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Properties

Label 16.16.4401660273...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 101^{12}$
Root discriminant $71.24$
Ramified primes $5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-317375, 891400, 864770, -3315295, 268374, 2560624, -785803, -659347, 256088, 79762, -34929, -5016, 2362, 159, -78, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375)
 
gp: K = bnfinit(x^16 - 2*x^15 - 78*x^14 + 159*x^13 + 2362*x^12 - 5016*x^11 - 34929*x^10 + 79762*x^9 + 256088*x^8 - 659347*x^7 - 785803*x^6 + 2560624*x^5 + 268374*x^4 - 3315295*x^3 + 864770*x^2 + 891400*x - 317375, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 78 x^{14} + 159 x^{13} + 2362 x^{12} - 5016 x^{11} - 34929 x^{10} + 79762 x^{9} + 256088 x^{8} - 659347 x^{7} - 785803 x^{6} + 2560624 x^{5} + 268374 x^{4} - 3315295 x^{3} + 864770 x^{2} + 891400 x - 317375 \)

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:  \(440166027395300672133281640625=5^{8}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.24$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{65} a^{13} + \frac{6}{65} a^{12} - \frac{1}{13} a^{11} - \frac{27}{65} a^{9} + \frac{1}{5} a^{8} - \frac{2}{13} a^{7} - \frac{5}{13} a^{6} + \frac{6}{65} a^{5} - \frac{19}{65} a^{4} + \frac{2}{13} a^{3} + \frac{4}{13} a^{2} - \frac{3}{13} a - \frac{1}{13}$, $\frac{1}{325} a^{14} + \frac{2}{325} a^{13} + \frac{2}{65} a^{12} - \frac{6}{325} a^{11} - \frac{27}{325} a^{10} + \frac{121}{325} a^{9} + \frac{24}{65} a^{8} + \frac{2}{325} a^{7} - \frac{24}{325} a^{6} - \frac{108}{325} a^{5} - \frac{1}{65} a^{4} + \frac{149}{325} a^{3} - \frac{19}{65} a^{2} + \frac{24}{65} a + \frac{6}{13}$, $\frac{1}{3985156594548265063937838677875} a^{15} + \frac{4073295958163310430549638747}{3985156594548265063937838677875} a^{14} + \frac{5341711510803964905444978998}{797031318909653012787567735575} a^{13} - \frac{332454283702744276974949484601}{3985156594548265063937838677875} a^{12} - \frac{359760770401671210490114426937}{3985156594548265063937838677875} a^{11} + \frac{330282723577436894715783392181}{3985156594548265063937838677875} a^{10} + \frac{2131438443885707842305150447}{797031318909653012787567735575} a^{9} + \frac{542366131733591579219333407817}{3985156594548265063937838677875} a^{8} + \frac{596255847141360225678314987421}{3985156594548265063937838677875} a^{7} + \frac{1841141010679184896576253230562}{3985156594548265063937838677875} a^{6} - \frac{8774949524586112074134029129}{31881252756386120511502709423} a^{5} + \frac{1832060098324133675701024506454}{3985156594548265063937838677875} a^{4} + \frac{175475930094431533986154733809}{797031318909653012787567735575} a^{3} - \frac{150647909706662548363721198301}{797031318909653012787567735575} a^{2} + \frac{2997711180885305325507749038}{159406263781930602557513547115} a + \frac{10294198917331760117818254519}{31881252756386120511502709423}$

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:  \( 18951197490.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{505}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.4.51005.1 x2, 4.4.2525.1 x2, 8.8.65037750625.1, 8.8.132690018825125.1 x4

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$101$101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$