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Properties

Label 16.0.11256645592...6249.7
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 53^{14}$
Root discriminant $116.34$
Ramified primes $13, 53$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4962179, -4096578, 7510735, -4063001, -2654985, 1246730, 565267, -122429, 114330, -54485, 10357, -4675, 1004, -191, 52, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 52*x^14 - 191*x^13 + 1004*x^12 - 4675*x^11 + 10357*x^10 - 54485*x^9 + 114330*x^8 - 122429*x^7 + 565267*x^6 + 1246730*x^5 - 2654985*x^4 - 4063001*x^3 + 7510735*x^2 - 4096578*x + 4962179)
 
gp: K = bnfinit(x^16 - 4*x^15 + 52*x^14 - 191*x^13 + 1004*x^12 - 4675*x^11 + 10357*x^10 - 54485*x^9 + 114330*x^8 - 122429*x^7 + 565267*x^6 + 1246730*x^5 - 2654985*x^4 - 4063001*x^3 + 7510735*x^2 - 4096578*x + 4962179, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 52 x^{14} - 191 x^{13} + 1004 x^{12} - 4675 x^{11} + 10357 x^{10} - 54485 x^{9} + 114330 x^{8} - 122429 x^{7} + 565267 x^{6} + 1246730 x^{5} - 2654985 x^{4} - 4063001 x^{3} + 7510735 x^{2} - 4096578 x + 4962179 \)

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:  \(1125664559289128829386632937086249=13^{8}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.34$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{611} a^{14} - \frac{119}{611} a^{13} + \frac{190}{611} a^{12} + \frac{230}{611} a^{11} - \frac{14}{47} a^{10} + \frac{48}{611} a^{9} + \frac{118}{611} a^{8} + \frac{225}{611} a^{7} + \frac{301}{611} a^{6} + \frac{187}{611} a^{5} + \frac{139}{611} a^{4} + \frac{108}{611} a^{3} + \frac{290}{611} a^{2} + \frac{62}{611} a + \frac{14}{611}$, $\frac{1}{14339903508679933085061818650887103840682995681390783} a^{15} + \frac{1068407742042885501540864123162136875518606215910}{14339903508679933085061818650887103840682995681390783} a^{14} + \frac{3812119168844707324654247364370436412497948229244421}{14339903508679933085061818650887103840682995681390783} a^{13} - \frac{5503384013615271396700074668335933656482006208195031}{14339903508679933085061818650887103840682995681390783} a^{12} - \frac{2491209705644478868775982527413462801636481018442941}{14339903508679933085061818650887103840682995681390783} a^{11} + \frac{5374405786914499604695365150744649681770699836657168}{14339903508679933085061818650887103840682995681390783} a^{10} + \frac{3076100470564625645650413325300946770117855034509076}{14339903508679933085061818650887103840682995681390783} a^{9} + \frac{3931173744428407446024350040409353864377821808039905}{14339903508679933085061818650887103840682995681390783} a^{8} - \frac{4336809269767750717089978590991142530265259172086424}{14339903508679933085061818650887103840682995681390783} a^{7} + \frac{2860308631615828338230384018529783112200142775709819}{14339903508679933085061818650887103840682995681390783} a^{6} + \frac{3667167889794193607934681578639742555472453273394875}{14339903508679933085061818650887103840682995681390783} a^{5} + \frac{3954310125528170012191762401634336895794992742794250}{14339903508679933085061818650887103840682995681390783} a^{4} - \frac{2016307481293557616553773897769794545617407204051}{23469563843993343838071716286230939182787226974453} a^{3} - \frac{1135015870963298261150459892348276052470852912506839}{14339903508679933085061818650887103840682995681390783} a^{2} - \frac{4863505304602616607046710228631102263885320226154886}{14339903508679933085061818650887103840682995681390783} a - \frac{4798296530812077238485007519828836849571170442001638}{14339903508679933085061818650887103840682995681390783}$

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:  \( 445433494.742 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.48695101400413.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$