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Properties

Label 16.0.11981707062...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{12}\cdot 61^{8}$
Root discriminant $36.93$
Ramified primes $2, 5, 61$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1076, -7276, 23226, -39494, 44991, -34912, 20318, -9382, 3932, -1594, 592, -296, 166, -78, 34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076)
 
gp: K = bnfinit(x^16 - 8*x^15 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 34 x^{14} - 78 x^{13} + 166 x^{12} - 296 x^{11} + 592 x^{10} - 1594 x^{9} + 3932 x^{8} - 9382 x^{7} + 20318 x^{6} - 34912 x^{5} + 44991 x^{4} - 39494 x^{3} + 23226 x^{2} - 7276 x + 1076 \)

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:  \(11981707062330062500000000=2^{8}\cdot 5^{12}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.93$
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:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{336} a^{13} - \frac{5}{84} a^{12} + \frac{5}{168} a^{11} + \frac{13}{336} a^{10} - \frac{17}{168} a^{9} + \frac{1}{168} a^{8} - \frac{73}{336} a^{7} + \frac{19}{84} a^{6} + \frac{29}{168} a^{5} - \frac{137}{336} a^{4} - \frac{15}{56} a^{3} - \frac{17}{56} a^{2} - \frac{5}{12} a - \frac{11}{84}$, $\frac{1}{116482464} a^{14} + \frac{47315}{38827488} a^{13} - \frac{5227}{510888} a^{12} - \frac{72239}{1848928} a^{11} - \frac{2177543}{116482464} a^{10} + \frac{1593}{12164} a^{9} + \frac{826829}{5546784} a^{8} + \frac{2607743}{116482464} a^{7} - \frac{2085785}{9706872} a^{6} + \frac{3848287}{38827488} a^{5} + \frac{44933471}{116482464} a^{4} + \frac{1490133}{3235624} a^{3} - \frac{22018615}{58241232} a^{2} + \frac{138304}{404453} a + \frac{14425139}{29120616}$, $\frac{1}{11972489432922144} a^{15} - \frac{36334507}{11972489432922144} a^{14} - \frac{308304724247}{665138301829008} a^{13} - \frac{189166583696591}{3990829810974048} a^{12} - \frac{381128409488747}{11972489432922144} a^{11} + \frac{290731126640167}{5986244716461072} a^{10} + \frac{137310229917521}{570118544424864} a^{9} + \frac{2951180439341783}{11972489432922144} a^{8} + \frac{23632038922241}{855177816637296} a^{7} + \frac{22093938115511}{190039514808288} a^{6} + \frac{83506487206709}{1710355633274592} a^{5} - \frac{644922887957545}{5986244716461072} a^{4} + \frac{2767980301570811}{5986244716461072} a^{3} + \frac{78084529745009}{427588908318648} a^{2} - \frac{9394590642355}{22504679385192} a + \frac{271317111140951}{1496561179115268}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 160289.989776 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{305}) \), 4.4.465125.1 x2, \(\Q(\sqrt{5}, \sqrt{61})\), 4.4.7625.1 x2, 8.0.3461460250000.1 x2, 8.0.692292050000.2 x2, 8.8.216341265625.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$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}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$