LMFDB - Number field 18.6.60874624883546499579904.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8)

gp: K = bnfinit(y^18 - 4*y^17 + 2*y^16 + 10*y^15 - 27*y^14 + 70*y^13 - 103*y^12 + 12*y^11 + 98*y^10 - 76*y^9 + 106*y^8 - 208*y^7 + 336*y^5 - 152*y^4 - 176*y^3 + 104*y^2 + 16*y - 8, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8)

$ x^{18} - 4 x^{17} + 2 x^{16} + 10 x^{15} - 27 x^{14} + 70 x^{13} - 103 x^{12} + 12 x^{11} + 98 x^{10} + \cdots - 8 $ x^18 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8

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:	$[6, 6]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$60874624883546499579904$ 60874624883546499579904 $\medspace = 2^{42}\cdot 7^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$18.44$	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:	$2^{11/4}7^{2/3}\approx 24.616776431006123$
Ramified primes:	$2$, $7$ 2, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$3$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{284}a^{16}-\frac{1}{142}a^{15}-\frac{17}{284}a^{14}+\frac{3}{142}a^{13}+\frac{27}{284}a^{12}+\frac{17}{142}a^{11}-\frac{7}{142}a^{10}-\frac{25}{71}a^{9}-\frac{105}{284}a^{8}-\frac{16}{71}a^{7}-\frac{20}{71}a^{6}+\frac{6}{71}a^{5}-\frac{15}{142}a^{4}-\frac{21}{71}a^{3}-\frac{3}{71}a^{2}-\frac{19}{71}a+\frac{33}{71}$, $\frac{1}{20486074359748}a^{17}-\frac{35091258163}{20486074359748}a^{16}-\frac{273291610130}{5121518589937}a^{15}-\frac{182355469771}{20486074359748}a^{14}+\frac{4754566958811}{20486074359748}a^{13}+\frac{4319976871985}{20486074359748}a^{12}+\frac{472633929847}{20486074359748}a^{11}-\frac{391644644849}{5121518589937}a^{10}-\frac{1937146799521}{10243037179874}a^{9}+\frac{2438595045927}{20486074359748}a^{8}-\frac{3924258540939}{10243037179874}a^{7}-\frac{1580116379967}{10243037179874}a^{6}+\frac{1498075054949}{5121518589937}a^{5}-\frac{1507215488333}{5121518589937}a^{4}-\frac{5607900391}{5121518589937}a^{3}-\frac{1386251139825}{5121518589937}a^{2}-\frac{1505249853248}{5121518589937}a-\frac{1476916335330}{5121518589937}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, 1/2*a^10 - 1/2*a^6 - 1/2*a^4, 1/2*a^11 - 1/2*a^7 - 1/2*a^5, 1/2*a^12 - 1/2*a^8 - 1/2*a^6, 1/2*a^13 - 1/2*a^9 - 1/2*a^7, 1/4*a^14 - 1/4*a^10 - 1/2*a^9 - 1/4*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5, 1/4*a^15 - 1/4*a^11 - 1/4*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^4, 1/284*a^16 - 1/142*a^15 - 17/284*a^14 + 3/142*a^13 + 27/284*a^12 + 17/142*a^11 - 7/142*a^10 - 25/71*a^9 - 105/284*a^8 - 16/71*a^7 - 20/71*a^6 + 6/71*a^5 - 15/142*a^4 - 21/71*a^3 - 3/71*a^2 - 19/71*a + 33/71, 1/20486074359748*a^17 - 35091258163/20486074359748*a^16 - 273291610130/5121518589937*a^15 - 182355469771/20486074359748*a^14 + 4754566958811/20486074359748*a^13 + 4319976871985/20486074359748*a^12 + 472633929847/20486074359748*a^11 - 391644644849/5121518589937*a^10 - 1937146799521/10243037179874*a^9 + 2438595045927/20486074359748*a^8 - 3924258540939/10243037179874*a^7 - 1580116379967/10243037179874*a^6 + 1498075054949/5121518589937*a^5 - 1507215488333/5121518589937*a^4 - 5607900391/5121518589937*a^3 - 1386251139825/5121518589937*a^2 - 1505249853248/5121518589937*a - 1476916335330/5121518589937

