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Properties

Label 16.0.32349497931...001.19
Degree $16$
Signature $[0, 8]$
Discriminant $31^{8}\cdot 41^{14}$
Root discriminant $143.50$
Ramified primes $31, 41$
Class number $128$ (GRH)
Class group $[2, 2, 32]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1031411207, 648497959, 140732982, -116122039, -86737142, -10471450, 7432505, 3051194, 274964, -160157, -39834, 3290, 1977, 28, -66, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 66*x^14 + 28*x^13 + 1977*x^12 + 3290*x^11 - 39834*x^10 - 160157*x^9 + 274964*x^8 + 3051194*x^7 + 7432505*x^6 - 10471450*x^5 - 86737142*x^4 - 116122039*x^3 + 140732982*x^2 + 648497959*x + 1031411207)
 
gp: K = bnfinit(x^16 - 2*x^15 - 66*x^14 + 28*x^13 + 1977*x^12 + 3290*x^11 - 39834*x^10 - 160157*x^9 + 274964*x^8 + 3051194*x^7 + 7432505*x^6 - 10471450*x^5 - 86737142*x^4 - 116122039*x^3 + 140732982*x^2 + 648497959*x + 1031411207, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 66 x^{14} + 28 x^{13} + 1977 x^{12} + 3290 x^{11} - 39834 x^{10} - 160157 x^{9} + 274964 x^{8} + 3051194 x^{7} + 7432505 x^{6} - 10471450 x^{5} - 86737142 x^{4} - 116122039 x^{3} + 140732982 x^{2} + 648497959 x + 1031411207 \)

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:  \(32349497931606921267167334562710001=31^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.50$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{124} a^{14} + \frac{1}{31} a^{13} + \frac{11}{124} a^{12} + \frac{27}{124} a^{11} + \frac{14}{31} a^{10} - \frac{27}{124} a^{9} - \frac{45}{124} a^{8} + \frac{6}{31} a^{7} + \frac{47}{124} a^{6} + \frac{23}{124} a^{5} + \frac{8}{31} a^{4} - \frac{37}{124} a^{3} + \frac{7}{124} a^{2} + \frac{29}{62} a - \frac{37}{124}$, $\frac{1}{10155538108613418184008424312162330910107290581051891673716} a^{15} - \frac{1478830257746895480986815415496797385041471473530551355}{2538884527153354546002106078040582727526822645262972918429} a^{14} + \frac{1021769551109505044156337709339711443093500822894354059333}{10155538108613418184008424312162330910107290581051891673716} a^{13} + \frac{129101236809484986666503313888838776419821148049257198453}{10155538108613418184008424312162330910107290581051891673716} a^{12} - \frac{333719227293030413711894872848634370202255125658335476614}{2538884527153354546002106078040582727526822645262972918429} a^{11} - \frac{16067558315198513612340375370219669311365047227854430423}{327598003503658651097045945553623577745396470356512634636} a^{10} - \frac{1315561801150423950931157656808532344456828883310304464255}{10155538108613418184008424312162330910107290581051891673716} a^{9} + \frac{582837912993537712562969930643481663123725961181979498011}{2538884527153354546002106078040582727526822645262972918429} a^{8} + \frac{1239342139225492836933678208759031470629668023385937217721}{10155538108613418184008424312162330910107290581051891673716} a^{7} + \frac{5067183206664539235583152475748251808649571122130012694217}{10155538108613418184008424312162330910107290581051891673716} a^{6} + \frac{1049712455011424184468677594016744199112823504154567800334}{2538884527153354546002106078040582727526822645262972918429} a^{5} - \frac{964512417370713838783756173895201136751250221509361825983}{10155538108613418184008424312162330910107290581051891673716} a^{4} - \frac{3700755871901389616819089581580511150194374843382705874671}{10155538108613418184008424312162330910107290581051891673716} a^{3} + \frac{1763415815923432855553367051270997318648452283746129191989}{5077769054306709092004212156081165455053645290525945836858} a^{2} + \frac{2584157581112408777464597153063293799153827691019606896777}{10155538108613418184008424312162330910107290581051891673716} a + \frac{293963216159070370658248268924759044955748361667469250825}{5077769054306709092004212156081165455053645290525945836858}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{32}$, which has order $128$ (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:  \( 309377418.474 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.2136551.1, 4.0.66233081.2, 4.2.52111.1, 8.0.179859661768855001.2 x2, 8.0.4386821018752561.4

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$