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Properties

Label 16.8.33746583222...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{4}\cdot 13^{4}\cdot 101^{12}$
Root discriminant $255.87$
Ramified primes $2, 5, 13, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![162435025, 0, 2290997460, 0, -421149516, 0, -4903364, 0, 2770854, 0, 16396, 0, -4140, 0, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 28*x^14 - 4140*x^12 + 16396*x^10 + 2770854*x^8 - 4903364*x^6 - 421149516*x^4 + 2290997460*x^2 + 162435025)
 
gp: K = bnfinit(x^16 - 28*x^14 - 4140*x^12 + 16396*x^10 + 2770854*x^8 - 4903364*x^6 - 421149516*x^4 + 2290997460*x^2 + 162435025, 1)
 

Normalized defining polynomial

\( x^{16} - 28 x^{14} - 4140 x^{12} + 16396 x^{10} + 2770854 x^{8} - 4903364 x^{6} - 421149516 x^{4} + 2290997460 x^{2} + 162435025 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(337465832223268533088336601531023360000=2^{24}\cdot 5^{4}\cdot 13^{4}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $255.87$
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:  $2, 5, 13, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{32} a^{9} - \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{4} + \frac{1}{16} a^{3} + \frac{1}{16} a^{2} + \frac{15}{32} a - \frac{11}{32}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{1}{32} a^{6} + \frac{1}{32} a^{5} - \frac{1}{32} a^{4} - \frac{15}{64} a^{3} + \frac{15}{64} a^{2} - \frac{17}{64} a + \frac{17}{64}$, $\frac{1}{320} a^{12} + \frac{1}{64} a^{8} + \frac{1}{20} a^{6} - \frac{13}{320} a^{4} - \frac{3}{20} a^{2} - \frac{5}{64}$, $\frac{1}{640} a^{13} - \frac{1}{640} a^{12} + \frac{1}{128} a^{9} - \frac{1}{128} a^{8} + \frac{1}{40} a^{7} - \frac{1}{40} a^{6} - \frac{13}{640} a^{5} + \frac{13}{640} a^{4} - \frac{3}{40} a^{3} + \frac{3}{40} a^{2} - \frac{5}{128} a + \frac{5}{128}$, $\frac{1}{36080761988072211501828480} a^{14} + \frac{1127139845357594932949}{2405384132538147433455232} a^{12} + \frac{35180138801252243591235}{2405384132538147433455232} a^{10} - \frac{604971209021037023195489}{36080761988072211501828480} a^{8} - \frac{624005777453781013994813}{36080761988072211501828480} a^{6} - \frac{2414503045331026523342383}{36080761988072211501828480} a^{4} + \frac{275629418512310300712451}{7216152397614442300365696} a^{2} + \frac{1193602814377092187450513}{7216152397614442300365696}$, $\frac{1}{183939724615192134236321591040} a^{15} - \frac{1}{72161523976144423003656960} a^{14} - \frac{6049917318570919542352869}{12262648307679475615754772736} a^{13} + \frac{31948427844120578982993}{24053841325381474334552320} a^{12} + \frac{59267764402553132792426323}{12262648307679475615754772736} a^{11} - \frac{35180138801252243591235}{4810768265076294866910464} a^{10} - \frac{2229155785878543725565820199}{183939724615192134236321591040} a^{9} - \frac{1086314509169847890952721}{72161523976144423003656960} a^{8} - \frac{5313516208521086924658238493}{183939724615192134236321591040} a^{7} + \frac{2428043876857391589086237}{72161523976144423003656960} a^{6} + \frac{3960267934675912327325913647}{183939724615192134236321591040} a^{5} + \frac{5458817338074619368809161}{72161523976144423003656960} a^{4} - \frac{5999053070148194657165818061}{36787944923038426847264318208} a^{3} + \frac{11250119603263722522077713}{72161523976144423003656960} a^{2} + \frac{6333029074577412033775519111}{36787944923038426847264318208} a - \frac{5816450444098844286122287}{14432304795228884600731392}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 9308502801110 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.132613.1, 8.8.229628038978008320.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
101Data not computed