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Properties

Label 16.4.26968313608...625.33
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 101^{10}$
Root discriminant $59.83$
Ramified primes $5, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![278501, 182831, -135503, -47230, -52860, -42171, -15289, -13295, 374, -1845, 529, -198, 125, -10, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 13*x^14 - 10*x^13 + 125*x^12 - 198*x^11 + 529*x^10 - 1845*x^9 + 374*x^8 - 13295*x^7 - 15289*x^6 - 42171*x^5 - 52860*x^4 - 47230*x^3 - 135503*x^2 + 182831*x + 278501)
 
gp: K = bnfinit(x^16 - 2*x^15 + 13*x^14 - 10*x^13 + 125*x^12 - 198*x^11 + 529*x^10 - 1845*x^9 + 374*x^8 - 13295*x^7 - 15289*x^6 - 42171*x^5 - 52860*x^4 - 47230*x^3 - 135503*x^2 + 182831*x + 278501, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 13 x^{14} - 10 x^{13} + 125 x^{12} - 198 x^{11} + 529 x^{10} - 1845 x^{9} + 374 x^{8} - 13295 x^{7} - 15289 x^{6} - 42171 x^{5} - 52860 x^{4} - 47230 x^{3} - 135503 x^{2} + 182831 x + 278501 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26968313608671985107666015625=5^{12}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.83$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{85} a^{13} + \frac{1}{85} a^{12} + \frac{36}{85} a^{11} - \frac{32}{85} a^{10} - \frac{6}{17} a^{9} + \frac{6}{17} a^{8} - \frac{9}{85} a^{7} + \frac{3}{17} a^{6} + \frac{23}{85} a^{5} - \frac{33}{85} a^{4} + \frac{9}{85} a^{3} + \frac{4}{17} a^{2} - \frac{19}{85} a - \frac{8}{85}$, $\frac{1}{3995} a^{14} + \frac{4}{3995} a^{13} - \frac{33}{799} a^{12} + \frac{824}{3995} a^{11} + \frac{23}{85} a^{10} - \frac{1488}{3995} a^{9} + \frac{125}{799} a^{8} - \frac{199}{3995} a^{7} - \frac{2}{47} a^{6} - \frac{360}{799} a^{5} - \frac{209}{3995} a^{4} + \frac{1186}{3995} a^{3} + \frac{1503}{3995} a^{2} + \frac{1584}{3995} a - \frac{287}{799}$, $\frac{1}{13305185061307306065194757411752435} a^{15} - \frac{379621661950059727097439155846}{13305185061307306065194757411752435} a^{14} - \frac{7284177337871389577639209791338}{2661037012261461213038951482350487} a^{13} - \frac{420438487177390644311670599456582}{13305185061307306065194757411752435} a^{12} + \frac{2378239554121016842331007882121273}{13305185061307306065194757411752435} a^{11} + \frac{578338398447587868919453952260575}{2661037012261461213038951482350487} a^{10} + \frac{4534288012467811102455711708240288}{13305185061307306065194757411752435} a^{9} + \frac{4550773194523349959994294850945422}{13305185061307306065194757411752435} a^{8} - \frac{672473643790148513661073529177153}{13305185061307306065194757411752435} a^{7} + \frac{5471185693216123810578027928159988}{13305185061307306065194757411752435} a^{6} - \frac{226057097258751321869391739475441}{578486307013361133269337278771845} a^{5} - \frac{1276321811114950553339045537760854}{2661037012261461213038951482350487} a^{4} - \frac{4276917926156950384980296931065521}{13305185061307306065194757411752435} a^{3} + \frac{1132498243121114281957537105626907}{13305185061307306065194757411752435} a^{2} + \frac{193075943558400068424316185336427}{578486307013361133269337278771845} a - \frac{6283053350728778799825039004186139}{13305185061307306065194757411752435}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 53508961.8507 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.4.1625943765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed