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Properties

Label 16.0.61168212980...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{12}\cdot 11^{8}\cdot 571^{4}$
Root discriminant $306.66$
Ramified primes $2, 5, 11, 571$
Class number $186274560$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 1455270]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T467)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10362339067001, -5348496829980, 5126676453884, -1183776575800, 744323000990, -98903155760, 53612475676, -3943978960, 2032000719, -58510080, 37499896, -283240, 279960, -460, 884, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 884*x^14 - 460*x^13 + 279960*x^12 - 283240*x^11 + 37499896*x^10 - 58510080*x^9 + 2032000719*x^8 - 3943978960*x^7 + 53612475676*x^6 - 98903155760*x^5 + 744323000990*x^4 - 1183776575800*x^3 + 5126676453884*x^2 - 5348496829980*x + 10362339067001)
 
gp: K = bnfinit(x^16 + 884*x^14 - 460*x^13 + 279960*x^12 - 283240*x^11 + 37499896*x^10 - 58510080*x^9 + 2032000719*x^8 - 3943978960*x^7 + 53612475676*x^6 - 98903155760*x^5 + 744323000990*x^4 - 1183776575800*x^3 + 5126676453884*x^2 - 5348496829980*x + 10362339067001, 1)
 

Normalized defining polynomial

\( x^{16} + 884 x^{14} - 460 x^{13} + 279960 x^{12} - 283240 x^{11} + 37499896 x^{10} - 58510080 x^{9} + 2032000719 x^{8} - 3943978960 x^{7} + 53612475676 x^{6} - 98903155760 x^{5} + 744323000990 x^{4} - 1183776575800 x^{3} + 5126676453884 x^{2} - 5348496829980 x + 10362339067001 \)

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:  \(6116821298063232221593993216000000000000=2^{40}\cdot 5^{12}\cdot 11^{8}\cdot 571^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $306.66$
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:  $2, 5, 11, 571$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1713} a^{12} + \frac{661}{1713} a^{11} + \frac{218}{571} a^{10} - \frac{452}{1713} a^{9} - \frac{463}{1713} a^{8} + \frac{47}{571} a^{7} + \frac{170}{571} a^{6} - \frac{155}{571} a^{5} + \frac{181}{1713} a^{4} - \frac{251}{571} a^{3} + \frac{172}{1713} a^{2} - \frac{332}{1713} a - \frac{292}{1713}$, $\frac{1}{18843} a^{13} - \frac{1}{18843} a^{12} - \frac{8678}{18843} a^{11} + \frac{8554}{18843} a^{10} + \frac{1946}{6281} a^{9} + \frac{20}{18843} a^{8} - \frac{10}{571} a^{7} + \frac{33}{571} a^{6} + \frac{8236}{18843} a^{5} + \frac{6187}{18843} a^{4} - \frac{4964}{18843} a^{3} + \frac{4001}{18843} a^{2} - \frac{45}{571} a + \frac{4874}{18843}$, $\frac{1}{18843} a^{14} + \frac{8423}{18843} a^{11} - \frac{128}{18843} a^{10} + \frac{2294}{18843} a^{9} - \frac{5128}{18843} a^{8} - \frac{9}{571} a^{7} + \frac{7510}{18843} a^{6} - \frac{7753}{18843} a^{5} + \frac{8153}{18843} a^{4} + \frac{757}{6281} a^{3} + \frac{6707}{18843} a^{2} + \frac{4940}{18843} a - \frac{4432}{18843}$, $\frac{1}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{15} - \frac{8924155480034764624106927993501000420726741670020890722740828631260768479378634564}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{14} - \frac{2454621999112763750807099687978452648235709115697564571237417520444443703865601450}{123228646346582449576152616414424463345173826365962488593428364343363271188725047808879} a^{13} - \frac{56771822264819161833344304441018903735406449861228623242149985982844551303489505529}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{12} + \frac{51276895214230164003323994153819252415101551980202081232752156211272598415954444346013}{123228646346582449576152616414424463345173826365962488593428364343363271188725047808879} a^{11} + \frac{54309861251156524604992784240282699032964333592815178015785289186521745358173488963604}{123228646346582449576152616414424463345173826365962488593428364343363271188725047808879} a^{10} + \frac{68944540543673429356721648375370775210216861770289047884410616464409563726713288856549}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{9} - \frac{88080502912374312134389746248117711947239765908907880528645299652242790233347829608965}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{8} - \frac{3853881092178414698788587763976795868855235231758909625530878719763686835092359218624}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{7} + \frac{14354087885556912065514796965503143462858095396304512238049602755262569173346756162}{215811990099093606963489695997240741410111779975415916976231811459480334831392377949} a^{6} - \frac{6350331952853721985105340653925704850656579642336511011233331242510044131580175221329}{33607812639977031702587077203933944548683770827080678707298644820917255778743194856967} a^{5} + \frac{33147392053456183313859682309302874738547747117343029215175884445681966839125449539910}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{4} + \frac{30108262835727262607869269141818334241803646320038881716289018050755385953334901604471}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{3} - \frac{38340163343177063537639923625985932021723609364631052505459322652480712012988567690196}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a^{2} - \frac{155961484674458745076288456735029775677150717889790231025462924304017174131127888798224}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637} a + \frac{54038525610234339585911080679446679838262930211073687809338710060216247022236884568982}{369685939039747348728457849243273390035521479097887465780285093030089813566175143426637}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1455270}$, which has order $186274560$ (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:  \( 130320.792052 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T467):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.8000.1, 4.4.17600.1, 4.4.22000.1, 8.8.123904000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$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}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
571Data not computed