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

Trivial group, which has order $1$

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:	$11$	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:	$\frac{186694716619}{5121518589937}a^{17}-\frac{666676318234}{5121518589937}a^{16}+\frac{39210960531}{5121518589937}a^{15}+\frac{4131317845517}{10243037179874}a^{14}-\frac{4261268700386}{5121518589937}a^{13}+\frac{21754628550011}{10243037179874}a^{12}-\frac{13300903151094}{5121518589937}a^{11}-\frac{6951075150710}{5121518589937}a^{10}+\frac{20517274663622}{5121518589937}a^{9}-\frac{7430652889087}{5121518589937}a^{8}+\frac{13776096688246}{5121518589937}a^{7}-\frac{28458985512240}{5121518589937}a^{6}-\frac{19955522006912}{5121518589937}a^{5}+\frac{130577149675455}{10243037179874}a^{4}-\frac{1354017689352}{5121518589937}a^{3}-\frac{47995425879114}{5121518589937}a^{2}+\frac{10657037188140}{5121518589937}a+\frac{4412713783295}{5121518589937}$, $\frac{721717999241}{10243037179874}a^{17}-\frac{5607957431147}{20486074359748}a^{16}+\frac{1282706365441}{10243037179874}a^{15}+\frac{13632193499671}{20486074359748}a^{14}-\frac{9260582882097}{5121518589937}a^{13}+\frac{99308364781503}{20486074359748}a^{12}-\frac{36267215763225}{5121518589937}a^{11}+\frac{10586180186347}{10243037179874}a^{10}+\frac{29166098362520}{5121518589937}a^{9}-\frac{91400257698645}{20486074359748}a^{8}+\frac{40989575781814}{5121518589937}a^{7}-\frac{76298392452052}{5121518589937}a^{6}+\frac{347341589728}{5121518589937}a^{5}+\frac{215842402929329}{10243037179874}a^{4}-\frac{42744224135292}{5121518589937}a^{3}-\frac{48741486441528}{5121518589937}a^{2}+\frac{242274508540}{72134064647}a+\frac{1449363580001}{5121518589937}$, $\frac{2263526927635}{10243037179874}a^{17}-\frac{7080711007913}{10243037179874}a^{16}-\frac{3171408918749}{20486074359748}a^{15}+\frac{42470739961319}{20486074359748}a^{14}-\frac{43000419669083}{10243037179874}a^{13}+\frac{60408998167109}{5121518589937}a^{12}-\frac{254445374568669}{20486074359748}a^{11}-\frac{165699515240931}{20486074359748}a^{10}+\frac{304005856824863}{20486074359748}a^{9}-\frac{79266180446639}{20486074359748}a^{8}+\frac{93874661966591}{5121518589937}a^{7}-\frac{288456871914937}{10243037179874}a^{6}-\frac{255548932104259}{10243037179874}a^{5}+\frac{272345597523614}{5121518589937}a^{4}+\frac{60229539342507}{5121518589937}a^{3}-\frac{161929876895052}{5121518589937}a^{2}-\frac{1913098717047}{5121518589937}a+\frac{18263557701083}{5121518589937}$, $\frac{823676959035}{10243037179874}a^{17}-\frac{1102896318880}{5121518589937}a^{16}-\frac{1876610900731}{10243037179874}a^{15}+\frac{15640242119215}{20486074359748}a^{14}-\frac{12121513973387}{10243037179874}a^{13}+\frac{36219611873329}{10243037179874}a^{12}-\frac{11724227975853}{5121518589937}a^{11}-\frac{118119126637815}{20486074359748}a^{10}+\frac{24552020538485}{5121518589937}a^{9}+\frac{19147585145615}{20486074359748}a^{8}+\frac{52575983000815}{10243037179874}a^{7}-\frac{25191537501532}{5121518589937}a^{6}-\frac{169633474588559}{10243037179874}a^{5}+\frac{90750997078180}{5121518589937}a^{4}+\frac{67093460126688}{5121518589937}a^{3}-\frac{59743236854259}{5121518589937}a^{2}-\frac{138525892605}{72134064647}a-\frac{319607751041}{5121518589937}$, $\frac{902613271119}{10243037179874}a^{17}-\frac{1197544297735}{5121518589937}a^{16}-\frac{1867443384515}{10243037179874}a^{15}+\frac{15877329814679}{20486074359748}a^{14}-\frac{6701743819723}{5121518589937}a^{13}+\frac{20365692764794}{5121518589937}a^{12}-\frac{14579412521538}{5121518589937}a^{11}-\frac{105510821623539}{20486