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Properties

Label 16.8.12746025680...9401.2
Degree $16$
Signature $[8, 4]$
Discriminant $31^{10}\cdot 41^{15}$
Root discriminant $278.03$
Ramified primes $31, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![118357793, 35090087210, -10350960621, 4277390699, 3143223321, -2264266519, -416803896, 177151636, 27762331, -5440409, -1157237, 94770, 22314, -475, -234, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 234*x^14 - 475*x^13 + 22314*x^12 + 94770*x^11 - 1157237*x^10 - 5440409*x^9 + 27762331*x^8 + 177151636*x^7 - 416803896*x^6 - 2264266519*x^5 + 3143223321*x^4 + 4277390699*x^3 - 10350960621*x^2 + 35090087210*x + 118357793)
 
gp: K = bnfinit(x^16 - x^15 - 234*x^14 - 475*x^13 + 22314*x^12 + 94770*x^11 - 1157237*x^10 - 5440409*x^9 + 27762331*x^8 + 177151636*x^7 - 416803896*x^6 - 2264266519*x^5 + 3143223321*x^4 + 4277390699*x^3 - 10350960621*x^2 + 35090087210*x + 118357793, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 234 x^{14} - 475 x^{13} + 22314 x^{12} + 94770 x^{11} - 1157237 x^{10} - 5440409 x^{9} + 27762331 x^{8} + 177151636 x^{7} - 416803896 x^{6} - 2264266519 x^{5} + 3143223321 x^{4} + 4277390699 x^{3} - 10350960621 x^{2} + 35090087210 x + 118357793 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1274602568003244304847660149105336749401=31^{10}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $278.03$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{15} - \frac{53657500123803020189389115540360491153068551499283240729657779393202403611942432}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{14} + \frac{9691515251101957337484318617065526241090719964137647132634235153960042517958854}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{13} - \frac{13670330115646152539354649504610782842812975095868179059236457133651806331636812}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{12} + \frac{243891621397386298898773792257444546935834633723586460059860925643951251832290660}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{11} - \frac{209281757370219607153853881571663712518874210678749726140766073298227209794411514}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{10} - \frac{371425932794360534065414374318116195901614258518877498179896674232763561337032155}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{9} - \frac{45025838399030309929217248131666609607630109226427490865059764607212678157430280}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{8} + \frac{55056455434950310085094902364602126104191464353145787232381829321580786681882340}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{7} + \frac{289983906504908961865503629694568827127259875821988224295663754408244456205073097}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{6} - \frac{278693307240384117663619878494493953946908406016764675822892493895755140846979591}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{5} + \frac{231258780985874981410533907899605797202557656164111745327478139023253252595031533}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{4} - \frac{306291449966983862853240045274899881868799891375025312683812563552670509152502728}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{3} + \frac{70553925840530422180544926450793314398961029256022023058499311416958424943119575}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a^{2} - \frac{302604357349686249963343488450320222961561655256128349415717905557481848174140968}{803463874843428877688260598403430517966146584550529710417492259284464647308503697} a + \frac{15315272722741943232570276113328783883541980927702377514948898058209976031793474}{34933211949714299029924373843627413824615068893501291757282272142802810752543639}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 17546247672000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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

Sibling fields

Degree 16 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
31Data not computed
41Data not computed