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Properties

Label 16.0.15359445630...5041.6
Degree $16$
Signature $[0, 8]$
Discriminant $41^{15}\cdot 61^{14}$
Root discriminant $1186.17$
Ramified primes $41, 61$
Class number $47900560$ (GRH)
Class group $[4, 11975140]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1463534766472091, -3706153303430478, 2766467958123332, -276270386009564, 82741220348795, 12621694069014, 2107575937554, 243389781077, 20144084946, 1241998870, 26545517, 89446, 204662, 8167, -466, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 466*x^14 + 8167*x^13 + 204662*x^12 + 89446*x^11 + 26545517*x^10 + 1241998870*x^9 + 20144084946*x^8 + 243389781077*x^7 + 2107575937554*x^6 + 12621694069014*x^5 + 82741220348795*x^4 - 276270386009564*x^3 + 2766467958123332*x^2 - 3706153303430478*x + 1463534766472091)
 
gp: K = bnfinit(x^16 - 3*x^15 - 466*x^14 + 8167*x^13 + 204662*x^12 + 89446*x^11 + 26545517*x^10 + 1241998870*x^9 + 20144084946*x^8 + 243389781077*x^7 + 2107575937554*x^6 + 12621694069014*x^5 + 82741220348795*x^4 - 276270386009564*x^3 + 2766467958123332*x^2 - 3706153303430478*x + 1463534766472091, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 466 x^{14} + 8167 x^{13} + 204662 x^{12} + 89446 x^{11} + 26545517 x^{10} + 1241998870 x^{9} + 20144084946 x^{8} + 243389781077 x^{7} + 2107575937554 x^{6} + 12621694069014 x^{5} + 82741220348795 x^{4} - 276270386009564 x^{3} + 2766467958123332 x^{2} - 3706153303430478 x + 1463534766472091 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15359445630083788418852698524374845584773320185041=41^{15}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1186.17$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $41, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{332} a^{14} - \frac{13}{166} a^{13} + \frac{27}{332} a^{12} + \frac{67}{332} a^{11} + \frac{45}{166} a^{10} - \frac{3}{332} a^{9} + \frac{123}{332} a^{8} - \frac{10}{83} a^{7} + \frac{9}{332} a^{6} - \frac{163}{332} a^{5} + \frac{47}{166} a^{4} - \frac{59}{332} a^{3} + \frac{123}{332} a^{2} + \frac{30}{83} a + \frac{93}{332}$, $\frac{1}{1562434085343895708142186231710311413775008462664620030192146009697358632343420205889960920454894220559641945895677164} a^{15} - \frac{4986570817473102654333981116007831348978989016343576453901393449617444217938914840120013581166298952121998510519}{7301093856747176206271898279020146793341161040488878645757691634099806693193552363971780002125673927848794139699426} a^{14} + \frac{22209451896907243876545286330781270111882993347612629789414731584741793731152866617557020165421267804036194939716617}{781217042671947854071093115855155706887504231332310015096073004848679316171710102944980460227447110279820972947838582} a^{13} - \frac{48335629491348051290384827217011021954854105230932370719906875078556991914909811557110139269584594573328250843774405}{390608521335973927035546557927577853443752115666155007548036502424339658085855051472490230113723555139910486473919291} a^{12} - \frac{94047030633274889189572381186656809621351050027565408522807683445892047339659963998377737444692603917034231524045470}{390608521335973927035546557927577853443752115666155007548036502424339658085855051472490230113723555139910486473919291} a^{11} - \frac{163592687632907292196875097330214364306352417551099778178591104063181292618598686947978056648282400406758510937267347}{390608521335973927035546557927577853443752115666155007548036502424339658085855051472490230113723555139910486473919291} a^{10} - \frac{187045426287489839613172609131017336554392291879125996988795379493475705635346780534122238907226438650547142091720988}{390608521335973927035546557927577853443752115666155007548036502424339658085855051472490230113723555139910486473919291} a^{9} - \frac{189095934783234910131478447990457696583483652000217730110345418066226036779514308347213421375348876310363047691345194}{390608521335973927035546557927577853443752115666155007548036502424339658085855051472490230113723555139910486473919291} a^{8} + \frac{342639113472820329821362680514425769168047739892788387769806807741806000290771015707029688557839372445663923767141925}{781217042671947854071093115855155706887504231332310015096073004848679316171710102944980460227447110279820972947838582} a^{7} - \frac{386531567463741609911357247216519168416880573663234209687502710211337477813673944855633346129321923311027382285178021}{781217042671947854071093115855155706887504231332310015096073004848679316171710102944980460227447110279820972947838582} a^{6} + \frac{227537556588901180955290194379059407415094918890463726063295502888394365124588820315253249415884177719028927276487241}{781217042671947854071093115855155706887504231332310015096073004848679316171710102944980460227447110279820972947838582} a^{5} + \frac{119751467921344892042150199425761629680124907335232058711562897755082255523834695628043725329807811878667012115596}{251518687273647087595329399824583292623150106674922735059907599758106669726886704103342067040388638209858651947147} a^{4} + \frac{160559693932793321791092413019040715650939891741701584060325166027592164392396920839078208526640640087609731379073073}{390608521335973927035546557927577853443752115666155007548036502424339658085855051472490230113723555139910486473919291} a^{3} - \frac{142355401396016588014429654856564152915472885513066370485139269611872196942779874565641623900388422080383776395978450}{390608521335973927035546557927577853443752115666155007548036502424339658085855051472490230113723555139910486473919291} a^{2} + \frac{345967072681059809875424231991950947556312476179976673437755643009611030269533599677982224399082072365552066676248661}{781217042671947854071093115855155706887504231332310015096073004848679316171710102944980460227447110279820972947838582} a + \frac{757827739032354820453523633333319588695908827958494959412986282455815736834311187265339289383612350609710141197933861}{1562434085343895708142186231710311413775008462664620030192146009697358632343420205889960920454894220559641945895677164}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}\times C_{11975140}$, which has order $47900560$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16603475989.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.256455041.2, 8.8.10033813098753844365041.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
41Data not computed
$61$61.8.7.3$x^{8} + 122$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.3$x^{8} + 122$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$