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Properties

Label 16.8.35826423586...4961.3
Degree $16$
Signature $[8, 4]$
Discriminant $13^{14}\cdot 157^{10}$
Root discriminant $222.40$
Ramified primes $13, 157$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1902420484, 290985398, -2543361055, -127242000, 775454875, 58581679, -31636488, 5714488, -2350140, -280059, 64298, -26624, 6058, -671, 129, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 129*x^14 - 671*x^13 + 6058*x^12 - 26624*x^11 + 64298*x^10 - 280059*x^9 - 2350140*x^8 + 5714488*x^7 - 31636488*x^6 + 58581679*x^5 + 775454875*x^4 - 127242000*x^3 - 2543361055*x^2 + 290985398*x + 1902420484)
 
gp: K = bnfinit(x^16 - 6*x^15 + 129*x^14 - 671*x^13 + 6058*x^12 - 26624*x^11 + 64298*x^10 - 280059*x^9 - 2350140*x^8 + 5714488*x^7 - 31636488*x^6 + 58581679*x^5 + 775454875*x^4 - 127242000*x^3 - 2543361055*x^2 + 290985398*x + 1902420484, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 129 x^{14} - 671 x^{13} + 6058 x^{12} - 26624 x^{11} + 64298 x^{10} - 280059 x^{9} - 2350140 x^{8} + 5714488 x^{7} - 31636488 x^{6} + 58581679 x^{5} + 775454875 x^{4} - 127242000 x^{3} - 2543361055 x^{2} + 290985398 x + 1902420484 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35826423586415848519139534914245144961=13^{14}\cdot 157^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $222.40$
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:  $13, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{26} a^{8} - \frac{3}{26} a^{7} - \frac{5}{26} a^{6} - \frac{3}{13} a^{5} + \frac{1}{13} a^{4} - \frac{11}{26} a^{3} + \frac{11}{26} a^{2} + \frac{3}{26} a - \frac{2}{13}$, $\frac{1}{26} a^{9} - \frac{1}{26} a^{7} + \frac{5}{26} a^{6} - \frac{3}{26} a^{5} + \frac{4}{13} a^{4} - \frac{9}{26} a^{3} + \frac{5}{13} a^{2} - \frac{4}{13} a - \frac{6}{13}$, $\frac{1}{52} a^{10} - \frac{1}{52} a^{9} - \frac{5}{26} a^{7} - \frac{1}{4} a^{6} - \frac{2}{13} a^{5} + \frac{6}{13} a^{4} - \frac{5}{52} a^{3} - \frac{7}{52} a^{2} - \frac{7}{26} a + \frac{2}{13}$, $\frac{1}{52} a^{11} - \frac{1}{52} a^{9} - \frac{1}{52} a^{7} - \frac{19}{52} a^{6} + \frac{2}{13} a^{5} - \frac{1}{4} a^{4} - \frac{9}{26} a^{3} - \frac{15}{52} a^{2} + \frac{6}{13} a + \frac{5}{13}$, $\frac{1}{81640} a^{12} - \frac{35}{4082} a^{11} - \frac{16}{10205} a^{10} + \frac{85}{16328} a^{9} - \frac{1219}{81640} a^{8} + \frac{2569}{16328} a^{7} + \frac{1759}{16328} a^{6} + \frac{4439}{16328} a^{5} - \frac{3706}{10205} a^{4} - \frac{281}{8164} a^{3} - \frac{10997}{81640} a^{2} + \frac{1329}{8164} a - \frac{6023}{20410}$, $\frac{1}{81640} a^{13} - \frac{36}{10205} a^{11} + \frac{63}{16328} a^{10} - \frac{449}{81640} a^{9} + \frac{213}{16328} a^{8} - \frac{1673}{16328} a^{7} + \frac{253}{1256} a^{6} - \frac{1187}{20410} a^{5} + \frac{3039}{8164} a^{4} - \frac{20207}{81640} a^{3} + \frac{406}{2041} a^{2} - \frac{4684}{10205} a + \frac{877}{2041}$, $\frac{1}{4245280} a^{14} - \frac{3}{849056} a^{13} + \frac{1}{326560} a^{12} - \frac{151}{65312} a^{11} + \frac{493}{163280} a^{10} - \frac{1221}{65312} a^{9} - \frac{2803}{326560} a^{8} - \frac{8453}{65312} a^{7} + \frac{19627}{163280} a^{6} + \frac{7921}{65312} a^{5} - \frac{491}{65312} a^{4} + \frac{29967}{65312} a^{3} - \frac{65743}{326560} a^{2} - \frac{59925}{424528} a + \frac{212707}{1061320}$, $\frac{1}{4506540487785517443825178546124123119027379360} a^{15} - \frac{202465754755612580575570808711744853423}{2253270243892758721912589273062061559513689680} a^{14} - \frac{3400443172701476797143108078984399619}{2253270243892758721912589273062061559513689680} a^{13} + \frac{709952003859076845550834365761112738053}{173328480299442978608660713312466273808745360} a^{12} + \frac{768773104913582379357807424240294244560567}{346656960598885957217321426624932547617490720} a^{11} - \frac{2694393514821420866399113998023562677877787}{346656960598885957217321426624932547617490720} a^{10} - \frac{36386685413999584503805008105308179179885}{2476121147134899694409438761606661054410648} a^{9} + \frac{247138584195413594861253058869449111941823}{17332848029944297860866071331246627380874536} a^{8} + \frac{17578743342927728336612747248662740213108349}{346656960598885957217321426624932547617490720} a^{7} + \frac{158695569246525195696749404831165841518316571}{346656960598885957217321426624932547617490720} a^{6} + \frac{401297312389308190700913557534126173670087}{1456541851255823349652611036239212384947440} a^{5} + \frac{31277913469588005032132978563164025446439651}{86664240149721489304330356656233136904372680} a^{4} + \frac{628246447284500341350031053272517445706887}{1666620002879259409698660704927560325084090} a^{3} + \frac{13085051331043790421044629934743912395227051}{28704079540035142954300500293784223688072480} a^{2} + \frac{25379204533728460272173538508535760806745475}{64379149825507392054645407801773187414676848} a - \frac{19710239475344218413679554909382280086801205}{225327024389275872191258927306206155951368968}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 15803632252900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2041}) \), \(\Q(\sqrt{157}) \), 4.4.344929.1 x2, 4.4.54153853.1 x2, \(\Q(\sqrt{13}, \sqrt{157})\), 8.8.2932639794745609.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$157$157.8.4.1$x^{8} + 739470 x^{4} - 3869893 x^{2} + 136703970225$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
157.8.6.2$x^{8} + 1727 x^{4} + 887364$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$