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Properties

Label 16.0.52554919692...6256.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 17^{6}\cdot 89^{2}$
Root discriminant $26.31$
Ramified primes $2, 17, 89$
Class number $2$
Class group $[2]$
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4132, -17608, 30324, -28280, 20066, -16304, 12564, -7096, 3866, -2164, 954, -392, 173, -56, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 18*x^14 - 56*x^13 + 173*x^12 - 392*x^11 + 954*x^10 - 2164*x^9 + 3866*x^8 - 7096*x^7 + 12564*x^6 - 16304*x^5 + 20066*x^4 - 28280*x^3 + 30324*x^2 - 17608*x + 4132)
 
gp: K = bnfinit(x^16 - 4*x^15 + 18*x^14 - 56*x^13 + 173*x^12 - 392*x^11 + 954*x^10 - 2164*x^9 + 3866*x^8 - 7096*x^7 + 12564*x^6 - 16304*x^5 + 20066*x^4 - 28280*x^3 + 30324*x^2 - 17608*x + 4132, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 18 x^{14} - 56 x^{13} + 173 x^{12} - 392 x^{11} + 954 x^{10} - 2164 x^{9} + 3866 x^{8} - 7096 x^{7} + 12564 x^{6} - 16304 x^{5} + 20066 x^{4} - 28280 x^{3} + 30324 x^{2} - 17608 x + 4132 \)

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:  \(52554919692301559136256=2^{38}\cdot 17^{6}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a$, $\frac{1}{604675973939443577945545674} a^{15} + \frac{1793329586871280756163555}{28794093997116360854549794} a^{14} - \frac{22356090144147119852364883}{604675973939443577945545674} a^{13} - \frac{4042069356705862601125073}{201558657979814525981848558} a^{12} + \frac{23100415014233472421621243}{201558657979814525981848558} a^{11} + \frac{105077472050114170467560297}{604675973939443577945545674} a^{10} + \frac{11298023972395327944703781}{28794093997116360854549794} a^{9} + \frac{1849936667632462768375565}{26290259736497546867197638} a^{8} - \frac{135136067132133447410758330}{302337986969721788972772837} a^{7} - \frac{17241591430502449705656977}{100779328989907262990924279} a^{6} + \frac{54434333495525455958098618}{302337986969721788972772837} a^{5} + \frac{21213061227259443834273467}{100779328989907262990924279} a^{4} + \frac{15332032624979157688059089}{100779328989907262990924279} a^{3} - \frac{54590302321403634321378782}{302337986969721788972772837} a^{2} + \frac{23179546729836272518847470}{100779328989907262990924279} a - \frac{76096740036326933103629395}{302337986969721788972772837}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{11790805950398156015}{280852751481395066393658} a^{15} - \frac{8999718928335661822}{20060910820099647599547} a^{14} + \frac{338990090290285646605}{280852751481395066393658} a^{13} - \frac{914214629900012380672}{140426375740697533196829} a^{12} + \frac{5124347880833501903077}{280852751481395066393658} a^{11} - \frac{2800937731621038642149}{46808791913565844398943} a^{10} + \frac{1537187162669077788053}{13373940546733098399698} a^{9} - \frac{588121198820792354780}{2035164865807210626041} a^{8} + \frac{99517340879809155583733}{140426375740697533196829} a^{7} - \frac{132983757636515337830641}{140426375740697533196829} a^{6} + \frac{280556661433519976264126}{140426375740697533196829} a^{5} - \frac{511703427704111960543572}{140426375740697533196829} a^{4} + \frac{427365524475442469832257}{140426375740697533196829} a^{3} - \frac{194376623455022611013459}{46808791913565844398943} a^{2} + \frac{368025969727107332565397}{46808791913565844398943} a - \frac{225847532465926616298829}{46808791913565844398943} \) (order $8$)
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:  \( 618627.431608 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.0.1088.2, 4.4.4352.1, \(\Q(\zeta_{8})\), 8.0.18939904.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.22.87$x^{8} + 4 x^{7} + 6 x^{4} + 12 x^{2} + 2$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$