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Properties

Label 16.0.11925500621...6176.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 71^{8}$
Root discriminant $116.76$
Ramified primes $2, 3, 71$
Class number $1188096$ (GRH)
Class group $[4, 8, 37128]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24635195998, -8135298584, 9376658828, -2596926520, 1590906842, -370616376, 157206132, -30603224, 9888039, -1578616, 404700, -50904, 10486, -952, 156, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 156*x^14 - 952*x^13 + 10486*x^12 - 50904*x^11 + 404700*x^10 - 1578616*x^9 + 9888039*x^8 - 30603224*x^7 + 157206132*x^6 - 370616376*x^5 + 1590906842*x^4 - 2596926520*x^3 + 9376658828*x^2 - 8135298584*x + 24635195998)
 
gp: K = bnfinit(x^16 - 8*x^15 + 156*x^14 - 952*x^13 + 10486*x^12 - 50904*x^11 + 404700*x^10 - 1578616*x^9 + 9888039*x^8 - 30603224*x^7 + 157206132*x^6 - 370616376*x^5 + 1590906842*x^4 - 2596926520*x^3 + 9376658828*x^2 - 8135298584*x + 24635195998, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 156 x^{14} - 952 x^{13} + 10486 x^{12} - 50904 x^{11} + 404700 x^{10} - 1578616 x^{9} + 9888039 x^{8} - 30603224 x^{7} + 157206132 x^{6} - 370616376 x^{5} + 1590906842 x^{4} - 2596926520 x^{3} + 9376658828 x^{2} - 8135298584 x + 24635195998 \)

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:  \(1192550062163720610398944075186176=2^{48}\cdot 3^{8}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.76$
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:  $2, 3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3408=2^{4}\cdot 3\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{3408}(1,·)$, $\chi_{3408}(709,·)$, $\chi_{3408}(3265,·)$, $\chi_{3408}(2699,·)$, $\chi_{3408}(143,·)$, $\chi_{3408}(851,·)$, $\chi_{3408}(853,·)$, $\chi_{3408}(1561,·)$, $\chi_{3408}(3407,·)$, $\chi_{3408}(995,·)$, $\chi_{3408}(1703,·)$, $\chi_{3408}(1705,·)$, $\chi_{3408}(2413,·)$, $\chi_{3408}(1847,·)$, $\chi_{3408}(2555,·)$, $\chi_{3408}(2557,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} - \frac{6}{23} a^{11} - \frac{11}{23} a^{10} - \frac{5}{23} a^{9} - \frac{5}{23} a^{8} + \frac{7}{23} a^{7} - \frac{8}{23} a^{6} + \frac{10}{23} a^{5} - \frac{8}{23} a^{4} - \frac{4}{23} a^{3} - \frac{1}{23} a^{2} + \frac{7}{23} a + \frac{11}{23}$, $\frac{1}{23} a^{13} - \frac{1}{23} a^{11} - \frac{2}{23} a^{10} + \frac{11}{23} a^{9} + \frac{11}{23} a^{7} + \frac{8}{23} a^{6} + \frac{6}{23} a^{5} - \frac{6}{23} a^{4} - \frac{2}{23} a^{3} + \frac{1}{23} a^{2} + \frac{7}{23} a - \frac{3}{23}$, $\frac{1}{1354666603518963985663} a^{14} - \frac{1}{193523800502709140809} a^{13} + \frac{5638572097489382977}{1354666603518963985663} a^{12} - \frac{33831432584936297771}{1354666603518963985663} a^{11} - \frac{353935051438242060022}{1354666603518963985663} a^{10} - \frac{629536484484801608482}{1354666603518963985663} a^{9} + \frac{363608788155148778183}{1354666603518963985663} a^{8} + \frac{85138770333564468186}{193523800502709140809} a^{7} + \frac{93521806629468208358}{193523800502709140809} a^{6} + \frac{316101068449894203230}{1354666603518963985663} a^{5} + \frac{186401459340663863168}{1354666603518963985663} a^{4} - \frac{260226821553724650211}{1354666603518963985663} a^{3} + \frac{314788918726483707654}{1354666603518963985663} a^{2} + \frac{195033548069759931135}{1354666603518963985663} a + \frac{77715295746667841333}{193523800502709140809}$, $\frac{1}{52223634376265074461427560319} a^{15} + \frac{838063}{2270592798968046715714241753} a^{14} + \frac{811565571981404817134362387}{52223634376265074461427560319} a^{13} - \frac{625304568755767520036901166}{52223634376265074461427560319} a^{12} - \frac{21164616648417764872136333586}{52223634376265074461427560319} a^{11} + \frac{16434898653042095549396179844}{52223634376265074461427560319} a^{10} + \frac{11938276266487453190352170005}{52223634376265074461427560319} a^{9} + \frac{25483041603626389380247304419}{52223634376265074461427560319} a^{8} + \frac{1648486574544585769656009966}{7460519196609296351632508617} a^{7} + \frac{5833471156265806105862328749}{52223634376265074461427560319} a^{6} - \frac{10933263653490693832961439480}{52223634376265074461427560319} a^{5} - \frac{409729044786603882013357514}{7460519196609296351632508617} a^{4} - \frac{17312386381879493507185853804}{52223634376265074461427560319} a^{3} - \frac{674006058154481655634161390}{52223634376265074461427560319} a^{2} - \frac{255471067832558449298749524}{52223634376265074461427560319} a + \frac{3603350745203153753473309782}{7460519196609296351632508617}$

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

Class group and class number

$C_{4}\times C_{8}\times C_{37128}$, which has order $1188096$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-71}) \), \(\Q(\sqrt{-213}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-142}) \), \(\Q(\sqrt{-426}) \), \(\Q(\sqrt{3}, \sqrt{-71})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-142})\), \(\Q(\sqrt{2}, \sqrt{-71})\), \(\Q(\sqrt{6}, \sqrt{-71})\), \(\Q(\sqrt{2}, \sqrt{-213})\), \(\Q(\sqrt{6}, \sqrt{-142})\), 4.4.18432.1, \(\Q(\zeta_{16})^+\), 4.0.92915712.5, 4.0.10323968.5, 8.0.134895774007296.29, \(\Q(\zeta_{48})^+\), 8.0.34533318145867776.53, 8.0.8633329536466944.86, 8.0.106584315265024.29, 8.0.34533318145867776.21, 8.0.34533318145867776.57

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$71$71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$