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Properties

Label 16.0.19680337591...1329.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 43^{8}$
Root discriminant $78.23$
Ramified primes $17, 43$
Class number $24$ (GRH)
Class group $[24]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87872, 209248, 52048, -122784, -29044, -42578, -35077, 65961, 5438, -17651, 6939, -3306, 865, -81, 22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 22*x^14 - 81*x^13 + 865*x^12 - 3306*x^11 + 6939*x^10 - 17651*x^9 + 5438*x^8 + 65961*x^7 - 35077*x^6 - 42578*x^5 - 29044*x^4 - 122784*x^3 + 52048*x^2 + 209248*x + 87872)
 
gp: K = bnfinit(x^16 - x^15 + 22*x^14 - 81*x^13 + 865*x^12 - 3306*x^11 + 6939*x^10 - 17651*x^9 + 5438*x^8 + 65961*x^7 - 35077*x^6 - 42578*x^5 - 29044*x^4 - 122784*x^3 + 52048*x^2 + 209248*x + 87872, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 22 x^{14} - 81 x^{13} + 865 x^{12} - 3306 x^{11} + 6939 x^{10} - 17651 x^{9} + 5438 x^{8} + 65961 x^{7} - 35077 x^{6} - 42578 x^{5} - 29044 x^{4} - 122784 x^{3} + 52048 x^{2} + 209248 x + 87872 \)

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:  \(1968033759133442969054057291329=17^{14}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.23$
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:  $17, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{11} + \frac{1}{8} a^{6} + \frac{3}{16} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{32} a^{6} + \frac{1}{8} a^{5} - \frac{7}{32} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{1664} a^{13} + \frac{3}{832} a^{12} + \frac{1}{208} a^{11} + \frac{51}{1664} a^{10} + \frac{11}{832} a^{9} - \frac{7}{104} a^{8} - \frac{5}{128} a^{7} - \frac{33}{832} a^{6} - \frac{11}{208} a^{5} - \frac{379}{1664} a^{4} + \frac{215}{832} a^{3} - \frac{3}{8} a^{2} + \frac{23}{208} a - \frac{31}{104}$, $\frac{1}{33280} a^{14} + \frac{3}{33280} a^{13} + \frac{99}{16640} a^{12} + \frac{443}{33280} a^{11} - \frac{651}{33280} a^{10} + \frac{171}{16640} a^{9} - \frac{2017}{33280} a^{8} + \frac{1169}{33280} a^{7} + \frac{783}{16640} a^{6} - \frac{6979}{33280} a^{5} - \frac{3529}{33280} a^{4} + \frac{1171}{3328} a^{3} - \frac{323}{832} a^{2} - \frac{1483}{4160} a + \frac{457}{2080}$, $\frac{1}{5786616451925754251223040} a^{15} - \frac{27591874788618526847}{5786616451925754251223040} a^{14} - \frac{163141487708158688389}{723327056490719281402880} a^{13} - \frac{19726654795593598987297}{5786616451925754251223040} a^{12} - \frac{5313257577201162937441}{5786616451925754251223040} a^{11} + \frac{40653201493570408647973}{1446654112981438562805760} a^{10} - \frac{39347300791915593823037}{5786616451925754251223040} a^{9} - \frac{703755522942719561358061}{5786616451925754251223040} a^{8} - \frac{113567911881002664543191}{1446654112981438562805760} a^{7} - \frac{940025042455649560787599}{5786616451925754251223040} a^{6} - \frac{872645009760169497350819}{5786616451925754251223040} a^{5} + \frac{11968116966311215638081}{144665411298143856280576} a^{4} + \frac{5939710120553497600777}{15227938031383563819008} a^{3} - \frac{24734302748100076991441}{55640542806978406261760} a^{2} + \frac{20188421994559932648599}{45207941030669955087680} a - \frac{13682402933881042767225}{36166352824535964070144}$

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

Class group and class number

$C_{24}$, which has order $24$ (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:  \( 57492245.5287 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-43}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-731}) \), \(\Q(\sqrt{17}, \sqrt{-43})\), 4.2.211259.1 x2, 4.0.9084137.1 x2, 8.0.82521545034769.1, 8.2.32624796874211.2 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$