↑

Properties

Label 16.0.23043642001...0976.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{72}\cdot 47^{4}$
Root discriminant $59.25$
Ramified primes $2, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![132098, -225632, 238672, -227392, 159520, -102272, 60832, -24064, 12692, -4720, 1136, -672, 120, -32, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 32*x^13 + 120*x^12 - 672*x^11 + 1136*x^10 - 4720*x^9 + 12692*x^8 - 24064*x^7 + 60832*x^6 - 102272*x^5 + 159520*x^4 - 227392*x^3 + 238672*x^2 - 225632*x + 132098)
 
gp: K = bnfinit(x^16 + 16*x^14 - 32*x^13 + 120*x^12 - 672*x^11 + 1136*x^10 - 4720*x^9 + 12692*x^8 - 24064*x^7 + 60832*x^6 - 102272*x^5 + 159520*x^4 - 227392*x^3 + 238672*x^2 - 225632*x + 132098, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 32 x^{13} + 120 x^{12} - 672 x^{11} + 1136 x^{10} - 4720 x^{9} + 12692 x^{8} - 24064 x^{7} + 60832 x^{6} - 102272 x^{5} + 159520 x^{4} - 227392 x^{3} + 238672 x^{2} - 225632 x + 132098 \)

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:  \(23043642001495833226013310976=2^{72}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.25$
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:  $2, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{70971636111254061729556865401485239} a^{15} - \frac{171395258235242766559526951441990}{10138805158750580247079552200212177} a^{14} - \frac{12306048684826066845018230911820679}{70971636111254061729556865401485239} a^{13} - \frac{2508333456346699258289500416641275}{10138805158750580247079552200212177} a^{12} - \frac{22288964567990583487055586486771135}{70971636111254061729556865401485239} a^{11} + \frac{28239530205593968942021657878550243}{70971636111254061729556865401485239} a^{10} - \frac{3038817429090074122042104081708086}{10138805158750580247079552200212177} a^{9} + \frac{5729904208310920569849580298551575}{70971636111254061729556865401485239} a^{8} + \frac{27196848985922336648758874548655476}{70971636111254061729556865401485239} a^{7} + \frac{1991278950144594006934258248216032}{4174802124191415395856286200087367} a^{6} + \frac{11288893129655370382520534465910679}{70971636111254061729556865401485239} a^{5} - \frac{12685047220866190250208211082118661}{70971636111254061729556865401485239} a^{4} + \frac{30666518844409423346687216764700728}{70971636111254061729556865401485239} a^{3} + \frac{22305675068143675973155368228711805}{70971636111254061729556865401485239} a^{2} + \frac{31457879732298138351769025758226562}{70971636111254061729556865401485239} a - \frac{11819572832489242222014735374020039}{70971636111254061729556865401485239}$

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:  \( -1 \) (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:  \( 40150114.3022 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.0.2147483648.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
47Data not computed