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Properties

Label 16.8.10729765818...1088.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{67}\cdot 17^{6}\cdot 548837153^{2}$
Root discriminant $652.25$
Ramified primes $2, 17, 548837153$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1113

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![696425773826392185608, 0, -12392713672696488160, 0, -4032319635459760, 0, 576976513783936, 0, -99624248436, 0, -2092104320, 0, -335372, 0, 2112, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2112*x^14 - 335372*x^12 - 2092104320*x^10 - 99624248436*x^8 + 576976513783936*x^6 - 4032319635459760*x^4 - 12392713672696488160*x^2 + 696425773826392185608)
 
gp: K = bnfinit(x^16 + 2112*x^14 - 335372*x^12 - 2092104320*x^10 - 99624248436*x^8 + 576976513783936*x^6 - 4032319635459760*x^4 - 12392713672696488160*x^2 + 696425773826392185608, 1)
 

Normalized defining polynomial

\( x^{16} + 2112 x^{14} - 335372 x^{12} - 2092104320 x^{10} - 99624248436 x^{8} + 576976513783936 x^{6} - 4032319635459760 x^{4} - 12392713672696488160 x^{2} + 696425773826392185608 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1072976581893572471636613847437598787096281088=2^{67}\cdot 17^{6}\cdot 548837153^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $652.25$
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:  $2, 17, 548837153$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{3434298812067367182013236764152853525933595937951289334662167297067109212} a^{14} + \frac{24731895579569899250787229652222995215194323046827465803654037309744691}{858574703016841795503309191038213381483398984487822333665541824266777303} a^{12} + \frac{65181236599200063491464057189605150614090205624837226980064084671523255}{858574703016841795503309191038213381483398984487822333665541824266777303} a^{10} - \frac{6626605480341790229670923027338090633238862328381204113153617959498469}{50504394295108340911959364178718434204905822616930725509737754368633959} a^{8} + \frac{9397969220067945401344338796000522595812540616806050330090693641006097}{50504394295108340911959364178718434204905822616930725509737754368633959} a^{6} - \frac{119417211682511207991857503945879474957883409736401807517280973584160307}{858574703016841795503309191038213381483398984487822333665541824266777303} a^{4} - \frac{236323214128196173764684595170074298516040785844239169846225128248315165}{858574703016841795503309191038213381483398984487822333665541824266777303} a^{2} - \frac{42446181774919213597884188226692472997315632482272610948319723}{92020727858976305330334231543392679549348625487332351769082503}$, $\frac{1}{3434298812067367182013236764152853525933595937951289334662167297067109212} a^{15} + \frac{24731895579569899250787229652222995215194323046827465803654037309744691}{858574703016841795503309191038213381483398984487822333665541824266777303} a^{13} + \frac{65181236599200063491464057189605150614090205624837226980064084671523255}{858574703016841795503309191038213381483398984487822333665541824266777303} a^{11} - \frac{6626605480341790229670923027338090633238862328381204113153617959498469}{50504394295108340911959364178718434204905822616930725509737754368633959} a^{9} + \frac{9397969220067945401344338796000522595812540616806050330090693641006097}{50504394295108340911959364178718434204905822616930725509737754368633959} a^{7} - \frac{119417211682511207991857503945879474957883409736401807517280973584160307}{858574703016841795503309191038213381483398984487822333665541824266777303} a^{5} - \frac{236323214128196173764684595170074298516040785844239169846225128248315165}{858574703016841795503309191038213381483398984487822333665541824266777303} a^{3} - \frac{42446181774919213597884188226692472997315632482272610948319723}{92020727858976305330334231543392679549348625487332351769082503} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 12672042406200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1113:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1113
Character table for t16n1113 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1

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 R $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
548837153Data not computed