074359748}a^{10}+\frac{42723526305533}{10243037179874}a^{9}+\frac{16347470533065}{20486074359748}a^{8}+\frac{33883797653730}{5121518589937}a^{7}-\frac{78912180589933}{10243037179874}a^{6}-\frac{145193517538681}{10243037179874}a^{5}+\frac{79248944136328}{5121518589937}a^{4}+\frac{71056465694550}{5121518589937}a^{3}-\frac{52622210259583}{5121518589937}a^{2}-\frac{24713168178805}{5121518589937}a+\frac{11119371571450}{5121518589937}$, $\frac{1117869480721}{20486074359748}a^{17}-\frac{3510012999143}{20486074359748}a^{16}-\frac{1034295229849}{20486074359748}a^{15}+\frac{11492853180983}{20486074359748}a^{14}-\frac{21827932060725}{20486074359748}a^{13}+\frac{58189239874137}{20486074359748}a^{12}-\frac{14237174704998}{5121518589937}a^{11}-\frac{29617947723905}{10243037179874}a^{10}+\frac{103703983852115}{20486074359748}a^{9}-\frac{37155141378641}{20486074359748}a^{8}+\frac{22931020781171}{5121518589937}a^{7}-\frac{69660615076463}{10243037179874}a^{6}-\frac{36847397518610}{5121518589937}a^{5}+\frac{163993476969739}{10243037179874}a^{4}+\frac{11174034323901}{5121518589937}a^{3}-\frac{51140272018043}{5121518589937}a^{2}+\frac{5384205901109}{5121518589937}a+\frac{8002692212497}{5121518589937}$, $\frac{1621779435653}{5121518589937}a^{17}-\frac{10576175249123}{10243037179874}a^{16}-\frac{2235386230821}{20486074359748}a^{15}+\frac{61659594132027}{20486074359748}a^{14}-\frac{64692420301665}{10243037179874}a^{13}+\frac{90746010001518}{5121518589937}a^{12}-\frac{409775844142225}{20486074359748}a^{11}-\frac{198026193524317}{20486074359748}a^{10}+\frac{456182518047317}{20486074359748}a^{9}-\frac{169484759111863}{20486074359748}a^{8}+\frac{151867904322203}{5121518589937}a^{7}-\frac{233484245917350}{5121518589937}a^{6}-\frac{157647656907163}{5121518589937}a^{5}+\frac{820495143366953}{10243037179874}a^{4}+\frac{48080757190422}{5121518589937}a^{3}-\frac{218151365235199}{5121518589937}a^{2}+\frac{7288323249139}{5121518589937}a+\frac{14391426731094}{5121518589937}$, $\frac{1277649435701}{20486074359748}a^{17}-\frac{5286498165065}{20486074359748}a^{16}+\frac{3367579216031}{20486074359748}a^{15}+\frac{11701753069045}{20486074359748}a^{14}-\frac{35310176416151}{20486074359748}a^{13}+\frac{95703319990757}{20486074359748}a^{12}-\frac{74459995721905}{10243037179874}a^{11}+\frac{22982347386925}{10243037179874}a^{10}+\frac{98363515264093}{20486074359748}a^{9}-\frac{102354344986809}{20486074359748}a^{8}+\frac{42229296312802}{5121518589937}a^{7}-\frac{76645734041780}{5121518589937}a^{6}+\frac{27349527995103}{10243037179874}a^{5}+\frac{191629715315281}{10243037179874}a^{4}-\frac{57513921626972}{5121518589937}a^{3}-\frac{28413640587336}{5121518589937}a^{2}+\frac{26647389110204}{5121518589937}a-\frac{6559027006900}{5121518589937}$, $\frac{1612509661146}{5121518589937}a^{17}-\frac{4979584979193}{5121518589937}a^{16}-\frac{5780499095969}{20486074359748}a^{15}+\frac{30436813937095}{10243037179874}a^{14}-\frac{29626896250144}{5121518589937}a^{13}+\frac{84657003082137}{5121518589937}a^{12}-\frac{4848391839373}{288536258588}a^{11}-\frac{133069273886461}{10243037179874}a^{10}+\frac{5931286807939}{288536258588}a^{9}-\frac{22456910074943}{5121518589937}a^{8}+\frac{275519631566177}{10243037179874}a^{7}-\frac{198340676051795}{5121518589937}a^{6}-\frac{199111920255529}{5121518589937}a^{5}+\frac{768539730986371}{10243037179874}a^{4}+\frac{110855922032552}{5121518589937}a^{3}-\frac{213753485843069}{5121518589937}a^{2}-\frac{20775734673829}{5121518589937}a+\frac{20032915506927}{5121518589937}$, $\frac{1016979089697}{5121518589937}a^{17}-\frac{6244829746065}