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Properties

Label 16.0.15136091767...4384.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 3^{8}\cdot 29^{8}$
Root discriminant $136.85$
Ramified primes $2, 3, 29$
Class number $2832336$ (GRH)
Class group $[3, 944112]$ (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![107162335102, -30416805424, 34371364936, -8216449296, 4892167536, -984991600, 403705168, -67859840, 21116169, -2900888, 715900, -76984, 15318, -1176, 188, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 188*x^14 - 1176*x^13 + 15318*x^12 - 76984*x^11 + 715900*x^10 - 2900888*x^9 + 21116169*x^8 - 67859840*x^7 + 403705168*x^6 - 984991600*x^5 + 4892167536*x^4 - 8216449296*x^3 + 34371364936*x^2 - 30416805424*x + 107162335102)
 
gp: K = bnfinit(x^16 - 8*x^15 + 188*x^14 - 1176*x^13 + 15318*x^12 - 76984*x^11 + 715900*x^10 - 2900888*x^9 + 21116169*x^8 - 67859840*x^7 + 403705168*x^6 - 984991600*x^5 + 4892167536*x^4 - 8216449296*x^3 + 34371364936*x^2 - 30416805424*x + 107162335102, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 188 x^{14} - 1176 x^{13} + 15318 x^{12} - 76984 x^{11} + 715900 x^{10} - 2900888 x^{9} + 21116169 x^{8} - 67859840 x^{7} + 403705168 x^{6} - 984991600 x^{5} + 4892167536 x^{4} - 8216449296 x^{3} + 34371364936 x^{2} - 30416805424 x + 107162335102 \)

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:  \(15136091767428192657568213587984384=2^{62}\cdot 3^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.85$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2784=2^{5}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{2784}(1,·)$, $\chi_{2784}(2437,·)$, $\chi_{2784}(1217,·)$, $\chi_{2784}(521,·)$, $\chi_{2784}(1741,·)$, $\chi_{2784}(1045,·)$, $\chi_{2784}(697,·)$, $\chi_{2784}(349,·)$, $\chi_{2784}(869,·)$, $\chi_{2784}(2609,·)$, $\chi_{2784}(2089,·)$, $\chi_{2784}(173,·)$, $\chi_{2784}(1565,·)$, $\chi_{2784}(1393,·)$, $\chi_{2784}(1913,·)$, $\chi_{2784}(2261,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{318857418358168412276129} a^{14} - \frac{7}{318857418358168412276129} a^{13} - \frac{3132005431644924276093}{318857418358168412276129} a^{12} + \frac{18792032589869545656649}{318857418358168412276129} a^{11} - \frac{86756124523782722624824}{318857418358168412276129} a^{10} - \frac{57337094479725634338125}{318857418358168412276129} a^{9} + \frac{27950075364643912598789}{318857418358168412276129} a^{8} + \frac{120077205550174745580890}{318857418358168412276129} a^{7} + \frac{50189843429628722711633}{318857418358168412276129} a^{6} - \frac{157452875450855730966030}{318857418358168412276129} a^{5} + \frac{108338208316476418906601}{318857418358168412276129} a^{4} - \frac{74420643739841598318336}{318857418358168412276129} a^{3} - \frac{108095685229061439657936}{318857418358168412276129} a^{2} - \frac{157010354754049707549341}{318857418358168412276129} a + \frac{101002036539195327989544}{318857418358168412276129}$, $\frac{1}{25787127221313812342444772898273} a^{15} + \frac{40436761}{25787127221313812342444772898273} a^{14} - \frac{774434523691957708013223926089}{25787127221313812342444772898273} a^{13} + \frac{4907176225417556724931712921255}{25787127221313812342444772898273} a^{12} - \frac{618678318645937359538752040732}{25787127221313812342444772898273} a^{11} + \frac{276169321025485786397218008258}{25787127221313812342444772898273} a^{10} - \frac{3675325338486747102804530664592}{25787127221313812342444772898273} a^{9} + \frac{3890570663916133327513131872357}{25787127221313812342444772898273} a^{8} - \frac{5169483696191738858089112657736}{25787127221313812342444772898273} a^{7} + \frac{10896007362716729796532555644918}{25787127221313812342444772898273} a^{6} - \frac{7404541070619417593127085032365}{25787127221313812342444772898273} a^{5} + \frac{10258691992560839103337057720326}{25787127221313812342444772898273} a^{4} - \frac{9189090095743056386912224978534}{25787127221313812342444772898273} a^{3} + \frac{7908225838611981724876640836901}{25787127221313812342444772898273} a^{2} - \frac{7928242558013331175344986717349}{25787127221313812342444772898273} a + \frac{2395673068135712514315540436874}{25787127221313812342444772898273}$

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

Class group and class number

$C_{3}\times C_{944112}$, which has order $2832336$ (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:  \( 15753.94986242651 \) (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

\(\Q(\sqrt{-174}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{2}, \sqrt{-87})\), \(\Q(\zeta_{16})^+\), 4.0.15501312.6, 8.0.240290673721344.80, \(\Q(\zeta_{32})^+\), 8.0.123028824945328128.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
3Data not computed
29Data not computed