Properties

Label 16.16.350...673.1
Degree $16$
Signature $[16, 0]$
Discriminant $3.508\times 10^{43}$
Root discriminant $526.70$
Ramified primes $13, 73$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 417*x^14 + 1025*x^13 + 63344*x^12 - 209215*x^11 - 4564647*x^10 + 17645096*x^9 + 164307949*x^8 - 710547742*x^7 - 2725760041*x^6 + 13536478427*x^5 + 12517302234*x^4 - 98492861490*x^3 + 80748956016*x^2 + 23338091769*x - 4383666169)
 
gp: K = bnfinit(x^16 - x^15 - 417*x^14 + 1025*x^13 + 63344*x^12 - 209215*x^11 - 4564647*x^10 + 17645096*x^9 + 164307949*x^8 - 710547742*x^7 - 2725760041*x^6 + 13536478427*x^5 + 12517302234*x^4 - 98492861490*x^3 + 80748956016*x^2 + 23338091769*x - 4383666169, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4383666169, 23338091769, 80748956016, -98492861490, 12517302234, 13536478427, -2725760041, -710547742, 164307949, 17645096, -4564647, -209215, 63344, 1025, -417, -1, 1]);
 

\( x^{16} - x^{15} - 417 x^{14} + 1025 x^{13} + 63344 x^{12} - 209215 x^{11} - 4564647 x^{10} + 17645096 x^{9} + 164307949 x^{8} - 710547742 x^{7} - 2725760041 x^{6} + 13536478427 x^{5} + 12517302234 x^{4} - 98492861490 x^{3} + 80748956016 x^{2} + 23338091769 x - 4383666169 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(35079212407887857597919870221838606532086673\)\(\medspace = 13^{14}\cdot 73^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $526.70$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 73$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} + \frac{3}{19} a^{12} + \frac{3}{19} a^{11} + \frac{7}{19} a^{10} - \frac{2}{19} a^{9} - \frac{6}{19} a^{8} - \frac{4}{19} a^{7} - \frac{3}{19} a^{6} + \frac{1}{19} a^{5} - \frac{5}{19} a^{4} - \frac{9}{19} a^{3} + \frac{2}{19} a^{2} + \frac{1}{19} a - \frac{8}{19}$, $\frac{1}{1843} a^{14} + \frac{32}{1843} a^{13} - \frac{385}{1843} a^{12} + \frac{75}{1843} a^{11} + \frac{182}{1843} a^{10} - \frac{843}{1843} a^{9} + \frac{259}{1843} a^{8} - \frac{328}{1843} a^{7} - \frac{523}{1843} a^{6} - \frac{755}{1843} a^{5} - \frac{648}{1843} a^{4} - \frac{791}{1843} a^{3} + \frac{249}{1843} a^{2} + \frac{914}{1843} a - \frac{251}{1843}$, $\frac{1}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{15} - \frac{2256734403702128011876052642986659931886680452247217678076417727432}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{14} - \frac{75411261339282247728421571758337066270133539690526582205696064252875}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{13} + \frac{7050925904439932944313271353016041914977279223713235865004141049425090}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{12} + \frac{1387716668998562261296528246502760799984055593820094058167687600390664}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{11} - \frac{285805674203463574354349146062341738584114918238331394239478891667937}{799663985800684946714986270604477041780082023380683386675427067508393} a^{10} - \frac{925143231519738512921854392567486073755398651323773416097018095316139}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{9} + \frac{1230300983754012705052026436699642876259832337989307727760295676338750}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{8} + \frac{3945941833591044406292972993611089116350403785829894170770291607625423}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{7} - \frac{3808358008447854081514058009056653859514559332645141774266164067026877}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{6} - \frac{4517215043997574123761322613822850161572860846368837678223531775882231}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{5} - \frac{4693132384868353498396459590748130040285545218536106176227581381718171}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{4} - \frac{2719252626451265385943137273551508263916162379173587320656751856153271}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{3} + \frac{6170425172310820516699011323675654674939065395732923668615416973821516}{15193615730213013987584739141485063793821558444232984346833114282659467} a^{2} - \frac{7254214174155475399071242754927601769045243474893517539594380783901267}{15193615730213013987584739141485063793821558444232984346833114282659467} a + \frac{6041851910796394993695167036538832049607043154948235275747728843150165}{15193615730213013987584739141485063793821558444232984346833114282659467}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 17231511043300000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 17231511043300000 \cdot 2}{2\sqrt{35079212407887857597919870221838606532086673}}\approx 0.190668248135673$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.65743873.1, 8.8.53323682598564071473.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ R $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
13Data not computed
73Data not computed