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Properties

Label 16.0.72138957898...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 5^{4}$
Root discriminant $17.42$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 32, 152, 496, 1116, 1680, 1456, 600, 222, -8, -56, -48, 46, 0, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 46*x^12 - 48*x^11 - 56*x^10 - 8*x^9 + 222*x^8 + 600*x^7 + 1456*x^6 + 1680*x^5 + 1116*x^4 + 496*x^3 + 152*x^2 + 32*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 46*x^12 - 48*x^11 - 56*x^10 - 8*x^9 + 222*x^8 + 600*x^7 + 1456*x^6 + 1680*x^5 + 1116*x^4 + 496*x^3 + 152*x^2 + 32*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 46 x^{12} - 48 x^{11} - 56 x^{10} - 8 x^{9} + 222 x^{8} + 600 x^{7} + 1456 x^{6} + 1680 x^{5} + 1116 x^{4} + 496 x^{3} + 152 x^{2} + 32 x + 4 \)

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:  \(72138957898383360000=2^{44}\cdot 3^{8}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.42$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{22} a^{13} - \frac{1}{11} a^{12} - \frac{1}{11} a^{11} + \frac{5}{22} a^{10} + \frac{2}{11} a^{9} - \frac{1}{22} a^{8} + \frac{2}{11} a^{7} + \frac{3}{11} a^{6} - \frac{1}{11} a^{5} - \frac{5}{11} a^{3} + \frac{3}{11} a^{2} + \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{682} a^{14} - \frac{13}{682} a^{13} - \frac{123}{682} a^{12} + \frac{52}{341} a^{11} + \frac{24}{341} a^{10} - \frac{72}{341} a^{9} + \frac{68}{341} a^{8} - \frac{107}{341} a^{7} - \frac{56}{341} a^{6} + \frac{15}{31} a^{5} - \frac{27}{341} a^{4} + \frac{47}{341} a^{3} + \frac{136}{341} a^{2} - \frac{166}{341} a - \frac{12}{31}$, $\frac{1}{1464485694665818} a^{15} - \frac{148759771119}{1464485694665818} a^{14} + \frac{12340539754417}{732242847332909} a^{13} + \frac{144162822103682}{732242847332909} a^{12} + \frac{66462226387737}{732242847332909} a^{11} + \frac{1380702125645}{133135063151438} a^{10} - \frac{13249686130551}{1464485694665818} a^{9} - \frac{202522428287981}{1464485694665818} a^{8} + \frac{354806316901496}{732242847332909} a^{7} - \frac{108894380883911}{732242847332909} a^{6} + \frac{200319407028427}{732242847332909} a^{5} + \frac{351457771249384}{732242847332909} a^{4} - \frac{360971439227216}{732242847332909} a^{3} + \frac{141208146968898}{732242847332909} a^{2} + \frac{103251108855603}{732242847332909} a + \frac{159528688842379}{732242847332909}$

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:  \( \frac{136195559252195}{732242847332909} a^{15} - \frac{1241107479731121}{1464485694665818} a^{14} + \frac{362755373244241}{732242847332909} a^{13} - \frac{276352247877137}{732242847332909} a^{12} + \frac{6436098038611149}{732242847332909} a^{11} - \frac{20227754775655067}{1464485694665818} a^{10} - \frac{1205042168479599}{732242847332909} a^{9} - \frac{2906648283557577}{1464485694665818} a^{8} + \frac{30244502938618569}{732242847332909} a^{7} + \frac{65118200012232315}{732242847332909} a^{6} + \frac{165820454364918766}{732242847332909} a^{5} + \frac{146168522302533028}{732242847332909} a^{4} + \frac{93459175828995584}{732242847332909} a^{3} + \frac{38344474496936710}{732242847332909} a^{2} + \frac{1011292137876580}{66567531575719} a + \frac{2383102230608707}{732242847332909} \) (order $24$)
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:  \( 17129.0836242 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:C_2$ (as 16T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{24})\), 8.0.8493465600.1, 8.8.8493465600.1

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 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$