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Properties

Label 16.0.17612050268...000.10
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $18.42$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -1000, 450, -1240, 1736, -572, 708, -976, 227, -136, 228, -36, 8, -24, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625)
 
gp: K = bnfinit(x^16 + 2*x^14 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 24 x^{13} + 8 x^{12} - 36 x^{11} + 228 x^{10} - 136 x^{9} + 227 x^{8} - 976 x^{7} + 708 x^{6} - 572 x^{5} + 1736 x^{4} - 1240 x^{3} + 450 x^{2} - 1000 x + 625 \)

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:  \(176120502681600000000=2^{36}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.42$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{1975} a^{14} - \frac{19}{395} a^{13} + \frac{2}{1975} a^{12} + \frac{11}{1975} a^{11} + \frac{663}{1975} a^{10} - \frac{346}{1975} a^{9} + \frac{198}{1975} a^{8} + \frac{979}{1975} a^{7} + \frac{472}{1975} a^{6} + \frac{284}{1975} a^{5} - \frac{322}{1975} a^{4} + \frac{843}{1975} a^{3} - \frac{499}{1975} a^{2} + \frac{73}{395} a - \frac{7}{79}$, $\frac{1}{50733414875} a^{15} + \frac{1106978}{10146682975} a^{14} + \frac{887020327}{50733414875} a^{13} + \frac{6463669506}{50733414875} a^{12} - \frac{15464168827}{50733414875} a^{11} + \frac{19055373284}{50733414875} a^{10} - \frac{6123318087}{50733414875} a^{9} - \frac{14655265791}{50733414875} a^{8} + \frac{18297044412}{50733414875} a^{7} + \frac{14012458104}{50733414875} a^{6} - \frac{2557388657}{50733414875} a^{5} - \frac{8266153777}{50733414875} a^{4} - \frac{8463474369}{50733414875} a^{3} + \frac{306995476}{2029336595} a^{2} - \frac{2617413}{2029336595} a - \frac{29431240}{405867319}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{3764225328}{50733414875} a^{15} - \frac{1075426871}{10146682975} a^{14} - \frac{15201206606}{50733414875} a^{13} + \frac{69267193662}{50733414875} a^{12} + \frac{69142392246}{50733414875} a^{11} + \frac{235228747643}{50733414875} a^{10} - \frac{537597767104}{50733414875} a^{9} - \frac{260682316757}{50733414875} a^{8} - \frac{1247795732776}{50733414875} a^{7} + \frac{2021600797493}{50733414875} a^{6} + \frac{225487607856}{50733414875} a^{5} + \frac{2592569776226}{50733414875} a^{4} - \frac{3274583465198}{50733414875} a^{3} + \frac{18056730218}{10146682975} a^{2} - \frac{69009682524}{2029336595} a + \frac{13791860111}{405867319} \) (order $12$)
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:  \( 7183.55685667 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_4$ (as 16T25):

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 $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), 4.0.1280.1, 4.0.2880.1, 4.0.320.1, 4.0.11520.1, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), 8.0.3317760000.2, 8.0.13271040000.3, 8.0.13271040000.4, 8.0.163840000.2, 8.0.13271040000.2, 8.0.132710400.4, 8.0.8294400.2

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$