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Properties

Label 16.0.93492089436...456.13
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{12}$
Root discriminant $15.33$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2^2\wr C_2$ (as 16T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![76, -592, 2200, -5136, 8396, -10144, 9264, -6344, 3094, -904, 36, 64, 20, -48, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 48*x^13 + 20*x^12 + 64*x^11 + 36*x^10 - 904*x^9 + 3094*x^8 - 6344*x^7 + 9264*x^6 - 10144*x^5 + 8396*x^4 - 5136*x^3 + 2200*x^2 - 592*x + 76)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 48*x^13 + 20*x^12 + 64*x^11 + 36*x^10 - 904*x^9 + 3094*x^8 - 6344*x^7 + 9264*x^6 - 10144*x^5 + 8396*x^4 - 5136*x^3 + 2200*x^2 - 592*x + 76, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 48 x^{13} + 20 x^{12} + 64 x^{11} + 36 x^{10} - 904 x^{9} + 3094 x^{8} - 6344 x^{7} + 9264 x^{6} - 10144 x^{5} + 8396 x^{4} - 5136 x^{3} + 2200 x^{2} - 592 x + 76 \)

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:  \(9349208943630483456=2^{44}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.33$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{814} a^{14} + \frac{15}{74} a^{13} - \frac{29}{407} a^{12} + \frac{27}{814} a^{11} - \frac{155}{814} a^{10} - \frac{3}{74} a^{9} - \frac{49}{407} a^{8} + \frac{113}{407} a^{7} - \frac{78}{407} a^{6} - \frac{39}{407} a^{5} - \frac{71}{407} a^{4} - \frac{53}{407} a^{3} + \frac{156}{407} a^{2} + \frac{147}{407} a + \frac{58}{407}$, $\frac{1}{1080771406} a^{15} - \frac{610639}{1080771406} a^{14} + \frac{8043435}{83136262} a^{13} + \frac{35897151}{1080771406} a^{12} + \frac{151704957}{1080771406} a^{11} - \frac{251284367}{1080771406} a^{10} - \frac{182717193}{1080771406} a^{9} - \frac{992489}{41568131} a^{8} - \frac{455577}{3778921} a^{7} + \frac{3889485}{41568131} a^{6} + \frac{64909563}{540385703} a^{5} - \frac{66774495}{540385703} a^{4} + \frac{11104169}{540385703} a^{3} + \frac{6835500}{540385703} a^{2} - \frac{246776984}{540385703} a - \frac{116423702}{540385703}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{22008}{34669} a^{15} + \frac{158351}{34669} a^{14} - \frac{487504}{34669} a^{13} + \frac{656704}{34669} a^{12} + \frac{106496}{34669} a^{11} - \frac{1338883}{34669} a^{10} - \frac{1886570}{34669} a^{9} + \frac{36836777}{69338} a^{8} - \frac{53119914}{34669} a^{7} + \frac{96058768}{34669} a^{6} - \frac{124579112}{34669} a^{5} + \frac{119764419}{34669} a^{4} - \frac{84588004}{34669} a^{3} + \frac{41663786}{34669} a^{2} - \frac{12811292}{34669} a + \frac{1877147}{34669} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4479.5852819 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), 4.0.27648.1, 4.4.27648.1, 4.0.3072.1, 4.0.3072.2, 4.0.432.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), 4.0.1728.1, 8.0.47775744.3, 8.0.764411904.5, 8.0.84934656.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$