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Properties

Label 16.8.70699102466...7153.3
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 89^{12}$
Root discriminant $412.66$
Ramified primes $17, 89$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10012855669, 93956112739, -241662173554, 293577770820, -142014209188, -6256536692, 2958541378, -737819626, 171044192, 9402260, -2342188, 251778, -41872, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 41872*x^12 + 251778*x^11 - 2342188*x^10 + 9402260*x^9 + 171044192*x^8 - 737819626*x^7 + 2958541378*x^6 - 6256536692*x^5 - 142014209188*x^4 + 293577770820*x^3 - 241662173554*x^2 + 93956112739*x - 10012855669)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 41872*x^12 + 251778*x^11 - 2342188*x^10 + 9402260*x^9 + 171044192*x^8 - 737819626*x^7 + 2958541378*x^6 - 6256536692*x^5 - 142014209188*x^4 + 293577770820*x^3 - 241662173554*x^2 + 93956112739*x - 10012855669, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 41872 x^{12} + 251778 x^{11} - 2342188 x^{10} + 9402260 x^{9} + 171044192 x^{8} - 737819626 x^{7} + 2958541378 x^{6} - 6256536692 x^{5} - 142014209188 x^{4} + 293577770820 x^{3} - 241662173554 x^{2} + 93956112739 x - 10012855669 \)

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:  \(706991024666918541719408792174287819307153=17^{15}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $412.66$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{89} a^{4} - \frac{2}{89} a^{3} - \frac{43}{89} a^{2} + \frac{44}{89} a + \frac{39}{89}$, $\frac{1}{89} a^{5} + \frac{42}{89} a^{3} - \frac{42}{89} a^{2} + \frac{38}{89} a - \frac{11}{89}$, $\frac{1}{89} a^{6} + \frac{42}{89} a^{3} - \frac{25}{89} a^{2} + \frac{10}{89} a - \frac{36}{89}$, $\frac{1}{89} a^{7} - \frac{30}{89} a^{3} + \frac{36}{89} a^{2} - \frac{15}{89} a - \frac{36}{89}$, $\frac{1}{7921} a^{8} - \frac{4}{7921} a^{7} + \frac{7}{7921} a^{6} - \frac{7}{7921} a^{5} - \frac{29}{7921} a^{4} + \frac{65}{7921} a^{3} - \frac{3020}{7921} a^{2} + \frac{2987}{7921} a + \frac{3123}{7921}$, $\frac{1}{7921} a^{9} - \frac{9}{7921} a^{7} + \frac{21}{7921} a^{6} + \frac{32}{7921} a^{5} + \frac{38}{7921} a^{4} + \frac{800}{7921} a^{3} - \frac{816}{7921} a^{2} - \frac{1394}{7921} a - \frac{858}{7921}$, $\frac{1}{7921} a^{10} - \frac{15}{7921} a^{7} + \frac{6}{7921} a^{6} - \frac{25}{7921} a^{5} + \frac{5}{7921} a^{4} - \frac{2901}{7921} a^{3} - \frac{3387}{7921} a^{2} + \frac{1639}{7921} a + \frac{2564}{7921}$, $\frac{1}{7921} a^{11} + \frac{35}{7921} a^{7} - \frac{9}{7921} a^{6} - \frac{11}{7921} a^{5} - \frac{43}{7921} a^{4} - \frac{3747}{7921} a^{3} - \frac{1386}{7921} a^{2} + \frac{3314}{7921} a + \frac{31}{7921}$, $\frac{1}{704969} a^{12} - \frac{6}{704969} a^{11} - \frac{28}{704969} a^{10} + \frac{17}{704969} a^{9} - \frac{8}{704969} a^{8} + \frac{398}{704969} a^{7} - \frac{2561}{704969} a^{6} + \frac{3793}{704969} a^{5} - \frac{1825}{704969} a^{4} + \frac{175963}{704969} a^{3} + \frac{296591}{704969} a^{2} + \frac{72790}{704969} a + \frac{72135}{704969}$, $\frac{1}{704969} a^{13} + \frac{25}{704969} a^{11} + \frac{27}{704969} a^{10} + \frac{5}{704969} a^{9} - \frac{6}{704969} a^{8} + \frac{2497}{704969} a^{7} + \frac{175}{704969} a^{6} - \frac{694}{704969} a^{5} + \frac{2677}{704969} a^{4} + \frac{35881}{704969} a^{3} - \frac{173749}{704969} a^{2} + \frac{148336}{704969} a + \frac{22876}{704969}$, $\frac{1}{2615197288994624620250804099} a^{14} - \frac{7}{2615197288994624620250804099} a^{13} - \frac{67692169454120785221}{2615197288994624620250804099} a^{12} + \frac{406153016724724711417}{2615197288994624620250804099} a^{11} - \frac{55891439661353237695326}{2615197288994624620250804099} a^{10} - \frac{54425862049017123093145}{2615197288994624620250804099} a^{9} - \frac{164392687502218579581123}{2615197288994624620250804099} a^{8} + \frac{1979073578594369033854092}{2615197288994624620250804099} a^{7} - \frac{9621390116519851921763818}{2615197288994624620250804099} a^{6} + \frac{2125869052266084822955029}{2615197288994624620250804099} a^{5} - \frac{484213031674904830046896}{2615197288994624620250804099} a^{4} - \frac{311110369173600794777323742}{2615197288994624620250804099} a^{3} + \frac{1290064363163510751059731882}{2615197288994624620250804099} a^{2} - \frac{534556653839694886108554730}{2615197288994624620250804099} a + \frac{1064484715707295840942064922}{2615197288994624620250804099}$, $\frac{1}{1331135420098263931707659286391} a^{15} + \frac{19}{102395032315251071669819945107} a^{14} - \frac{155873530635789851933381}{1331135420098263931707659286391} a^{13} - \frac{892268083692603682128673}{1331135420098263931707659286391} a^{12} - \frac{20385551389427016188478076}{1331135420098263931707659286391} a^{11} + \frac{34449601973159057609191739}{1331135420098263931707659286391} a^{10} - \frac{14793558480028635789496260}{1331135420098263931707659286391} a^{9} - \frac{76739749391696988873621394}{1331135420098263931707659286391} a^{8} + \frac{1815773254843586445203167768}{1331135420098263931707659286391} a^{7} - \frac{6180312830497349696657104085}{1331135420098263931707659286391} a^{6} - \frac{7084349116648519950635966989}{1331135420098263931707659286391} a^{5} - \frac{4946719817370790910966768276}{1331135420098263931707659286391} a^{4} - \frac{39064464133491398542610507519}{102395032315251071669819945107} a^{3} - \frac{259813706130473943176175658041}{1331135420098263931707659286391} a^{2} - \frac{325219987103960158634789835233}{1331135420098263931707659286391} a + \frac{640474686343413789396955369923}{1331135420098263931707659286391}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 119950898824000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

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 $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.25745567912986193.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$89$89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$