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Properties

Label 16.0.84480512701...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 7^{12}$
Root discriminant $17.60$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, -1900, 4994, -8300, 9652, -8600, 6502, -4545, 2915, -1665, 863, -390, 167, -60, 21, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361)
 
gp: K = bnfinit(x^16 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 21 x^{14} - 60 x^{13} + 167 x^{12} - 390 x^{11} + 863 x^{10} - 1665 x^{9} + 2915 x^{8} - 4545 x^{7} + 6502 x^{6} - 8600 x^{5} + 9652 x^{4} - 8300 x^{3} + 4994 x^{2} - 1900 x + 361 \)

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:  \(84480512701416015625=5^{14}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.60$
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:  $5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{4}{11} a^{6} - \frac{2}{11} a^{5} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} + \frac{4}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{9} - \frac{5}{11} a^{7} + \frac{5}{11} a^{6} - \frac{2}{11} a^{5} - \frac{1}{11} a^{4} + \frac{5}{11} a^{2} - \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{10} - \frac{1}{11}$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{121} a^{13} + \frac{1}{121} a^{12} + \frac{2}{121} a^{11} - \frac{3}{121} a^{10} + \frac{4}{121} a^{9} + \frac{3}{121} a^{8} + \frac{21}{121} a^{7} + \frac{19}{121} a^{6} - \frac{36}{121} a^{5} - \frac{37}{121} a^{4} + \frac{18}{121} a^{3} - \frac{3}{11} a^{2} - \frac{21}{121} a + \frac{39}{121}$, $\frac{1}{121} a^{14} + \frac{1}{121} a^{12} - \frac{5}{121} a^{11} - \frac{4}{121} a^{10} - \frac{1}{121} a^{9} - \frac{4}{121} a^{8} + \frac{20}{121} a^{7} + \frac{3}{11} a^{6} + \frac{43}{121} a^{5} + \frac{5}{11} a^{4} - \frac{29}{121} a^{3} - \frac{10}{121} a^{2} - \frac{28}{121} a + \frac{49}{121}$, $\frac{1}{31927387789} a^{15} - \frac{130977284}{31927387789} a^{14} - \frac{5091029}{31927387789} a^{13} - \frac{964810959}{31927387789} a^{12} + \frac{1428416956}{31927387789} a^{11} + \frac{767808249}{31927387789} a^{10} + \frac{1227629566}{31927387789} a^{9} - \frac{707241553}{31927387789} a^{8} + \frac{7011204866}{31927387789} a^{7} + \frac{12823937573}{31927387789} a^{6} - \frac{10076014085}{31927387789} a^{5} + \frac{681230400}{2902489799} a^{4} + \frac{125545345}{1680388831} a^{3} + \frac{10631161944}{31927387789} a^{2} + \frac{5603726199}{31927387789} a - \frac{709468879}{1680388831}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{636632551}{31927387789} a^{15} + \frac{2864587133}{31927387789} a^{14} - \frac{11934579568}{31927387789} a^{13} + \frac{32070288634}{31927387789} a^{12} - \frac{89756675916}{31927387789} a^{11} + \frac{201159205318}{31927387789} a^{10} - \frac{443855865113}{31927387789} a^{9} + \frac{823219991343}{31927387789} a^{8} - \frac{1415787238940}{31927387789} a^{7} + \frac{2121531755396}{31927387789} a^{6} - \frac{2976694803706}{31927387789} a^{5} + \frac{347074939690}{2902489799} a^{4} - \frac{211148594117}{1680388831} a^{3} + \frac{2986228431833}{31927387789} a^{2} - \frac{1342705929392}{31927387789} a + \frac{16111982569}{1680388831} \) (order $10$)
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:  \( 19929.8787165 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

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 $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 4.4.6125.1, \(\Q(\zeta_{5})\), 8.0.37515625.1, 8.4.9191328125.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
7Data not computed