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Properties

Label 16.4.28320098225...1441.2
Degree $16$
Signature $[4, 6]$
Discriminant $7^{8}\cdot 53^{12}$
Root discriminant $51.97$
Ramified primes $7, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4529, -9429, -5634, 7963, 11734, 12322, -10717, -1846, -4908, 3145, 506, 572, -207, -64, -34, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 34*x^14 - 64*x^13 - 207*x^12 + 572*x^11 + 506*x^10 + 3145*x^9 - 4908*x^8 - 1846*x^7 - 10717*x^6 + 12322*x^5 + 11734*x^4 + 7963*x^3 - 5634*x^2 - 9429*x + 4529)
 
gp: K = bnfinit(x^16 - 34*x^14 - 64*x^13 - 207*x^12 + 572*x^11 + 506*x^10 + 3145*x^9 - 4908*x^8 - 1846*x^7 - 10717*x^6 + 12322*x^5 + 11734*x^4 + 7963*x^3 - 5634*x^2 - 9429*x + 4529, 1)
 

Normalized defining polynomial

\( x^{16} - 34 x^{14} - 64 x^{13} - 207 x^{12} + 572 x^{11} + 506 x^{10} + 3145 x^{9} - 4908 x^{8} - 1846 x^{7} - 10717 x^{6} + 12322 x^{5} + 11734 x^{4} + 7963 x^{3} - 5634 x^{2} - 9429 x + 4529 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2832009822518079193000591441=7^{8}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.97$
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:  $7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{26} a^{14} + \frac{1}{13} a^{13} + \frac{9}{26} a^{11} + \frac{3}{13} a^{10} - \frac{2}{13} a^{9} - \frac{11}{26} a^{8} - \frac{6}{13} a^{6} - \frac{11}{26} a^{5} - \frac{5}{13} a^{4} + \frac{6}{13} a^{3} + \frac{5}{26} a^{2} + \frac{1}{13}$, $\frac{1}{85568749980516360012631802961400234} a^{15} + \frac{562047143949602042875240721265328}{42784374990258180006315901480700117} a^{14} + \frac{18302981765166462674282362242874113}{85568749980516360012631802961400234} a^{13} - \frac{9892785972095153113575382170067927}{42784374990258180006315901480700117} a^{12} - \frac{12907832867007768480403264854824399}{42784374990258180006315901480700117} a^{11} - \frac{19201000338125416571617727935482341}{85568749980516360012631802961400234} a^{10} + \frac{104071959931636841879392226016216}{3291105768481398462024300113900009} a^{9} + \frac{418679537219968299504695403882133}{42784374990258180006315901480700117} a^{8} + \frac{26584409736633692805064852847657187}{85568749980516360012631802961400234} a^{7} - \frac{13241546893575148488490723829432750}{42784374990258180006315901480700117} a^{6} + \frac{7910029935611755765443525239165837}{42784374990258180006315901480700117} a^{5} - \frac{37758707990463074850217824448097939}{85568749980516360012631802961400234} a^{4} - \frac{5095049909006308106505335691584502}{42784374990258180006315901480700117} a^{3} + \frac{14674945797872926707160382522760066}{42784374990258180006315901480700117} a^{2} - \frac{34581768295002373269166816116688017}{85568749980516360012631802961400234} a + \frac{19186231816177194431802689988749481}{85568749980516360012631802961400234}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 32212346.4505 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.7294973.1, 4.2.19663.1, 4.2.1042139.1, 8.2.2706434983.1, 8.2.7602375867247.2, 8.4.53216631070729.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$