{10243037179874}a^{16}-\frac{3300182003719}{20486074359748}a^{15}+\frac{36860128249977}{20486074359748}a^{14}-\frac{37365645210819}{10243037179874}a^{13}+\frac{54064417876723}{5121518589937}a^{12}-\frac{223367462544523}{20486074359748}a^{11}-\frac{144315120336023}{20486074359748}a^{10}+\frac{237885413954443}{20486074359748}a^{9}-\frac{65363208063305}{20486074359748}a^{8}+\frac{93455806481798}{5121518589937}a^{7}-\frac{126843060271786}{5121518589937}a^{6}-\frac{113909660529937}{5121518589937}a^{5}+\frac{440754628047025}{10243037179874}a^{4}+\frac{59290766231003}{5121518589937}a^{3}-\frac{118478794963380}{5121518589937}a^{2}-\frac{7739222209014}{5121518589937}a+\frac{6984426435677}{5121518589937}$, $\frac{2839740988085}{10243037179874}a^{17}-\frac{15603557695817}{20486074359748}a^{16}-\frac{10462823168117}{20486074359748}a^{15}+\frac{51485447988239}{20486074359748}a^{14}-\frac{44967022768515}{10243037179874}a^{13}+\frac{265522858916571}{20486074359748}a^{12}-\frac{204398278908999}{20486074359748}a^{11}-\frac{164040984120073}{10243037179874}a^{10}+\frac{325792827668685}{20486074359748}a^{9}-\frac{21328308810137}{20486074359748}a^{8}+\frac{233179463042465}{10243037179874}a^{7}-\frac{131504881622410}{5121518589937}a^{6}-\frac{458369857786085}{10243037179874}a^{5}+\frac{578373459626959}{10243037179874}a^{4}+\frac{168097579546361}{5121518589937}a^{3}-\frac{157477539535544}{5121518589937}a^{2}-\frac{45046555276609}{5121518589937}a+\frac{9320995574032}{5121518589937}$ 186694716619/5121518589937a^17 - 666676318234/5121518589937a^16 + 39210960531/5121518589937a^15 + 4131317845517/10243037179874a^14 - 4261268700386/5121518589937a^13 + 21754628550011/10243037179874a^12 - 13300903151094/5121518589937a^11 - 6951075150710/5121518589937a^10 + 20517274663622/5121518589937a^9 - 7430652889087/5121518589937a^8 + 13776096688246/5121518589937a^7 - 28458985512240/5121518589937a^6 - 19955522006912/5121518589937a^5 + 130577149675455/10243037179874a^4 - 1354017689352/5121518589937a^3 - 47995425879114/5121518589937a^2 + 10657037188140/5121518589937a + 4412713783295/5121518589937, 721717999241/10243037179874a^17 - 5607957431147/20486074359748a^16 + 1282706365441/10243037179874a^15 + 13632193499671/20486074359748a^14 - 9260582882097/5121518589937a^13 + 99308364781503/20486074359748a^12 - 36267215763225/5121518589937a^11 + 10586180186347/10243037179874a^10 + 29166098362520/5121518589937a^9 - 91400257698645/20486074359748a^8 + 40989575781814/5121518589937a^7 - 76298392452052/5121518589937a^6 + 347341589728/5121518589937a^5 + 215842402929329/10243037179874a^4 - 42744224135292/5121518589937a^3 - 48741486441528/5121518589937a^2 + 242274508540/72134064647a + 1449363580001/5121518589937, 2263526927635/10243037179874a^17 - 7080711007913/10243037179874a^16 - 3171408918749/20486074359748a^15 + 42470739961319/20486074359748a^14 - 43000419669083/10243037179874a^13 + 60408998167109/5121518589937a^12 - 254445374568669/20486074359748a^11 - 165699515240931/20486074359748a^10 + 304005856824863/20486074359748a^9 - 79266180446639/20486074359748a^8 + 93874661966591/5121518589937a^7 - 288456871914937/10243037179874a^6 - 255548932104259/10243037179874a^5 + 272345597523614/5121518589937a^4 + 60229539342507/5121518589937a^3 - 161929876895052/5121518589937a^2 - 1913098717047/5121518589937a + 18263557701083/5121518589937, 823676959035/10243037179874a^17 - 1102896318880/5121518589937a^16 - 1876610900731/10243037179874a^15 + 15640242119215/20486074359748a^14 - 12121513973387/10243037179874a^13 + 36219611873329/10243037179874a^12 - 11724227975853/5121518589937a^11 - 118119126637815/20486074359748a^10 + 24552020538485/5121518589937a^9 + 19147585145615/20486074359748a^8 + 52575983000815/10243037179874a^7 - 25191537501532/5121518589937a^6 - 169633474588559/10243037179874a^5 + 90750997078180/5121518589937a^4 + 67093460126688/5121518589937a^3 - 59743236854259/5121518589937a^2 - 138525892605/72134064647a - 319607751041/5121518589937, 902613271119/10243037179874a^17 - 1197544297735/5121518589937a^16 - 1867443384515/10243037179874a^15 + 15877329814679/20486074359748a^14 - 6701743819723/5121518589937a^13 + 20365692764794/5121518589937a^12 - 14579412521538/5121518589937a^11 - 105510821623539/20486074359748a^10 + 42723526305533/10243037179874a^9 + 16347470533065/20486074359748a^8 + 33883797653730/5121518589937a^7 - 78912180589933/10243037179874a^6 - 145193517538681/10243037179874a^5 + 79248944136328/5121518589937a^4 + 71056465694550/5121518589937a^3 - 52622210259583/5121518589937a^2 - 24713168178805/5121518589937a + 11119371571450/5121518589937, 1117869480721/20486074359748a^17 - 3510012999143/20486074359748a^16 - 1034295229849/20486074359748a^15 + 11492853180983/20486074359748a^14 - 21827932060725/20486074359748a^13 + 58189239874137/20486074359748a^12 - 14237174704998/5121518589937a^11 - 29617947723905/10243037179874a^10 + 103703983852115/20486074359748a^9 - 37155141378641/20486074359748a^8 + 22931020781171/5121518589937a^7 - 69660615076463/10243037179874a^6 - 36847397518610/5121518589937a^5 + 163993476969739/10243037179874a^4 + 11174034323901/5121518589937a^3 - 51140272018043/5121518589937a^2 + 5384205901109/5121518589937a + 8002692212497/5121518589937, 1621779435653/5121518589937a^17 - 10576175249123/10243037179874a^16 - 2235386230821/20486074359748a^15 + 61659594132027/20486074359748a^14 - 64692420301665/10243037179874a^13 + 90746010001518/5121518589937a^12 - 409775844142225/20486074359748a^11 - 198026193524317/20486074359748a^10 + 456182518047317/20486074359748a^9 - 169484759111863/20486074359748a^8 + 151867904322203/5121518589937a^7 - 233484245917350/5121518589937a^6 - 157647656907163/5121518589937a^5 + 820495143366953/10243037179874a^4 + 48080757190422/5121518589937a^3 - 218151365235199/5121518589937a^2 + 7288323249139/5121518589937a + 14391426731094/5121518589937, 1277649435701/20486074359748a^17 - 5286498165065/20486074359748a^16 + 3367579216031/20486074359748a^15 + 11701753069045/20486074359748a^14 - 35310176416151/20486074359748a^13 + 95703319990757/20486074359748a^12 - 74459995721905/10243037179874a^11 + 22982347386925/10243037179874a^10 + 98363515264093/20486074359748a^9 - 102354344986809/20486074359748a^8 + 42229296312802/5121518589937a^7 - 76645734041780/5121518589937a^6 + 27349527995103/10243037179874a^5 + 191629715315281/10243037179874a^4 - 57513921626972/5121518589937a^3 - 28413640587336/5121518589937a^2 + 26647389110204/5121518589937a - 6559027006900/5121518589937, 1612509661146/5121518589937a^17 - 4979584979193/5121518589937a^16 - 5780499095969/20486074359748a^15 + 30436813937095/10243037179874a^14 - 29626896250144/5121518589937a^13 + 84657003082137/5121518589937a^12 - 4848391839373/288536258588a^11 - 133069273886461/10243037179874a^10 + 5931286807939/288536258588a^9 - 22456910074943/5121518589937a^8 + 275519631566177/10243037179874a^7 - 198340676051795/5121518589937a^6 - 199111920255529/5121518589937a^5 + 768539730986371/10243037179874a^4 + 110855922032552/5121518589937a^3 - 213753485843069/5121518589937a^2 - 20775734673829/5121518589937a + 20032915506927/5121518589937, 1016979089697/5121518589937a^17 - 6244829746065/10243037179874a^16 - 