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Properties

Label 16.0.43153395314...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 17^{14}$
Root discriminant $46.20$
Ramified primes $3, 5, 17$
Class number $340$ (GRH)
Class group $[2, 170]$ (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![28561, -72501, 147875, -158392, 155828, -91287, 69368, -30030, 21507, -5340, 3528, -447, 448, -22, 25, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 25*x^14 - 22*x^13 + 448*x^12 - 447*x^11 + 3528*x^10 - 5340*x^9 + 21507*x^8 - 30030*x^7 + 69368*x^6 - 91287*x^5 + 155828*x^4 - 158392*x^3 + 147875*x^2 - 72501*x + 28561)
 
gp: K = bnfinit(x^16 - x^15 + 25*x^14 - 22*x^13 + 448*x^12 - 447*x^11 + 3528*x^10 - 5340*x^9 + 21507*x^8 - 30030*x^7 + 69368*x^6 - 91287*x^5 + 155828*x^4 - 158392*x^3 + 147875*x^2 - 72501*x + 28561, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 25 x^{14} - 22 x^{13} + 448 x^{12} - 447 x^{11} + 3528 x^{10} - 5340 x^{9} + 21507 x^{8} - 30030 x^{7} + 69368 x^{6} - 91287 x^{5} + 155828 x^{4} - 158392 x^{3} + 147875 x^{2} - 72501 x + 28561 \)

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:  \(431533953146964646550390625=3^{8}\cdot 5^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.20$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(134,·)$, $\chi_{255}(16,·)$, $\chi_{255}(19,·)$, $\chi_{255}(86,·)$, $\chi_{255}(94,·)$, $\chi_{255}(101,·)$, $\chi_{255}(59,·)$, $\chi_{255}(229,·)$, $\chi_{255}(166,·)$, $\chi_{255}(104,·)$, $\chi_{255}(106,·)$, $\chi_{255}(49,·)$, $\chi_{255}(179,·)$, $\chi_{255}(251,·)$, $\chi_{255}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} + \frac{2}{13} a^{9} + \frac{1}{13} a^{8} + \frac{1}{13} a^{7} - \frac{4}{13} a^{6} - \frac{6}{13} a^{5} + \frac{5}{13} a^{4} - \frac{5}{13} a^{3} - \frac{3}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{13} a^{11} - \frac{3}{13} a^{9} - \frac{1}{13} a^{8} - \frac{6}{13} a^{7} + \frac{2}{13} a^{6} + \frac{4}{13} a^{5} - \frac{2}{13} a^{4} - \frac{6}{13} a^{3} + \frac{1}{13} a^{2} - \frac{3}{13} a$, $\frac{1}{26} a^{12} + \frac{5}{26} a^{9} + \frac{5}{13} a^{8} - \frac{4}{13} a^{7} + \frac{5}{26} a^{6} + \frac{3}{13} a^{5} - \frac{2}{13} a^{4} - \frac{1}{26} a^{3} - \frac{6}{13} a^{2} - \frac{1}{13} a - \frac{1}{2}$, $\frac{1}{26} a^{13} - \frac{1}{26} a^{10} - \frac{1}{13} a^{9} + \frac{6}{13} a^{8} - \frac{1}{26} a^{7} + \frac{2}{13} a^{6} + \frac{3}{13} a^{5} - \frac{5}{26} a^{4} - \frac{4}{13} a^{3} - \frac{5}{13} a^{2} - \frac{9}{26} a$, $\frac{1}{15886} a^{14} - \frac{33}{7943} a^{13} + \frac{45}{7943} a^{12} - \frac{139}{15886} a^{11} + \frac{55}{7943} a^{10} + \frac{2058}{7943} a^{9} + \frac{2605}{15886} a^{8} - \frac{1396}{7943} a^{7} + \frac{32}{169} a^{6} + \frac{11}{1222} a^{5} - \frac{63}{611} a^{4} - \frac{1216}{7943} a^{3} - \frac{5281}{15886} a^{2} - \frac{127}{611} a + \frac{16}{47}$, $\frac{1}{948700101825439281355252514} a^{15} + \frac{25375485572471354816105}{948700101825439281355252514} a^{14} - \frac{2050979794770710769856313}{474350050912719640677626257} a^{13} + \frac{1409443579553728265604540}{474350050912719640677626257} a^{12} + \frac{10333140789865778593965093}{948700101825439281355252514} a^{11} + \frac{15286152852207765581088364}{474350050912719640677626257} a^{10} - \frac{6722570509963984488748849}{474350050912719640677626257} a^{9} - \frac{126323462510372673552241999}{948700101825439281355252514} a^{8} + \frac{62152036592533737727672115}{474350050912719640677626257} a^{7} + \frac{4985099439580963162260646}{36488465454824587744432789} a^{6} - \frac{33922973188403898231174991}{72976930909649175488865578} a^{5} - \frac{128301673576206757918383320}{474350050912719640677626257} a^{4} + \frac{204311551933182345383592749}{474350050912719640677626257} a^{3} + \frac{445045192817134084049205}{72976930909649175488865578} a^{2} - \frac{1228915574658370403388830}{2806805034986506749571753} a + \frac{145896165878139361548407}{431816159228693346087962}$

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

Class group and class number

$C_{2}\times C_{170}$, which has order $340$ (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:  \( -\frac{314721171004102506247}{20185108549477431518196862} a^{15} + \frac{253111397148211142417}{20185108549477431518196862} a^{14} - \frac{4015376594832740480054}{10092554274738715759098431} a^{13} + \frac{2787370995092314901401}{10092554274738715759098431} a^{12} - \frac{144990392857497515187259}{20185108549477431518196862} a^{11} + \frac{58189144918406468622138}{10092554274738715759098431} a^{10} - \frac{588512129359385318508423}{10092554274738715759098431} a^{9} + \frac{1531713499242667306232883}{20185108549477431518196862} a^{8} - \frac{3560131881833325812181872}{10092554274738715759098431} a^{7} + \frac{342073423858513919231527}{776350328826055058392187} a^{6} - \frac{1836984993473608462578351}{1552700657652110116784374} a^{5} + \frac{14055901433558182486315511}{10092554274738715759098431} a^{4} - \frac{26357405875521748082955917}{10092554274738715759098431} a^{3} + \frac{3380086126056756981190783}{1552700657652110116784374} a^{2} - \frac{160061546678788675099100}{59719256063542696799399} a + \frac{12276115466246143654171}{9187577855929645661446} \) (order $6$)
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:  \( 81485.0410293661 \) (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

\(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.4913.1, 4.0.44217.1, 8.0.1955143089.1, 8.0.20773395320625.1, 8.8.256461670625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$5$5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
17Data not computed