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Properties

Label 16.0.79100591003...0881.5
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 149^{14}$
Root discriminant $1517.53$
Ramified primes $29, 149$
Class number $11734105680$ (GRH)
Class group $[2, 24222, 242220]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![414401005855515456, 9966977583135840, 4213301148768960, -1395404638505784, 158678336760608, -16152690635942, 3069805875379, -378903900861, 20065856528, -763954787, 42103059, -1712558, 85309, -7911, 404, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 404*x^14 - 7911*x^13 + 85309*x^12 - 1712558*x^11 + 42103059*x^10 - 763954787*x^9 + 20065856528*x^8 - 378903900861*x^7 + 3069805875379*x^6 - 16152690635942*x^5 + 158678336760608*x^4 - 1395404638505784*x^3 + 4213301148768960*x^2 + 9966977583135840*x + 414401005855515456)
 
gp: K = bnfinit(x^16 - 5*x^15 + 404*x^14 - 7911*x^13 + 85309*x^12 - 1712558*x^11 + 42103059*x^10 - 763954787*x^9 + 20065856528*x^8 - 378903900861*x^7 + 3069805875379*x^6 - 16152690635942*x^5 + 158678336760608*x^4 - 1395404638505784*x^3 + 4213301148768960*x^2 + 9966977583135840*x + 414401005855515456, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 404 x^{14} - 7911 x^{13} + 85309 x^{12} - 1712558 x^{11} + 42103059 x^{10} - 763954787 x^{9} + 20065856528 x^{8} - 378903900861 x^{7} + 3069805875379 x^{6} - 16152690635942 x^{5} + 158678336760608 x^{4} - 1395404638505784 x^{3} + 4213301148768960 x^{2} + 9966977583135840 x + 414401005855515456 \)

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:  \(791005910030165719912925444172568170567213001250881=29^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1517.53$
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:  $29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} + \frac{1}{12} a^{6} + \frac{1}{12} a^{4} + \frac{3}{8} a^{3} - \frac{7}{24} a^{2} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{24} a^{4} + \frac{5}{12} a^{3} - \frac{5}{24} a^{2} + \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{6} a^{6} - \frac{5}{24} a^{5} - \frac{1}{6} a^{4} + \frac{5}{24} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{144} a^{12} - \frac{1}{72} a^{11} - \frac{1}{72} a^{10} - \frac{1}{144} a^{9} - \frac{1}{24} a^{8} + \frac{1}{18} a^{7} - \frac{1}{16} a^{6} + \frac{1}{72} a^{5} - \frac{5}{72} a^{4} - \frac{47}{144} a^{3} + \frac{7}{72} a^{2} + \frac{1}{12} a$, $\frac{1}{3168} a^{13} + \frac{1}{528} a^{12} + \frac{5}{528} a^{11} + \frac{5}{288} a^{10} - \frac{25}{1584} a^{9} + \frac{23}{792} a^{8} - \frac{233}{3168} a^{7} + \frac{301}{1584} a^{6} - \frac{1}{48} a^{5} + \frac{401}{3168} a^{4} - \frac{547}{1584} a^{3} - \frac{19}{99} a^{2} + \frac{13}{132} a + \frac{7}{22}$, $\frac{1}{519552} a^{14} - \frac{19}{519552} a^{13} + \frac{79}{86592} a^{12} + \frac{5905}{519552} a^{11} - \frac{481}{57728} a^{10} - \frac{127}{11808} a^{9} - \frac{10057}{519552} a^{8} - \frac{37397}{519552} a^{7} - \frac{50227}{259776} a^{6} + \frac{94451}{519552} a^{5} + \frac{99761}{519552} a^{4} + \frac{469}{5412} a^{3} + \frac{11237}{129888} a^{2} - \frac{1567}{3608} a - \frac{153}{3608}$, $\frac{1}{6545115644784426952647391396621170349044746122219265101342828099154670393075977209586530524616495270377216} a^{15} + \frac{5257846333634338515970728126407859271842465347671744900429589906147997745296598308923031094065513205}{6545115644784426952647391396621170349044746122219265101342828099154670393075977209586530524616495270377216} a^{14} - \frac{145929615208210181763907003537775897668444829184995751146900705405358162440447920996089324951507778943}{3272557822392213476323695698310585174522373061109632550671414049577335196537988604793265262308247635188608} a^{13} - \frac{19287467882954531464633743584732896808562121446032666261202371532403207405470595705882600506040806719}{80803896849190456205523350575570004309194396570608211127689235792032967815752805056623833637240682350336} a^{12} + \frac{131559915982953679614959703679820491850085220110798157235104801403561884528502246415738377726783864379439}{6545115644784426952647391396621170349044746122219265101342828099154670393075977209586530524616495270377216} a^{11} - \frac{1184124427385892918926639927918081369618169437043369608318829805832083330094348171852783917761638659489}{148752628290555158014713440832299326114653320959528752303246093162606145297181300217875693741283983417664} a^{10} + \frac{33762505058590518551769333063323383265563432248256330232213998511603196517869114240252607362384165186125}{2181705214928142317549130465540390116348248707406421700447609366384890131025325736528843508205498423459072} a^{9} - \frac{3398166523442679461183786296784892185121203468770125104111761224014201772587691049166839259625013753181}{6545115644784426952647391396621170349044746122219265101342828099154670393075977209586530524616495270377216} a^{8} + \frac{119281381651635610815921230883816858217287752200068163892613142564993801970268836932540161268445687857369}{3272557822392213476323695698310585174522373061109632550671414049577335196537988604793265262308247635188608} a^{7} - \frac{11793023165136592491418138535889282606326161655796209713588227694186394684354226907886361871782756894781}{80803896849190456205523350575570004309194396570608211127689235792032967815752805056623833637240682350336} a^{6} + \frac{523649321387361837198980301308721210052701503312802754085163019300090057827242753634403723705873635568345}{6545115644784426952647391396621170349044746122219265101342828099154670393075977209586530524616495270377216} a^{5} - \frac{59163611862542251583253472596235734561413538570136511917138963793746904401721756329512855543910548293737}{818139455598053369080923924577646293630593265277408137667853512394333799134497151198316315577061908797152} a^{4} + \frac{3553873608805497730387974050989448563148578593005598209049735207065054074547826693321707680100065234631}{20712391280963376432428453786775855534951728234871092092857050946691994914797396232868767482963592627776} a^{3} + \frac{424873593045742506158708620409838286407591759605411084604174008925980917129647333198345236209058101865}{1108590048235844673551387431677027498144435318804076067300614515439476692594169581569534302949948385904} a^{2} + \frac{698881842875147254022923329041502318821048334510673930111150065075936867406437668034651959585854259963}{45452191977669631615606884698758127423921848070967118759325195133018544396360952844350906420947883822064} a + \frac{151929039153929997943145807216374150044813716036602154032797755424864221510344302998450579266434617215}{631280444134300439105651176371640658665581223207876649435072154625257561060568789504873700290942830862}$

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

Class group and class number

$C_{2}\times C_{24222}\times C_{242220}$, which has order $11734105680$ (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:  \( 6427573726.31 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{149}) \), \(\Q(\sqrt{4321}) \), \(\Q(\sqrt{29}) \), 4.4.80677568161.2, 4.4.80677568161.1, \(\Q(\sqrt{29}, \sqrt{149})\), 8.0.28124827288894872783620641.2 x2, 8.0.28124827288894872783620641.1 x2, 8.8.6508870004372800921921.3

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$149$149.8.7.1$x^{8} - 149$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.1$x^{8} - 149$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$