3300182003719/20486074359748a^15 + 36860128249977/20486074359748a^14 - 37365645210819/10243037179874a^13 + 54064417876723/5121518589937a^12 - 223367462544523/20486074359748a^11 - 144315120336023/20486074359748a^10 + 237885413954443/20486074359748a^9 - 65363208063305/20486074359748a^8 + 93455806481798/5121518589937a^7 - 126843060271786/5121518589937a^6 - 113909660529937/5121518589937a^5 + 440754628047025/10243037179874a^4 + 59290766231003/5121518589937a^3 - 118478794963380/5121518589937a^2 - 7739222209014/5121518589937a + 6984426435677/5121518589937, 2839740988085/10243037179874a^17 - 15603557695817/20486074359748a^16 - 10462823168117/20486074359748a^15 + 51485447988239/20486074359748a^14 - 44967022768515/10243037179874a^13 + 265522858916571/20486074359748a^12 - 204398278908999/20486074359748a^11 - 164040984120073/10243037179874a^10 + 325792827668685/20486074359748a^9 - 21328308810137/20486074359748a^8 + 233179463042465/10243037179874a^7 - 131504881622410/5121518589937a^6 - 458369857786085/10243037179874a^5 + 578373459626959/10243037179874a^4 + 168097579546361/5121518589937a^3 - 157477539535544/5121518589937a^2 - 45046555276609/5121518589937*a + 9320995574032/5121518589937	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:	$ 26652.5198107 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 26652.5198107 \cdot 1}{2\cdot\sqrt{60874624883546499579904}}\cr\approx \mathstrut & 0.212691098492 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8)

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 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8, 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 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8);

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 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8);

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

$C_3^2:C_{12}$ (as 18T44):

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 108

The 18 conjugacy class representatives for $C_3^2:C_{12}$

Character table for $C_3^2:C_{12}$

Intermediate fields

$\Q(\sqrt{2}) $, $\Q(\zeta_{7})^+$, 6.6.1229312.1, 6.2.802816.1

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 12 siblings:	data not computed
Degree 18 sibling:	data not computed
Degree 27 sibling:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	12.0.20624432955392.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$	${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$	R	${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$	${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$	${\href{/padicField/17.3.0.1}{3} }^{6}$	${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }$	${\href{/padicField/23.3.0.1}{3} }^{6}$	${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$	${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$	${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$	${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$	${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$	${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$	${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$

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	2.6.9.1	$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
$2$ 2	2.12.33.344	$x^{12} + 56 x^{10} - 392 x^{9} + 226 x^{8} - 640 x^{7} - 2480 x^{6} + 3968 x^{5} + 1276 x^{4} - 384 x^{3} + 9280 x^{2} + 24224 x + 31544$	$4$	$3$	$33$	$C_{12}$	$[3, 4]^{3}$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$7$ 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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