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Properties

Label 16.0.35120559063...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{14}\cdot 11^{8}$
Root discriminant $45.61$
Ramified primes $2, 5, 11$
Class number $192$ (GRH)
Class group $[2, 2, 2, 24]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21025, -20300, 50300, -39050, 58760, -37690, 39070, -19660, 15771, -6074, 4022, -1092, 540, -138, 42, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 42*x^14 - 138*x^13 + 540*x^12 - 1092*x^11 + 4022*x^10 - 6074*x^9 + 15771*x^8 - 19660*x^7 + 39070*x^6 - 37690*x^5 + 58760*x^4 - 39050*x^3 + 50300*x^2 - 20300*x + 21025)
 
gp: K = bnfinit(x^16 - 6*x^15 + 42*x^14 - 138*x^13 + 540*x^12 - 1092*x^11 + 4022*x^10 - 6074*x^9 + 15771*x^8 - 19660*x^7 + 39070*x^6 - 37690*x^5 + 58760*x^4 - 39050*x^3 + 50300*x^2 - 20300*x + 21025, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 42 x^{14} - 138 x^{13} + 540 x^{12} - 1092 x^{11} + 4022 x^{10} - 6074 x^{9} + 15771 x^{8} - 19660 x^{7} + 39070 x^{6} - 37690 x^{5} + 58760 x^{4} - 39050 x^{3} + 50300 x^{2} - 20300 x + 21025 \)

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:  \(351205590630400000000000000=2^{28}\cdot 5^{14}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.61$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{2}{5} a^{11} + \frac{2}{15} a^{10} - \frac{1}{5} a^{9} + \frac{1}{3} a^{8} + \frac{1}{5} a^{7} + \frac{2}{15} a^{6} + \frac{1}{15} a^{5} + \frac{2}{5} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{13} - \frac{4}{15} a^{11} - \frac{2}{5} a^{10} + \frac{2}{15} a^{9} + \frac{1}{5} a^{8} + \frac{1}{3} a^{7} - \frac{2}{15} a^{6} - \frac{1}{5} a^{5} + \frac{1}{15} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{15} a^{14} - \frac{1}{3} a^{10} + \frac{2}{5} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{2}{5} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{65654555586742221804851041986465} a^{15} + \frac{1352905014063347928040475331644}{65654555586742221804851041986465} a^{14} + \frac{887085668926245313781984460769}{65654555586742221804851041986465} a^{13} - \frac{790749672335584697996278805584}{65654555586742221804851041986465} a^{12} - \frac{2997302769176640401026561271269}{21884851862247407268283680662155} a^{11} - \frac{7646819085595265468395034472147}{21884851862247407268283680662155} a^{10} - \frac{5542954559838902691426984248297}{21884851862247407268283680662155} a^{9} - \frac{2340605174406872404908157087433}{65654555586742221804851041986465} a^{8} - \frac{6595196677053012282767564967309}{21884851862247407268283680662155} a^{7} + \frac{27450897879804444621292229110244}{65654555586742221804851041986465} a^{6} - \frac{1695430647973360426245891604304}{21884851862247407268283680662155} a^{5} + \frac{13816779788726837809220356671556}{65654555586742221804851041986465} a^{4} - \frac{5025756740644671425342986332865}{13130911117348444360970208397293} a^{3} + \frac{1521348087782806598140192065148}{13130911117348444360970208397293} a^{2} - \frac{1337689046594322952115492406513}{4376970372449481453656736132431} a - \frac{98372869963501861868570713948}{452790038529256702102420979217}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{24}$, which has order $192$ (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:  \( 28495.2195483 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.22000.1, 4.4.4400.1, 8.0.4685120000000.11, 8.0.4685120000000.1, 8.8.7744000000.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$