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Properties

Label 16.16.5508664624...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 41^{12}$
Root discriminant $54.18$
Ramified primes $5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1021, 20159, -51826, -55591, 127991, 75917, -114529, -56263, 44732, 19533, -8509, -2982, 906, 196, -51, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 51*x^14 + 196*x^13 + 906*x^12 - 2982*x^11 - 8509*x^10 + 19533*x^9 + 44732*x^8 - 56263*x^7 - 114529*x^6 + 75917*x^5 + 127991*x^4 - 55591*x^3 - 51826*x^2 + 20159*x + 1021)
 
gp: K = bnfinit(x^16 - 4*x^15 - 51*x^14 + 196*x^13 + 906*x^12 - 2982*x^11 - 8509*x^10 + 19533*x^9 + 44732*x^8 - 56263*x^7 - 114529*x^6 + 75917*x^5 + 127991*x^4 - 55591*x^3 - 51826*x^2 + 20159*x + 1021, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 51 x^{14} + 196 x^{13} + 906 x^{12} - 2982 x^{11} - 8509 x^{10} + 19533 x^{9} + 44732 x^{8} - 56263 x^{7} - 114529 x^{6} + 75917 x^{5} + 127991 x^{4} - 55591 x^{3} - 51826 x^{2} + 20159 x + 1021 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5508664624112838398681640625=5^{12}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.18$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{9} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{10} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{6} + \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{7} + \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a$, $\frac{1}{40} a^{12} + \frac{1}{40} a^{11} - \frac{1}{40} a^{10} - \frac{1}{40} a^{9} - \frac{1}{20} a^{8} - \frac{1}{8} a^{7} - \frac{3}{20} a^{6} - \frac{3}{40} a^{5} + \frac{1}{20} a^{4} - \frac{13}{40} a^{3} - \frac{3}{40} a^{2} + \frac{1}{4} a + \frac{17}{40}$, $\frac{1}{40} a^{13} - \frac{1}{20} a^{11} - \frac{1}{40} a^{9} + \frac{1}{40} a^{8} - \frac{9}{40} a^{7} + \frac{3}{40} a^{6} - \frac{3}{40} a^{5} - \frac{19}{40} a^{4} + \frac{9}{20} a^{3} - \frac{7}{40} a^{2} - \frac{1}{8} a - \frac{13}{40}$, $\frac{1}{200} a^{14} - \frac{1}{200} a^{13} + \frac{1}{100} a^{12} + \frac{3}{100} a^{11} - \frac{9}{200} a^{10} + \frac{1}{100} a^{9} - \frac{1}{100} a^{8} + \frac{8}{25} a^{7} + \frac{31}{100} a^{6} + \frac{4}{25} a^{5} + \frac{89}{200} a^{4} + \frac{51}{200} a^{3} - \frac{1}{100} a^{2} + \frac{21}{50} a + \frac{9}{200}$, $\frac{1}{279924287790561984200} a^{15} - \frac{62594345781920997}{55984857558112396840} a^{14} + \frac{376965477749881323}{139962143895280992100} a^{13} + \frac{486340759376583063}{279924287790561984200} a^{12} - \frac{2693236422284161347}{69981071947640496050} a^{11} + \frac{2471981654325822713}{279924287790561984200} a^{10} + \frac{2510105884407092229}{55984857558112396840} a^{9} - \frac{1147794397011490949}{139962143895280992100} a^{8} - \frac{8045296667283801799}{279924287790561984200} a^{7} - \frac{42928193176119596253}{139962143895280992100} a^{6} + \frac{31660054628227056343}{139962143895280992100} a^{5} + \frac{1580218921256054233}{11196971511622479368} a^{4} - \frac{127427443041517543611}{279924287790561984200} a^{3} + \frac{38328480094676336057}{279924287790561984200} a^{2} + \frac{43220235448446791403}{279924287790561984200} a + \frac{48972482276296938169}{279924287790561984200}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 336719650.335 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

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 $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 4.4.210125.1 x2, 4.4.5125.1 x2, 4.4.68921.1, 4.4.1723025.1, 8.8.44152515625.1, 8.8.2968815150625.1, 8.8.74220378765625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$