Properties

Label 16.8.918...041.1
Degree $16$
Signature $[8, 4]$
Discriminant $9.189\times 10^{46}$
Root discriminant $861.40$
Ramified primes $41, 79$
Class number $1$ (GRH)
Class group trivial (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 - 4*x^15 - 13*x^14 + 3302*x^13 - 404406*x^12 + 658728*x^11 - 23802748*x^10 + 148149418*x^9 + 3248725095*x^8 - 13221233432*x^7 + 23779425890*x^6 + 491265567999*x^5 - 4182022415068*x^4 - 12864900236774*x^3 + 35128190290939*x^2 + 189418201227147*x + 234513710016293)
 
gp: K = bnfinit(x^16 - 4*x^15 - 13*x^14 + 3302*x^13 - 404406*x^12 + 658728*x^11 - 23802748*x^10 + 148149418*x^9 + 3248725095*x^8 - 13221233432*x^7 + 23779425890*x^6 + 491265567999*x^5 - 4182022415068*x^4 - 12864900236774*x^3 + 35128190290939*x^2 + 189418201227147*x + 234513710016293, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![234513710016293, 189418201227147, 35128190290939, -12864900236774, -4182022415068, 491265567999, 23779425890, -13221233432, 3248725095, 148149418, -23802748, 658728, -404406, 3302, -13, -4, 1]);
 

\( x^{16} - 4 x^{15} - 13 x^{14} + 3302 x^{13} - 404406 x^{12} + 658728 x^{11} - 23802748 x^{10} + 148149418 x^{9} + 3248725095 x^{8} - 13221233432 x^{7} + 23779425890 x^{6} + 491265567999 x^{5} - 4182022415068 x^{4} - 12864900236774 x^{3} + 35128190290939 x^{2} + 189418201227147 x + 234513710016293 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(91893109235660349920453207629584126297211854041\)\(\medspace = 41^{15}\cdot 79^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $861.40$
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:  $41, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{42149} a^{14} + \frac{15721}{42149} a^{13} - \frac{107}{42149} a^{12} - \frac{4259}{42149} a^{11} + \frac{5581}{42149} a^{10} - \frac{19290}{42149} a^{9} - \frac{17996}{42149} a^{8} - \frac{13432}{42149} a^{7} + \frac{17039}{42149} a^{6} - \frac{19043}{42149} a^{5} + \frac{4898}{42149} a^{4} - \frac{4575}{42149} a^{3} + \frac{14562}{42149} a^{2} - \frac{13300}{42149} a - \frac{13772}{42149}$, $\frac{1}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{15} + \frac{37070953533966561728975664610391933940185709104870875219056337497167223103513231794932928884570937272}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{14} + \frac{91678416927303674605738296942158574777914120829461715995654101457657352856412209048352015118146361706084286}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{13} - \frac{56111800643725958240320095755898347403569025243907153353877573562843495396801397222813705998242104365028652}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{12} - \frac{55998513253392395654402464119027364933502058762467995240760437203677745253920034865430927102128257688296599}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{11} - \frac{66045897342602720913566668390907689105737699123008095169003148723752185894985349525560675428932957957207102}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{10} + \frac{36402527281145707645078524600884845311886544127888172843343218077885115702849574330506943480410050745741979}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{9} + \frac{15271202128392920155470278864782160269360015888782140187847559845980371874705602639660645013418780414286355}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{8} + \frac{71028306238928706163200199826185694534844193001289425866442462311256828338120720609645518215689468710653030}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{7} + \frac{99512445007510222657739153198525984983227091621198610313338670837072120559521934629140397483443512114313335}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{6} - \frac{40316424058015411923230103035085158860417553912347167882962670287768585156441256282167919775851564232543556}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{5} + \frac{25531612702467465018812647208049061193006251096582553983326218328334755133474718243553852325100938392435838}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{4} - \frac{87237900728240415047109036314825454946557479542036623473419353975850658143904381878957329994688940413968176}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{3} - \frac{69594735173719148159380356053559473284197111448056986974924254804430512851652438344297236649864775493992424}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a^{2} - \frac{96693467740535313462407330114156612021183994914182422193637057812419916314823001635912201417949917101655756}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443} a - \frac{29519156216196940085161353464670244256132958347749110123035707943695899748958737751994493477301556677611163}{216551214318960148149370730099655097303082922795033124977672053569466848339668029964074706388800637021746443}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 230175324509000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 230175324509000000 \cdot 1}{2\sqrt{91893109235660349920453207629584126297211854041}}\approx 0.151477013501370$ (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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.7585694742761134361.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
41Data not computed
79Data not computed