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Properties

Label 16.0.16777216000...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 5^{12}$
Root discriminant $15.91$
Ramified primes $2, 5$
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![25, -200, 800, -2000, 3440, -4240, 3840, -2680, 1686, -1144, 768, -400, 176, -80, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 176*x^12 - 400*x^11 + 768*x^10 - 1144*x^9 + 1686*x^8 - 2680*x^7 + 3840*x^6 - 4240*x^5 + 3440*x^4 - 2000*x^3 + 800*x^2 - 200*x + 25)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 176*x^12 - 400*x^11 + 768*x^10 - 1144*x^9 + 1686*x^8 - 2680*x^7 + 3840*x^6 - 4240*x^5 + 3440*x^4 - 2000*x^3 + 800*x^2 - 200*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 80 x^{13} + 176 x^{12} - 400 x^{11} + 768 x^{10} - 1144 x^{9} + 1686 x^{8} - 2680 x^{7} + 3840 x^{6} - 4240 x^{5} + 3440 x^{4} - 2000 x^{3} + 800 x^{2} - 200 x + 25 \)

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:  \(16777216000000000000=2^{36}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.91$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{8} + \frac{3}{8}$, $\frac{1}{8} a^{9} + \frac{3}{8} a$, $\frac{1}{8} a^{10} + \frac{3}{8} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{3} - \frac{3}{16} a^{2} - \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{80} a^{12} + \frac{1}{40} a^{11} + \frac{1}{40} a^{10} + \frac{1}{80} a^{8} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{11}{80} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{16}$, $\frac{1}{80} a^{13} - \frac{1}{40} a^{11} - \frac{1}{20} a^{10} + \frac{1}{80} a^{9} - \frac{1}{40} a^{8} + \frac{1}{10} a^{7} + \frac{19}{80} a^{5} + \frac{1}{10} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{7}{16} a - \frac{3}{8}$, $\frac{1}{80} a^{14} - \frac{1}{16} a^{10} - \frac{1}{40} a^{9} - \frac{1}{16} a^{6} - \frac{1}{10} a^{4} + \frac{5}{16} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{35439200} a^{15} + \frac{106917}{35439200} a^{14} - \frac{142673}{35439200} a^{13} - \frac{2559}{1417568} a^{12} - \frac{761099}{35439200} a^{11} - \frac{118007}{7087840} a^{10} - \frac{184777}{35439200} a^{9} + \frac{515121}{35439200} a^{8} - \frac{3833949}{35439200} a^{7} + \frac{1006867}{7087840} a^{6} + \frac{1609993}{7087840} a^{5} + \frac{1080511}{7087840} a^{4} + \frac{1138579}{7087840} a^{3} + \frac{186671}{1417568} a^{2} + \frac{350669}{1417568} a - \frac{188165}{1417568}$

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:  \( -\frac{2017337}{17719600} a^{15} + \frac{7532113}{8859800} a^{14} - \frac{56643719}{17719600} a^{13} + \frac{6599473}{885980} a^{12} - \frac{287478377}{17719600} a^{11} + \frac{65975907}{1771960} a^{10} - \frac{1211135231}{17719600} a^{9} + \frac{423315837}{4429900} a^{8} - \frac{2552405947}{17719600} a^{7} + \frac{411819051}{1771960} a^{6} - \frac{226661101}{708784} a^{5} + \frac{57155769}{177196} a^{4} - \frac{826329343}{3543920} a^{3} + \frac{41019545}{354392} a^{2} - \frac{27017797}{708784} a + \frac{1146117}{177196} \) (order $4$)
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:  \( 1626.80823745 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 4.2.1600.1 x2, 4.0.1280.1 x2, 4.0.8000.1 x2, 4.2.2000.1 x2, 4.0.8000.2, 4.4.8000.1, 8.0.40960000.2, 8.0.64000000.3, 8.0.1024000000.2, 8.0.1024000000.3 x2, 8.4.1024000000.2 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$