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Properties

Label 16.0.60300391495...625.23
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 89^{12}$
Root discriminant $96.89$
Ramified primes $5, 89$
Class number $144$ (GRH)
Class group $[2, 6, 12]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![498229211, 248573337, -168241441, -92662748, 40514783, 11501862, -5269299, -1189269, 564003, 93377, -40746, -5644, 1913, 224, -59, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 59*x^14 + 224*x^13 + 1913*x^12 - 5644*x^11 - 40746*x^10 + 93377*x^9 + 564003*x^8 - 1189269*x^7 - 5269299*x^6 + 11501862*x^5 + 40514783*x^4 - 92662748*x^3 - 168241441*x^2 + 248573337*x + 498229211)
 
gp: K = bnfinit(x^16 - 4*x^15 - 59*x^14 + 224*x^13 + 1913*x^12 - 5644*x^11 - 40746*x^10 + 93377*x^9 + 564003*x^8 - 1189269*x^7 - 5269299*x^6 + 11501862*x^5 + 40514783*x^4 - 92662748*x^3 - 168241441*x^2 + 248573337*x + 498229211, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 59 x^{14} + 224 x^{13} + 1913 x^{12} - 5644 x^{11} - 40746 x^{10} + 93377 x^{9} + 564003 x^{8} - 1189269 x^{7} - 5269299 x^{6} + 11501862 x^{5} + 40514783 x^{4} - 92662748 x^{3} - 168241441 x^{2} + 248573337 x + 498229211 \)

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:  \(60300391495425327222539306640625=5^{12}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.89$
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:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{30} a^{8} - \frac{1}{15} a^{7} - \frac{2}{15} a^{6} + \frac{7}{15} a^{5} + \frac{3}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{30} a^{2} - \frac{3}{10} a + \frac{11}{30}$, $\frac{1}{30} a^{9} - \frac{4}{15} a^{7} + \frac{1}{5} a^{6} + \frac{7}{30} a^{5} - \frac{7}{30} a^{3} - \frac{11}{30} a^{2} - \frac{7}{30} a - \frac{4}{15}$, $\frac{1}{30} a^{10} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{4}{15} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{15}$, $\frac{1}{30} a^{11} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{15} a - \frac{1}{3}$, $\frac{1}{2503080} a^{12} - \frac{1}{834360} a^{11} - \frac{3233}{250308} a^{10} + \frac{205}{13906} a^{9} - \frac{574}{104295} a^{8} - \frac{410731}{2503080} a^{7} + \frac{341651}{834360} a^{6} - \frac{111527}{1251540} a^{5} + \frac{835571}{2503080} a^{4} - \frac{599987}{2503080} a^{3} - \frac{1230941}{2503080} a^{2} + \frac{15713}{55624} a - \frac{1195571}{2503080}$, $\frac{1}{2503080} a^{13} - \frac{32339}{2503080} a^{11} + \frac{1297}{139060} a^{10} + \frac{562}{104295} a^{9} - \frac{34879}{2503080} a^{8} - \frac{50353}{104295} a^{7} + \frac{1099649}{2503080} a^{6} - \frac{250771}{2503080} a^{5} + \frac{536183}{1251540} a^{4} - \frac{36095}{250308} a^{3} - \frac{205597}{417180} a^{2} + \frac{16067}{36810} a - \frac{111133}{278120}$, $\frac{1}{12515400} a^{14} - \frac{1}{12515400} a^{13} - \frac{15596}{1564425} a^{11} + \frac{11278}{1564425} a^{10} + \frac{42497}{12515400} a^{9} + \frac{124123}{12515400} a^{8} - \frac{34439}{208590} a^{7} + \frac{443591}{1390600} a^{6} + \frac{952541}{4171800} a^{5} + \frac{183013}{12515400} a^{4} - \frac{5649713}{12515400} a^{3} - \frac{373483}{834360} a^{2} + \frac{2850943}{6257700} a + \frac{302869}{1564425}$, $\frac{1}{24235634153501731369272660411299400} a^{15} - \frac{29826188708901619277773949}{969425366140069254770906416451976} a^{14} - \frac{604977311878420594150275313}{12117817076750865684636330205649700} a^{13} + \frac{508019899442768750001161507}{24235634153501731369272660411299400} a^{12} + \frac{126659200375974391098780304824527}{8078544717833910456424220137099800} a^{11} + \frac{88129300904041653142354431368657}{8078544717833910456424220137099800} a^{10} + \frac{18927214481884748485508009032279}{4847126830700346273854532082259880} a^{9} - \frac{162113134746628205589960651120311}{12117817076750865684636330205649700} a^{8} + \frac{1513242005090708022907997104695019}{4039272358916955228212110068549900} a^{7} + \frac{643280719403110914778847338079051}{2019636179458477614106055034274950} a^{6} - \frac{528579565881629142169515126869827}{1425625538441278315839568259488200} a^{5} + \frac{81995873038488413930194798942064}{201963617945847761410605503427495} a^{4} + \frac{902254850890837580296197371468641}{12117817076750865684636330205649700} a^{3} - \frac{8670966699090517526680878119811229}{24235634153501731369272660411299400} a^{2} - \frac{60541803298572657509126089982889}{158402837604586479537729806609800} a - \frac{4033416083825230907102742447161003}{24235634153501731369272660411299400}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{12}$, which has order $144$ (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:  \( 8274413.16122 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{89}) \), \(\Q(\sqrt{445}) \), \(\Q(\sqrt{5}) \), 4.4.704969.1, \(\Q(\sqrt{5}, \sqrt{89})\), 4.4.17624225.2, 4.0.990125.1 x2, 4.0.11125.1 x2, 8.8.310613306850625.1, 8.0.980347515625.4, 8.0.7765332671265625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$