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Properties

Label 16.0.25916019301...5625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{6}$
Root discriminant $18.87$
Ramified primes $5, 101$
Class number $2$
Class group $[2]$
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![25, -125, 235, -210, 201, -536, 1020, -1005, 431, 10, -21, -55, 46, -25, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 15*x^14 - 25*x^13 + 46*x^12 - 55*x^11 - 21*x^10 + 10*x^9 + 431*x^8 - 1005*x^7 + 1020*x^6 - 536*x^5 + 201*x^4 - 210*x^3 + 235*x^2 - 125*x + 25)
 
gp: K = bnfinit(x^16 - 6*x^15 + 15*x^14 - 25*x^13 + 46*x^12 - 55*x^11 - 21*x^10 + 10*x^9 + 431*x^8 - 1005*x^7 + 1020*x^6 - 536*x^5 + 201*x^4 - 210*x^3 + 235*x^2 - 125*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 15 x^{14} - 25 x^{13} + 46 x^{12} - 55 x^{11} - 21 x^{10} + 10 x^{9} + 431 x^{8} - 1005 x^{7} + 1020 x^{6} - 536 x^{5} + 201 x^{4} - 210 x^{3} + 235 x^{2} - 125 x + 25 \)

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:  \(259160193017822265625=5^{12}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.87$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{303446943299705} a^{15} + \frac{3227185871093}{303446943299705} a^{14} - \frac{18572026923504}{303446943299705} a^{13} - \frac{23018696592666}{60689388659941} a^{12} - \frac{55286005849044}{303446943299705} a^{11} - \frac{992021517776}{4273900609855} a^{10} + \frac{87593605145254}{303446943299705} a^{9} + \frac{86202518741091}{303446943299705} a^{8} - \frac{124313304000949}{303446943299705} a^{7} + \frac{18232613995264}{303446943299705} a^{6} - \frac{17829689650592}{60689388659941} a^{5} + \frac{137424711053294}{303446943299705} a^{4} - \frac{127113144115848}{303446943299705} a^{3} + \frac{135091370659749}{303446943299705} a^{2} + \frac{22400438888857}{60689388659941} a - \frac{21525443863274}{60689388659941}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{349373242799574}{303446943299705} a^{15} - \frac{358696480254528}{60689388659941} a^{14} + \frac{3681767562604556}{303446943299705} a^{13} - \frac{1105240159512283}{60689388659941} a^{12} + \frac{11260766831348779}{303446943299705} a^{11} - \frac{132828462565516}{4273900609855} a^{10} - \frac{15590897760910709}{303446943299705} a^{9} - \frac{10045461078913539}{303446943299705} a^{8} + \frac{142145216646936559}{303446943299705} a^{7} - \frac{227507173449003106}{303446943299705} a^{6} + \frac{31539532379104455}{60689388659941} a^{5} - \frac{49810044683295909}{303446943299705} a^{4} + \frac{5518652297192813}{60689388659941} a^{3} - \frac{49937669318138216}{303446943299705} a^{2} + \frac{7717219471316012}{60689388659941} a - \frac{1985941093157517}{60689388659941} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19060.3137934 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.2525.1, 4.0.12625.1, 8.8.16098453125.1, 8.0.643938125.1, 8.0.159390625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$