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Properties

Label 16.0.19239111134...6993.2
Degree $16$
Signature $[0, 8]$
Discriminant $47^{8}\cdot 97^{7}$
Root discriminant $50.73$
Ramified primes $47, 97$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![122768, 214956, 81135, 9928, 96796, 76302, -8350, -17395, 202, 607, -860, -54, 123, 23, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 23*x^13 + 123*x^12 - 54*x^11 - 860*x^10 + 607*x^9 + 202*x^8 - 17395*x^7 - 8350*x^6 + 76302*x^5 + 96796*x^4 + 9928*x^3 + 81135*x^2 + 214956*x + 122768)
 
gp: K = bnfinit(x^16 - 3*x^15 + 23*x^13 + 123*x^12 - 54*x^11 - 860*x^10 + 607*x^9 + 202*x^8 - 17395*x^7 - 8350*x^6 + 76302*x^5 + 96796*x^4 + 9928*x^3 + 81135*x^2 + 214956*x + 122768, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 23 x^{13} + 123 x^{12} - 54 x^{11} - 860 x^{10} + 607 x^{9} + 202 x^{8} - 17395 x^{7} - 8350 x^{6} + 76302 x^{5} + 96796 x^{4} + 9928 x^{3} + 81135 x^{2} + 214956 x + 122768 \)

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:  \(1923911113486862917838536993=47^{8}\cdot 97^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.73$
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:  $47, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{56} a^{14} + \frac{1}{28} a^{13} + \frac{1}{8} a^{12} + \frac{5}{28} a^{11} - \frac{2}{7} a^{10} - \frac{5}{28} a^{9} - \frac{11}{28} a^{8} + \frac{23}{56} a^{7} + \frac{19}{56} a^{6} + \frac{19}{56} a^{5} + \frac{9}{28} a^{4} + \frac{3}{8} a^{3} - \frac{19}{56} a^{2} + \frac{13}{28} a + \frac{3}{7}$, $\frac{1}{6346963919151015409879959091900156264} a^{15} + \frac{4246924815747340072876169184008103}{3173481959575507704939979545950078132} a^{14} - \frac{51318813209510056050579725755006055}{6346963919151015409879959091900156264} a^{13} - \frac{312089930580171998085915663280216959}{3173481959575507704939979545950078132} a^{12} + \frac{176510217123599690765739694337872061}{3173481959575507704939979545950078132} a^{11} + \frac{22557387652002915135163762382463325}{3173481959575507704939979545950078132} a^{10} + \frac{481868345801217253548923403305684197}{3173481959575507704939979545950078132} a^{9} - \frac{1373244271708267136620613353754772377}{6346963919151015409879959091900156264} a^{8} + \frac{599844024718618388794699768487976551}{6346963919151015409879959091900156264} a^{7} - \frac{516903547482106278369767967651620859}{6346963919151015409879959091900156264} a^{6} + \frac{358027413715605569774979647926440507}{793370489893876926234994886487519533} a^{5} + \frac{978512038397558980737417069256587043}{6346963919151015409879959091900156264} a^{4} - \frac{780749431468237885366491485836612383}{6346963919151015409879959091900156264} a^{3} + \frac{284233254881267158804847065469469211}{1586740979787753852469989772975039066} a^{2} + \frac{289507191142613524217146619564841923}{3173481959575507704939979545950078132} a + \frac{207262658139605035586710547874438837}{793370489893876926234994886487519533}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 2921943.17435 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.214273.1, 8.0.4453553097313.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.7.2$x^{8} - 2425$$8$$1$$7$$C_8$$[\ ]_{8}$