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Properties

Label 16.0.17415183620...4081.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 61^{12}$
Root discriminant $37.81$
Ramified primes $3, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5329, -20586, 47112, -83598, 112678, -95394, 57297, -15927, -4947, 8310, -939, -57, 250, -42, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 9*x^14 - 42*x^13 + 250*x^12 - 57*x^11 - 939*x^10 + 8310*x^9 - 4947*x^8 - 15927*x^7 + 57297*x^6 - 95394*x^5 + 112678*x^4 - 83598*x^3 + 47112*x^2 - 20586*x + 5329)
 
gp: K = bnfinit(x^16 - 3*x^15 - 9*x^14 - 42*x^13 + 250*x^12 - 57*x^11 - 939*x^10 + 8310*x^9 - 4947*x^8 - 15927*x^7 + 57297*x^6 - 95394*x^5 + 112678*x^4 - 83598*x^3 + 47112*x^2 - 20586*x + 5329, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 9 x^{14} - 42 x^{13} + 250 x^{12} - 57 x^{11} - 939 x^{10} + 8310 x^{9} - 4947 x^{8} - 15927 x^{7} + 57297 x^{6} - 95394 x^{5} + 112678 x^{4} - 83598 x^{3} + 47112 x^{2} - 20586 x + 5329 \)

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:  \(17415183620366462784744081=3^{8}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.81$
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:  $3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{156} a^{12} - \frac{1}{156} a^{11} - \frac{1}{13} a^{10} + \frac{19}{156} a^{9} + \frac{25}{156} a^{8} - \frac{11}{156} a^{7} + \frac{7}{26} a^{6} - \frac{10}{39} a^{5} - \frac{1}{39} a^{4} - \frac{11}{78} a^{3} + \frac{21}{52} a^{2} - \frac{31}{78} a + \frac{23}{156}$, $\frac{1}{56940} a^{13} - \frac{17}{28470} a^{12} - \frac{3931}{56940} a^{11} + \frac{3821}{56940} a^{10} + \frac{1}{130} a^{9} - \frac{1237}{28470} a^{8} + \frac{3617}{11388} a^{7} + \frac{7139}{28470} a^{6} + \frac{2763}{9490} a^{5} - \frac{7012}{14235} a^{4} - \frac{3319}{56940} a^{3} + \frac{17177}{56940} a^{2} - \frac{89}{56940} a + \frac{113}{260}$, $\frac{1}{341640} a^{14} + \frac{1}{170820} a^{13} + \frac{8}{8541} a^{12} - \frac{55}{949} a^{11} + \frac{853}{18980} a^{10} - \frac{2017}{113880} a^{9} + \frac{388}{4745} a^{8} - \frac{12903}{37960} a^{7} + \frac{9037}{18980} a^{6} + \frac{2485}{11388} a^{5} + \frac{15757}{37960} a^{4} + \frac{7937}{37960} a^{3} - \frac{73561}{170820} a^{2} + \frac{128123}{341640} a + \frac{149}{4680}$, $\frac{1}{235679292982995239093475240} a^{15} - \frac{22931111173196761063}{78559764327665079697825080} a^{14} - \frac{8801001313164257246}{5891982324574880977336881} a^{13} + \frac{97364837199204167643983}{58919823245748809773368810} a^{12} - \frac{192427502060239160033693}{6546647027305423308152090} a^{11} + \frac{1324386598694604369742319}{78559764327665079697825080} a^{10} - \frac{11899836163878670268246489}{78559764327665079697825080} a^{9} - \frac{12431002499332477308644017}{78559764327665079697825080} a^{8} - \frac{1794609281277846272076343}{26186588109221693232608360} a^{7} - \frac{305857658056354770996290}{1963994108191626992445627} a^{6} + \frac{24674127557884673280535073}{78559764327665079697825080} a^{5} + \frac{9740125848407270363040389}{39279882163832539848912540} a^{4} - \frac{108305059565148543526957703}{235679292982995239093475240} a^{3} - \frac{8232747443882453284651127}{78559764327665079697825080} a^{2} + \frac{49671298280856399596033363}{117839646491497619546737620} a - \frac{4754985718867836850561}{3228483465520482727307880}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{22423944268374794554}{151076469860894384034279} a^{15} - \frac{57126086280887794309}{151076469860894384034279} a^{14} - \frac{375403303600748637948}{251794116434823973390465} a^{13} - \frac{1748695035861013184563}{251794116434823973390465} a^{12} + \frac{8509928821033303525678}{251794116434823973390465} a^{11} + \frac{1518534446062004142757}{251794116434823973390465} a^{10} - \frac{33550940672315357417249}{251794116434823973390465} a^{9} + \frac{295751498090770584728607}{251794116434823973390465} a^{8} - \frac{10912624048120458438476}{50358823286964794678093} a^{7} - \frac{589054409280435075553519}{251794116434823973390465} a^{6} + \frac{1876600472060185123257461}{251794116434823973390465} a^{5} - \frac{2765718049918372867463276}{251794116434823973390465} a^{4} + \frac{9451541054930584745316506}{755382349304471920171395} a^{3} - \frac{5855240823531132218436038}{755382349304471920171395} a^{2} + \frac{1284118917967921314391627}{251794116434823973390465} a - \frac{4136415577319462201572}{3449234471709917443705} \) (order $6$)
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:  \( 2172533.23185 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-183}) \), \(\Q(\sqrt{-3}, \sqrt{61})\), 4.2.680943.1 x2, 4.0.2042829.1 x2, 4.0.549.1 x2, 4.2.11163.1 x2, 4.0.226981.1, 4.4.2042829.1, 8.0.4173150323241.2, 8.0.1121513121.2, 8.0.4173150323241.1, 8.0.463683369249.1 x2, 8.4.4173150323241.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$