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Properties

Label 16.0.79443526569...081.12
Degree $16$
Signature $[0, 8]$
Discriminant $67^{8}\cdot 89^{14}$
Root discriminant $415.67$
Ramified primes $67, 89$
Class number $768$ (GRH)
Class group $[2, 8, 48]$ (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![240631555264, -231648299712, 116964971452, -34355491586, 15687125679, 2031622908, 1738428696, 340919290, 74703292, 8435040, 1126568, 16860, -1072, -530, -16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 16*x^14 - 530*x^13 - 1072*x^12 + 16860*x^11 + 1126568*x^10 + 8435040*x^9 + 74703292*x^8 + 340919290*x^7 + 1738428696*x^6 + 2031622908*x^5 + 15687125679*x^4 - 34355491586*x^3 + 116964971452*x^2 - 231648299712*x + 240631555264)
 
gp: K = bnfinit(x^16 - 6*x^15 - 16*x^14 - 530*x^13 - 1072*x^12 + 16860*x^11 + 1126568*x^10 + 8435040*x^9 + 74703292*x^8 + 340919290*x^7 + 1738428696*x^6 + 2031622908*x^5 + 15687125679*x^4 - 34355491586*x^3 + 116964971452*x^2 - 231648299712*x + 240631555264, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 16 x^{14} - 530 x^{13} - 1072 x^{12} + 16860 x^{11} + 1126568 x^{10} + 8435040 x^{9} + 74703292 x^{8} + 340919290 x^{7} + 1738428696 x^{6} + 2031622908 x^{5} + 15687125679 x^{4} - 34355491586 x^{3} + 116964971452 x^{2} - 231648299712 x + 240631555264 \)

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:  \(794435265691380646985793918947192534284081=67^{8}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $415.67$
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:  $67, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{268} a^{10} - \frac{4}{67} a^{9} - \frac{29}{268} a^{8} - \frac{21}{134} a^{7} - \frac{7}{67} a^{6} + \frac{4}{67} a^{5} - \frac{49}{268} a^{4} - \frac{6}{67} a^{3} - \frac{43}{268} a^{2} + \frac{61}{134} a + \frac{33}{67}$, $\frac{1}{268} a^{11} - \frac{17}{268} a^{9} + \frac{15}{134} a^{8} - \frac{15}{134} a^{7} - \frac{15}{134} a^{6} - \frac{61}{268} a^{5} - \frac{1}{67} a^{4} - \frac{25}{268} a^{3} - \frac{15}{134} a^{2} + \frac{37}{134} a - \frac{8}{67}$, $\frac{1}{1072} a^{12} + \frac{93}{1072} a^{9} - \frac{27}{536} a^{8} - \frac{13}{67} a^{7} + \frac{267}{1072} a^{6} - \frac{1}{4} a^{5} - \frac{47}{268} a^{4} - \frac{505}{1072} a^{3} + \frac{241}{536} a^{2} + \frac{75}{268} a + \frac{23}{67}$, $\frac{1}{1072} a^{13} + \frac{1}{1072} a^{10} + \frac{39}{536} a^{9} + \frac{3}{67} a^{8} - \frac{157}{1072} a^{7} + \frac{41}{268} a^{6} - \frac{13}{268} a^{5} + \frac{251}{1072} a^{4} - \frac{129}{536} a^{3} - \frac{75}{268} a^{2} + \frac{25}{67} a - \frac{22}{67}$, $\frac{1}{13543788657672080} a^{14} - \frac{507627061847}{1354378865767208} a^{13} + \frac{418853574971}{13543788657672080} a^{12} + \frac{23649822441881}{13543788657672080} a^{11} - \frac{100755474691}{3385947164418020} a^{10} - \frac{943609721059217}{13543788657672080} a^{9} + \frac{1260703649242933}{13543788657672080} a^{8} + \frac{159520280140237}{6771894328836040} a^{7} - \frac{3205017262186563}{13543788657672080} a^{6} + \frac{121625486910835}{2708757731534416} a^{5} + \frac{83705131091089}{677189432883604} a^{4} - \frac{3618346695115367}{13543788657672080} a^{3} - \frac{2078325943354439}{6771894328836040} a^{2} + \frac{1279130737711053}{3385947164418020} a - \frac{135021479148627}{846486791104505}$, $\frac{1}{12210961902381195002505292334696087783888514896044473154240} a^{15} - \frac{66712157677867980294552829002736795731821}{6105480951190597501252646167348043891944257448022236577120} a^{14} + \frac{458512818848609780130359895633119677809463699901092707}{1526370237797649375313161541837010972986064362005559144280} a^{13} - \frac{2014896016960920028218182933005574729586365776098727723}{6105480951190597501252646167348043891944257448022236577120} a^{12} - \frac{228380139622203613839414383120356636151341910597559791}{763185118898824687656580770918505486493032181002779572140} a^{11} + \frac{2392845127790247955475724412367648732144188310282499339}{3052740475595298750626323083674021945972128724011118288560} a^{10} - \frac{119637908741631769005897810539036899429242794517938948087}{3052740475595298750626323083674021945972128724011118288560} a^{9} + \frac{14569737065493842716515570035609108856392431558676026863}{763185118898824687656580770918505486493032181002779572140} a^{8} + \frac{571997403522350879983529313694541896172402796832003272371}{3052740475595298750626323083674021945972128724011118288560} a^{7} + \frac{1167449227220652901098576533256173787934060097840056131183}{6105480951190597501252646167348043891944257448022236577120} a^{6} + \frac{40597856244704290554724485323902549067941134109357614425}{305274047559529875062632308367402194597212872401111828856} a^{5} - \frac{293821465624818585695091125128485885342300291515449250593}{3052740475595298750626323083674021945972128724011118288560} a^{4} - \frac{4793258850916267625081615774503015332410737691325472891389}{12210961902381195002505292334696087783888514896044473154240} a^{3} + \frac{2057904558467756957510186901819580586754119354860647597209}{6105480951190597501252646167348043891944257448022236577120} a^{2} - \frac{43886637111177591630520641037122298373926261789435463459}{3052740475595298750626323083674021945972128724011118288560} a - \frac{292495764066132499849375002308106549904274702670882813851}{763185118898824687656580770918505486493032181002779572140}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{48}$, which has order $768$ (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:  \( 2440693464320 \) (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

\(\Q(\sqrt{89}) \), \(\Q(\sqrt{-5963}) \), \(\Q(\sqrt{-67}) \), 4.4.704969.1, 4.0.3164605841.2, \(\Q(\sqrt{-67}, \sqrt{89})\), 8.4.198554462346029681.1 x2, 8.0.891310981471327238009.1 x2, 8.0.10014730128891317281.2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$