↑

Properties

Label 16.0.76765634560...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 37^{4}$
Root discriminant $23.33$
Ramified primes $2, 5, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4 \times D_4$ (as 16T19)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2401, -2058, -1617, 2436, 65, -1668, 807, 774, -740, -126, 321, -84, -31, 12, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 9*x^14 + 12*x^13 - 31*x^12 - 84*x^11 + 321*x^10 - 126*x^9 - 740*x^8 + 774*x^7 + 807*x^6 - 1668*x^5 + 65*x^4 + 2436*x^3 - 1617*x^2 - 2058*x + 2401)
 
gp: K = bnfinit(x^16 - 6*x^15 + 9*x^14 + 12*x^13 - 31*x^12 - 84*x^11 + 321*x^10 - 126*x^9 - 740*x^8 + 774*x^7 + 807*x^6 - 1668*x^5 + 65*x^4 + 2436*x^3 - 1617*x^2 - 2058*x + 2401, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 9 x^{14} + 12 x^{13} - 31 x^{12} - 84 x^{11} + 321 x^{10} - 126 x^{9} - 740 x^{8} + 774 x^{7} + 807 x^{6} - 1668 x^{5} + 65 x^{4} + 2436 x^{3} - 1617 x^{2} - 2058 x + 2401 \)

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:  \(7676563456000000000000=2^{24}\cdot 5^{12}\cdot 37^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.33$
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:  $2, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{28} a^{13} + \frac{1}{28} a^{12} - \frac{5}{28} a^{11} + \frac{5}{28} a^{10} - \frac{5}{14} a^{9} - \frac{2}{7} a^{7} - \frac{3}{7} a^{5} - \frac{5}{14} a^{4} + \frac{9}{28} a^{3} + \frac{5}{28} a^{2} + \frac{9}{28} a + \frac{1}{4}$, $\frac{1}{4312} a^{14} + \frac{25}{2156} a^{13} - \frac{41}{2156} a^{12} - \frac{183}{2156} a^{11} + \frac{85}{392} a^{10} + \frac{3}{308} a^{9} - \frac{25}{98} a^{8} - \frac{73}{154} a^{7} - \frac{117}{539} a^{6} + \frac{933}{2156} a^{5} - \frac{145}{4312} a^{4} - \frac{729}{2156} a^{3} + \frac{29}{98} a^{2} + \frac{7}{44} a + \frac{3}{88}$, $\frac{1}{1120163617075974856} a^{15} + \frac{45105567130607}{560081808537987428} a^{14} - \frac{8556061640348823}{560081808537987428} a^{13} + \frac{43392957559156463}{560081808537987428} a^{12} - \frac{110968825443262909}{1120163617075974856} a^{11} - \frac{2543970650224573}{11430240990571172} a^{10} + \frac{89643003920762521}{280040904268993714} a^{9} - \frac{2824045751358371}{5715120495285586} a^{8} - \frac{6482329660170425}{140020452134496857} a^{7} + \frac{168806156623363233}{560081808537987428} a^{6} - \frac{479477284311802817}{1120163617075974856} a^{5} + \frac{211203062110159937}{560081808537987428} a^{4} - \frac{37602274460252164}{140020452134496857} a^{3} + \frac{21852734173166451}{80011686933998204} a^{2} + \frac{4216665953962543}{22860481981142344} a - \frac{307738438444831}{816445785040798}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{7922172361235}{50916528048907948} a^{15} - \frac{40123849362417}{101833056097815896} a^{14} + \frac{4693802176164}{12729132012226987} a^{13} - \frac{142055425654581}{50916528048907948} a^{12} + \frac{327699875334583}{50916528048907948} a^{11} + \frac{2999024939571}{14547579442545128} a^{10} - \frac{502901020292215}{50916528048907948} a^{9} - \frac{72547939614751}{1818447430318141} a^{8} + \frac{2347387381304605}{12729132012226987} a^{7} - \frac{384347588637825}{12729132012226987} a^{6} - \frac{12078116718434159}{25458264024453974} a^{5} + \frac{24503869790909177}{101833056097815896} a^{4} + \frac{6331588197611309}{12729132012226987} a^{3} - \frac{1422674014322193}{1818447430318141} a^{2} - \frac{187522703500421}{1039112817324652} a + \frac{409022279186851}{296889376378472} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 63660.574765 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), 4.4.296000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.296000.2, 8.8.87616000000.1, 8.0.64000000.2, 8.0.3504640000.1

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 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$