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Properties

Label 16.0.25714283968...2113.1
Degree $16$
Signature $[0, 8]$
Discriminant $67^{8}\cdot 97^{15}$
Root discriminant $596.53$
Ramified primes $67, 97$
Class number $1073881378$ (GRH)
Class group $[1073881378]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16052774378388739, -7283791516402294, 2817137139500788, -486125527533824, 125839435016506, -11993916195144, 2679277607936, -149369605348, 31950552532, -1005651829, 220867782, -3460136, 851534, -4849, 1604, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 1604*x^14 - 4849*x^13 + 851534*x^12 - 3460136*x^11 + 220867782*x^10 - 1005651829*x^9 + 31950552532*x^8 - 149369605348*x^7 + 2679277607936*x^6 - 11993916195144*x^5 + 125839435016506*x^4 - 486125527533824*x^3 + 2817137139500788*x^2 - 7283791516402294*x + 16052774378388739)
 
gp: K = bnfinit(x^16 - x^15 + 1604*x^14 - 4849*x^13 + 851534*x^12 - 3460136*x^11 + 220867782*x^10 - 1005651829*x^9 + 31950552532*x^8 - 149369605348*x^7 + 2679277607936*x^6 - 11993916195144*x^5 + 125839435016506*x^4 - 486125527533824*x^3 + 2817137139500788*x^2 - 7283791516402294*x + 16052774378388739, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 1604 x^{14} - 4849 x^{13} + 851534 x^{12} - 3460136 x^{11} + 220867782 x^{10} - 1005651829 x^{9} + 31950552532 x^{8} - 149369605348 x^{7} + 2679277607936 x^{6} - 11993916195144 x^{5} + 125839435016506 x^{4} - 486125527533824 x^{3} + 2817137139500788 x^{2} - 7283791516402294 x + 16052774378388739 \)

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:  \(257142839682757276401140868988052161101602113=67^{8}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $596.53$
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:  $67, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(6499=67\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{6499}(1408,·)$, $\chi_{6499}(1,·)$, $\chi_{6499}(1540,·)$, $\chi_{6499}(1473,·)$, $\chi_{6499}(5964,·)$, $\chi_{6499}(269,·)$, $\chi_{6499}(1810,·)$, $\chi_{6499}(4823,·)$, $\chi_{6499}(5828,·)$, $\chi_{6499}(604,·)$, $\chi_{6499}(803,·)$, $\chi_{6499}(872,·)$, $\chi_{6499}(4086,·)$, $\chi_{6499}(6297,·)$, $\chi_{6499}(4153,·)$, $\chi_{6499}(5562,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{113} a^{14} - \frac{14}{113} a^{13} + \frac{27}{113} a^{12} - \frac{10}{113} a^{11} - \frac{50}{113} a^{10} - \frac{29}{113} a^{9} - \frac{23}{113} a^{8} + \frac{41}{113} a^{7} - \frac{22}{113} a^{6} - \frac{31}{113} a^{5} + \frac{22}{113} a^{4} - \frac{49}{113} a^{3} + \frac{36}{113} a^{2} + \frac{24}{113} a - \frac{41}{113}$, $\frac{1}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{15} - \frac{17082141173943938753955907186515266943104367666886009641371306150654575259588662627775819340195764063790331}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{14} - \frac{2778349787130724595025988839864201269123578589134404236903423797253931348665491913777479760386518797662175083}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{13} - \frac{4070459519676662225147577612456413790452648729652126791639415452714576312795687280274480716321186286710873612}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{12} - \frac{4099244038555276147812045333506965701089518518598866589242968889374589406563902954753519131017893952524303556}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{11} + \frac{2136839911531152334150507422565198544543642860187514536489345409752770964264699994442981606710878561888572108}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{10} + \frac{641527553028875293713849454111267482605820169723196074331519043060810324812755734582724398097360312764576817}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{9} + \frac{2372110915230886156241676596487060830012643029472257175154387561484320953153374720877883488741320480015367800}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{8} - \frac{1908127116156292798010481740380693730521264604453804916118434228225192444709445802183952408188612929426086827}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{7} - \frac{9122543616293737113870912837960283211124332096097419811206893064899182425566958200580332710992832542759878}{21246216253521976559797112844458937487469539689032124944640410980407326131792007648724120902228054126800717} a^{6} + \frac{2726111081240168583426185587093196035591278486691521432512052371745907620511894223750569434331399162795247029}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{5} + \frac{10043275358760533281462483475126001928549896887210862342996690875124257427911796357688590920902154105151849}{73139629403717246741248468110571032589607530433924748703231149304234069604133548454457371955457637657747601} a^{4} + \frac{630705904297892412050154837373320467730722542889931173268743604313640393905496538123456405987882692244086840}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{3} - \frac{3938699259543020395363105436555527097273264382470349396537726739188170329806696532407603268004664080708506938}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a^{2} - \frac{295621777784487898788026824526776936660540884236746079201955841517210485920723537262698856988975578536452952}{8264778122620048881761076896494526682625650939033496603465119871378449865267090975353683030966713055325478913} a + \frac{1098862412210239299348402678862083973055253427879758071507233715113347831034872891993527141403334672186923}{6372226771488087032969218887042811628855551996170776101360925112859252016397140304821652298355214383442929}$

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

Class group and class number

$C_{1073881378}$, which has order $1073881378$ (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:  \( 1675810.87182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

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

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
67Data not computed
97Data not computed