.6805279152.4, 6.0.2834352.1, 6.0.1037232.1, 9.1.324121835451456.12 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:	9.1.324121835451456.12
Minimal sibling:	9.1.324121835451456.12

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	${\href{/padicField/5.2.0.1}{2} }^{9}$	R	${\href{/padicField/11.2.0.1}{2} }^{9}$	${\href{/padicField/13.3.0.1}{3} }^{6}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.2.0.1}{2} }^{9}$	${\href{/padicField/29.2.0.1}{2} }^{9}$	${\href{/padicField/31.3.0.1}{3} }^{6}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.3.0.1}{3} }^{6}$	${\href{/padicField/47.2.0.1}{2} }^{9}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.2.0.1}{2} }^{9}$

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	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$3$ 3	3.6.11.8	$x^{6} + 18 x^{2} + 21$	$6$	$1$	$11$	$S_3$	$[5/2]_{2}$
	3.6.11.8	$x^{6} + 18 x^{2} + 21$	$6$	$1$	$11$	$S_3$	$[5/2]_{2}$
	3.6.11.8	$x^{6} + 18 x^{2} + 21$	$6$	$1$	$11$	$S_3$	$[5/2]_{2}$
$7$ 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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