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Properties

Label 16.0.91778367105...8144.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $36.32$
Ramified primes $2, 17, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![738433, -560472, -220724, 709236, -214944, -233928, 220534, -15272, -47271, 17832, 2946, -2848, 396, 184, -26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 26*x^14 + 184*x^13 + 396*x^12 - 2848*x^11 + 2946*x^10 + 17832*x^9 - 47271*x^8 - 15272*x^7 + 220534*x^6 - 233928*x^5 - 214944*x^4 + 709236*x^3 - 220724*x^2 - 560472*x + 738433)
 
gp: K = bnfinit(x^16 - 4*x^15 - 26*x^14 + 184*x^13 + 396*x^12 - 2848*x^11 + 2946*x^10 + 17832*x^9 - 47271*x^8 - 15272*x^7 + 220534*x^6 - 233928*x^5 - 214944*x^4 + 709236*x^3 - 220724*x^2 - 560472*x + 738433, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 26 x^{14} + 184 x^{13} + 396 x^{12} - 2848 x^{11} + 2946 x^{10} + 17832 x^{9} - 47271 x^{8} - 15272 x^{7} + 220534 x^{6} - 233928 x^{5} - 214944 x^{4} + 709236 x^{3} - 220724 x^{2} - 560472 x + 738433 \)

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:  \(9177836710508849627398144=2^{32}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.32$
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:  $2, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4264} a^{14} - \frac{17}{328} a^{13} - \frac{29}{533} a^{12} + \frac{439}{4264} a^{11} + \frac{11}{4264} a^{10} + \frac{81}{533} a^{9} + \frac{919}{4264} a^{8} + \frac{353}{4264} a^{7} - \frac{1693}{4264} a^{6} + \frac{745}{2132} a^{5} + \frac{93}{4264} a^{4} + \frac{1981}{4264} a^{3} - \frac{98}{533} a^{2} + \frac{1341}{4264} a - \frac{1167}{4264}$, $\frac{1}{6650362646288274814037468005452399906803896} a^{15} - \frac{482190216017459392504060697833738087635}{6650362646288274814037468005452399906803896} a^{14} + \frac{96378369109256822972149695768193064542509}{3325181323144137407018734002726199953401948} a^{13} + \frac{815852100972764605135166675861348096699539}{6650362646288274814037468005452399906803896} a^{12} - \frac{1065538387231231232735755898721953729514323}{6650362646288274814037468005452399906803896} a^{11} - \frac{587361761971051910448489682363756895031959}{3325181323144137407018734002726199953401948} a^{10} - \frac{1584000883073816500577694322582986209986953}{6650362646288274814037468005452399906803896} a^{9} - \frac{464539401184951176150565000423686985348709}{6650362646288274814037468005452399906803896} a^{8} + \frac{625451027998577474371192600289127481816761}{6650362646288274814037468005452399906803896} a^{7} + \frac{771658173251753455578635736093148203729531}{1662590661572068703509367001363099976700974} a^{6} - \frac{1477177910299768483561001492801241446412615}{6650362646288274814037468005452399906803896} a^{5} - \frac{1199776710349569938116875250285031592053149}{6650362646288274814037468005452399906803896} a^{4} - \frac{8043480544778910754988396527584087422093}{255783178703395185155287230978938457953996} a^{3} + \frac{2178317055669411852369867636106752906669889}{6650362646288274814037468005452399906803896} a^{2} - \frac{836164090744456680296277364980102843272793}{6650362646288274814037468005452399906803896} a + \frac{432573129076417403007128079333176973588365}{3325181323144137407018734002726199953401948}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{34049765162642203151767238291953}{8209827610784378246300492939874501304} a^{15} - \frac{16552193353575656111923199494068}{1026228451348047280787561617484312663} a^{14} - \frac{982668775276978736841301645245219}{8209827610784378246300492939874501304} a^{13} + \frac{6345553255048271294321927125437201}{8209827610784378246300492939874501304} a^{12} + \frac{2096867282289681955511202005129602}{1026228451348047280787561617484312663} a^{11} - \frac{106142556230230568281799469073023955}{8209827610784378246300492939874501304} a^{10} + \frac{33467505973671132599474510804006443}{8209827610784378246300492939874501304} a^{9} + \frac{191691701351249718992656131572863261}{2052456902696094561575123234968625326} a^{8} - \frac{714092715684790381323614048082384825}{4104913805392189123150246469937250652} a^{7} - \frac{2175006049085530196299974753707169653}{8209827610784378246300492939874501304} a^{6} + \frac{658801576939763326905108082361944995}{631525200829567557407730226144192408} a^{5} - \frac{835365096111486263678981961394955855}{4104913805392189123150246469937250652} a^{4} - \frac{18371514118245672199539431916727393585}{8209827610784378246300492939874501304} a^{3} + \frac{17954763454695034103227548012199938743}{8209827610784378246300492939874501304} a^{2} + \frac{1898845065653549822246997245662809432}{1026228451348047280787561617484312663} a - \frac{732768755368747522238524565294970253}{200239697824009225519524218045719544} \) (order $8$)
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:  \( 829553.598725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.4.4352.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
97Data not computed