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Properties

Label 16.0.29233126977...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 149^{12}$
Root discriminant $142.60$
Ramified primes $5, 149$
Class number $1000$ (GRH)
Class group $[10, 10, 10]$ (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![43046721, 23914845, 36669429, 4133430, 18173970, 820125, 14983866, -4592700, 7821279, 510300, 184986, -1125, 2770, -70, 69, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 69*x^14 - 70*x^13 + 2770*x^12 - 1125*x^11 + 184986*x^10 + 510300*x^9 + 7821279*x^8 - 4592700*x^7 + 14983866*x^6 + 820125*x^5 + 18173970*x^4 + 4133430*x^3 + 36669429*x^2 + 23914845*x + 43046721)
 
gp: K = bnfinit(x^16 - 5*x^15 + 69*x^14 - 70*x^13 + 2770*x^12 - 1125*x^11 + 184986*x^10 + 510300*x^9 + 7821279*x^8 - 4592700*x^7 + 14983866*x^6 + 820125*x^5 + 18173970*x^4 + 4133430*x^3 + 36669429*x^2 + 23914845*x + 43046721, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 69 x^{14} - 70 x^{13} + 2770 x^{12} - 1125 x^{11} + 184986 x^{10} + 510300 x^{9} + 7821279 x^{8} - 4592700 x^{7} + 14983866 x^{6} + 820125 x^{5} + 18173970 x^{4} + 4133430 x^{3} + 36669429 x^{2} + 23914845 x + 43046721 \)

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:  \(29233126977165462000179004150390625=5^{12}\cdot 149^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $142.60$
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:  $5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{1}{9} a^{5} + \frac{11}{27} a^{4} + \frac{1}{27} a^{3} + \frac{2}{9} a^{2}$, $\frac{1}{81} a^{8} + \frac{1}{81} a^{7} + \frac{1}{27} a^{6} + \frac{11}{81} a^{5} + \frac{1}{81} a^{4} + \frac{11}{27} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{243} a^{9} + \frac{1}{243} a^{8} + \frac{1}{81} a^{7} + \frac{11}{243} a^{6} + \frac{1}{243} a^{5} + \frac{38}{81} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{729} a^{10} + \frac{1}{729} a^{9} + \frac{1}{243} a^{8} + \frac{11}{729} a^{7} + \frac{1}{729} a^{6} + \frac{38}{243} a^{5} + \frac{8}{27} a^{4} + \frac{4}{27} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{2187} a^{11} + \frac{1}{2187} a^{10} + \frac{1}{729} a^{9} + \frac{11}{2187} a^{8} + \frac{1}{2187} a^{7} + \frac{38}{729} a^{6} + \frac{8}{81} a^{5} - \frac{23}{81} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{33841638} a^{12} + \frac{383}{16920819} a^{11} + \frac{176}{5640273} a^{10} - \frac{21625}{33841638} a^{9} + \frac{30155}{16920819} a^{8} + \frac{64975}{5640273} a^{7} + \frac{139013}{3760182} a^{6} + \frac{7237}{626697} a^{5} - \frac{98795}{208899} a^{4} - \frac{53759}{139266} a^{3} - \frac{8143}{23211} a^{2} - \frac{25}{7737} a + \frac{81}{5158}$, $\frac{1}{304574742} a^{13} + \frac{2}{152287371} a^{12} - \frac{4262}{50762457} a^{11} + \frac{9299}{304574742} a^{10} + \frac{261485}{152287371} a^{9} - \frac{78836}{16920819} a^{8} - \frac{156251}{33841638} a^{7} - \frac{70265}{1880091} a^{6} + \frac{128786}{1880091} a^{5} - \frac{3623}{139266} a^{4} + \frac{93218}{208899} a^{3} + \frac{1426}{7737} a^{2} + \frac{12781}{46422} a - \frac{850}{2579}$, $\frac{1}{1629615701710378638} a^{14} - \frac{1122707203}{814807850855189319} a^{13} + \frac{3174941071}{271602616951729773} a^{12} + \frac{140953500441515}{1629615701710378638} a^{11} + \frac{412194879326672}{814807850855189319} a^{10} - \frac{3114306883832}{10059356183397399} a^{9} - \frac{156692219499677}{181068411301153182} a^{8} - \frac{165308712722816}{10059356183397399} a^{7} + \frac{62612480599163}{10059356183397399} a^{6} + \frac{142937467758533}{2235412485199422} a^{5} - \frac{471457195243057}{1117706242599711} a^{4} + \frac{2790131272892}{124189582511079} a^{3} + \frac{44258198364601}{248379165022158} a^{2} - \frac{2073402570331}{4599614167077} a - \frac{418227810078}{1533204722359}$, $\frac{1}{1043685746544710288328462} a^{15} - \frac{41281}{521842873272355144164231} a^{14} - \frac{164751011684267}{173947624424118381388077} a^{13} + \frac{856135182303155}{1043685746544710288328462} a^{12} + \frac{71207092446994491686}{521842873272355144164231} a^{11} - \frac{16759978027557968369}{57982541474706127129359} a^{10} + \frac{72247929641766849445}{115965082949412254258718} a^{9} + \frac{24274431827788583315}{6442504608300680792151} a^{8} - \frac{25864668327878126812}{6442504608300680792151} a^{7} - \frac{34692699213344924213}{1431667690733484620478} a^{6} + \frac{35425587058598205710}{715833845366742310239} a^{5} + \frac{2209076564781518849}{26512364643212678157} a^{4} + \frac{43311916101987347749}{159074187859276068942} a^{3} + \frac{183347366030226559}{8837454881070892719} a^{2} - \frac{137602252490416877}{327313143743366397} a + \frac{30010024860779131}{109104381247788799}$

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

Class group and class number

$C_{10}\times C_{10}\times C_{10}$, which has order $1000$ (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:  \( -\frac{2149113658875905}{521842873272355144164231} a^{15} + \frac{7047005652107452}{521842873272355144164231} a^{14} - \frac{39699399316120321}{173947624424118381388077} a^{13} - \frac{179347821094037164}{521842873272355144164231} a^{12} - \frac{4870502322153102230}{521842873272355144164231} a^{11} - \frac{122252634130355225}{6442504608300680792151} a^{10} - \frac{40419951175121055341}{57982541474706127129359} a^{9} - \frac{22815409333297076710}{6442504608300680792151} a^{8} - \frac{206777727347054544437}{6442504608300680792151} a^{7} - \frac{23864416623057390982}{715833845366742310239} a^{6} + \frac{74145165763467048592}{715833845366742310239} a^{5} - \frac{42105784532021428073}{79537093929638034471} a^{4} + \frac{12177897637865423422}{79537093929638034471} a^{3} - \frac{232894891147631530}{2945818293690297573} a^{2} - \frac{202319072529369247}{981939431230099191} a - \frac{18978467875532993}{109104381247788799} \) (order $10$)
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:  \( 106425286.202 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{745}) \), \(\Q(\sqrt{5}, \sqrt{149})\), 4.4.16539745.1 x2, 4.4.82698725.1 x2, 4.0.2775125.1, \(\Q(\zeta_{5})\), 8.8.6839079116625625.1, 8.0.7701318765625.3, 8.0.170976977915640625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$149$149.8.6.1$x^{8} - 1043 x^{4} + 1798281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
149.8.6.1$x^{8} - 1043 x^{4} + 1798281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$