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Properties

Label 16.0.63864408381...7569.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 41^{14}$
Root discriminant $307.49$
Ramified primes $17, 41$
Class number $374935884$ (GRH)
Class group $[17, 22055052]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![401292214272, 361157329920, 167744328448, 53673234160, 13322733808, 2375245912, 259046624, 11757259, -2564467, -1270944, -143032, -14002, -2278, 320, 64, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 64*x^14 + 320*x^13 - 2278*x^12 - 14002*x^11 - 143032*x^10 - 1270944*x^9 - 2564467*x^8 + 11757259*x^7 + 259046624*x^6 + 2375245912*x^5 + 13322733808*x^4 + 53673234160*x^3 + 167744328448*x^2 + 361157329920*x + 401292214272)
 
gp: K = bnfinit(x^16 - x^15 + 64*x^14 + 320*x^13 - 2278*x^12 - 14002*x^11 - 143032*x^10 - 1270944*x^9 - 2564467*x^8 + 11757259*x^7 + 259046624*x^6 + 2375245912*x^5 + 13322733808*x^4 + 53673234160*x^3 + 167744328448*x^2 + 361157329920*x + 401292214272, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 64 x^{14} + 320 x^{13} - 2278 x^{12} - 14002 x^{11} - 143032 x^{10} - 1270944 x^{9} - 2564467 x^{8} + 11757259 x^{7} + 259046624 x^{6} + 2375245912 x^{5} + 13322733808 x^{4} + 53673234160 x^{3} + 167744328448 x^{2} + 361157329920 x + 401292214272 \)

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:  \(6386440838157604973770860952468990607569=17^{14}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $307.49$
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:  $17, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(697=17\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{697}(1,·)$, $\chi_{697}(132,·)$, $\chi_{697}(325,·)$, $\chi_{697}(519,·)$, $\chi_{697}(202,·)$, $\chi_{697}(409,·)$, $\chi_{697}(314,·)$, $\chi_{697}(288,·)$, $\chi_{697}(495,·)$, $\chi_{697}(178,·)$, $\chi_{697}(372,·)$, $\chi_{697}(565,·)$, $\chi_{697}(696,·)$, $\chi_{697}(378,·)$, $\chi_{697}(319,·)$, $\chi_{697}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{5} + \frac{5}{32} a^{3} - \frac{1}{8} a$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} + \frac{1}{128} a^{4} - \frac{1}{32} a^{2}$, $\frac{1}{384} a^{9} - \frac{1}{64} a^{7} - \frac{5}{128} a^{5} - \frac{1}{32} a^{3} + \frac{1}{12} a$, $\frac{1}{1536} a^{10} + \frac{1}{1536} a^{9} + \frac{3}{256} a^{7} + \frac{15}{512} a^{6} - \frac{5}{512} a^{5} - \frac{9}{256} a^{4} - \frac{11}{128} a^{3} + \frac{1}{192} a^{2} + \frac{1}{12} a$, $\frac{1}{3072} a^{11} + \frac{1}{1024} a^{9} + \frac{1}{512} a^{8} - \frac{15}{1024} a^{7} - \frac{7}{256} a^{6} + \frac{63}{1024} a^{5} - \frac{15}{512} a^{4} - \frac{85}{768} a^{3} + \frac{7}{128} a^{2} + \frac{1}{16} a$, $\frac{1}{6144} a^{12} - \frac{1}{6144} a^{10} - \frac{1}{1024} a^{9} + \frac{1}{2048} a^{8} + \frac{7}{512} a^{7} + \frac{35}{2048} a^{6} + \frac{15}{1024} a^{5} - \frac{19}{1536} a^{4} - \frac{7}{256} a^{3} - \frac{1}{192} a^{2}$, $\frac{1}{15704064} a^{13} - \frac{1}{36864} a^{12} - \frac{1061}{15704064} a^{11} - \frac{391}{1963008} a^{10} - \frac{75}{581632} a^{9} + \frac{925}{2617344} a^{8} - \frac{2689}{5234688} a^{7} + \frac{36515}{1308672} a^{6} - \frac{23867}{490752} a^{5} - \frac{829}{327168} a^{4} + \frac{106415}{981504} a^{3} - \frac{2689}{15336} a^{2} - \frac{35}{639} a - \frac{61}{213}$, $\frac{1}{125632512} a^{14} - \frac{1}{41877504} a^{13} - \frac{10007}{125632512} a^{12} - \frac{2075}{125632512} a^{11} - \frac{1549}{13959168} a^{10} + \frac{4301}{41877504} a^{9} + \frac{133193}{41877504} a^{8} - \frac{510859}{41877504} a^{7} - \frac{50489}{3926016} a^{6} + \frac{34951}{5234688} a^{5} - \frac{351955}{7852032} a^{4} - \frac{401029}{7852032} a^{3} - \frac{39709}{163584} a^{2} - \frac{93}{4544} a - \frac{5}{284}$, $\frac{1}{390550049849640951596640026650346165306916864} a^{15} + \frac{241236621994150053105882591592480753}{97637512462410237899160006662586541326729216} a^{14} - \frac{38268708546371205792665266644869273}{32545837487470079299720002220862180442243072} a^{13} + \frac{2589477946867598103253240237334615068169}{97637512462410237899160006662586541326729216} a^{12} - \frac{9147395984148101725395293137928089344275}{65091674974940158599440004441724360884486144} a^{11} - \frac{1324684088056857973682577572501984153899}{97637512462410237899160006662586541326729216} a^{10} + \frac{2741336796252288318698553654528146815513}{32545837487470079299720002220862180442243072} a^{9} - \frac{77007997808190397666269566014162486199179}{32545837487470079299720002220862180442243072} a^{8} + \frac{3049141190076283961276011184000974580382521}{390550049849640951596640026650346165306916864} a^{7} - \frac{429557047270935034895823608467948059791939}{48818756231205118949580003331293270663364608} a^{6} - \frac{161493294282525043158776454304011865661585}{16272918743735039649860001110431090221121536} a^{5} + \frac{744889153154001753654505782542192106064931}{12204689057801279737395000832823317665841152} a^{4} - \frac{1491173780945603712531231075904558728039025}{8136459371867519824930000555215545110560768} a^{3} - \frac{366999502353327989671744822651010496416411}{1525586132225159967174375104102914708230144} a^{2} - \frac{16863361666872984606930930394806974424959}{127132177685429997264531258675242892352512} a - \frac{670645520459520277634073307696538908345}{2648587035113124943011067889067560257344}$

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

Class group and class number

$C_{17}\times C_{22055052}$, which has order $374935884$ (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:  \( 3288669416.54 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{697}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{17}, \sqrt{41})\), 4.4.338608873.2, 4.4.338608873.1, 8.8.114655968874330129.1, 8.0.79915210305408099913.2, 8.0.79915210305408099913.3

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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
17Data not computed
41Data not computed