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Properties

Label 16.0.18797983274...0176.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{8}\cdot 89^{4}$
Root discriminant $50.66$
Ramified primes $2, 17, 89$
Class number $32$ (GRH)
Class group $[2, 2, 8]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T373)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![407601, -223464, 372830, -335548, 269360, -151328, 107994, -67708, 43061, -22600, 10174, -3708, 1164, -280, 60, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1164*x^12 - 3708*x^11 + 10174*x^10 - 22600*x^9 + 43061*x^8 - 67708*x^7 + 107994*x^6 - 151328*x^5 + 269360*x^4 - 335548*x^3 + 372830*x^2 - 223464*x + 407601)
 
gp: K = bnfinit(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1164*x^12 - 3708*x^11 + 10174*x^10 - 22600*x^9 + 43061*x^8 - 67708*x^7 + 107994*x^6 - 151328*x^5 + 269360*x^4 - 335548*x^3 + 372830*x^2 - 223464*x + 407601, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1164 x^{12} - 3708 x^{11} + 10174 x^{10} - 22600 x^{9} + 43061 x^{8} - 67708 x^{7} + 107994 x^{6} - 151328 x^{5} + 269360 x^{4} - 335548 x^{3} + 372830 x^{2} - 223464 x + 407601 \)

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:  \(1879798327454785434787250176=2^{32}\cdot 17^{8}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.66$
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:  $2, 17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{11} - \frac{1}{7} a^{10} - \frac{3}{14} a^{9} - \frac{3}{14} a^{8} - \frac{1}{2} a^{7} - \frac{3}{14} a^{5} - \frac{1}{2} a^{4} - \frac{5}{14} a^{3} - \frac{2}{7} a^{2} - \frac{1}{2} a - \frac{1}{14}$, $\frac{1}{336} a^{12} - \frac{1}{56} a^{11} - \frac{3}{112} a^{10} - \frac{17}{84} a^{9} - \frac{1}{21} a^{8} + \frac{5}{24} a^{7} - \frac{73}{336} a^{6} - \frac{11}{84} a^{5} + \frac{5}{56} a^{4} + \frac{5}{56} a^{3} + \frac{149}{336} a^{2} + \frac{13}{42} a - \frac{1}{112}$, $\frac{1}{336} a^{13} + \frac{1}{112} a^{11} - \frac{25}{168} a^{10} - \frac{4}{21} a^{9} - \frac{1}{168} a^{8} + \frac{11}{336} a^{7} + \frac{11}{168} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{4} + \frac{89}{336} a^{3} - \frac{17}{168} a^{2} + \frac{39}{112} a + \frac{17}{56}$, $\frac{1}{219562127127504} a^{14} - \frac{1}{31366018161072} a^{13} + \frac{69013192811}{73187375709168} a^{12} - \frac{1242237470507}{219562127127504} a^{11} - \frac{724189310221}{3920752270134} a^{10} - \frac{2700971726405}{109781063563752} a^{9} + \frac{45798405504265}{219562127127504} a^{8} + \frac{55115553652013}{219562127127504} a^{7} + \frac{31178375164879}{109781063563752} a^{6} - \frac{109323770702}{1524736993941} a^{5} - \frac{22987598769097}{219562127127504} a^{4} + \frac{549694575295}{1161704376336} a^{3} - \frac{32347342387247}{219562127127504} a^{2} + \frac{23489483255137}{73187375709168} a - \frac{203761842985}{6098947975764}$, $\frac{1}{4821364749592860336} a^{15} + \frac{2743}{1205341187398215084} a^{14} + \frac{586359503886613}{4821364749592860336} a^{13} + \frac{5636504200632061}{4821364749592860336} a^{12} + \frac{3425279004229949}{133926798599801676} a^{11} + \frac{952781989183856935}{4821364749592860336} a^{10} - \frac{523084518484902169}{4821364749592860336} a^{9} - \frac{253785658525996693}{2410682374796430168} a^{8} + \frac{281491836044155409}{803560791598810056} a^{7} - \frac{2238749813593621301}{4821364749592860336} a^{6} - \frac{267156644322280327}{4821364749592860336} a^{5} + \frac{271292523761941607}{602670593699107542} a^{4} - \frac{112430320709418095}{4821364749592860336} a^{3} - \frac{1698650728017130747}{4821364749592860336} a^{2} + \frac{8922389044287241}{114794398799830008} a - \frac{143528444588449373}{535707194399206704}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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:  \( 514221.672805 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T373):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), 4.0.387328.3, 4.0.387328.1, \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.43356641099776.24, 8.4.121788317696.1, 8.4.121788317696.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$