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Properties

Label 16.4.10787325443...625.33
Degree $16$
Signature $[4, 6]$
Discriminant $5^{10}\cdot 101^{10}$
Root discriminant $48.93$
Ramified primes $5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11125, -15500, -14825, 16850, 13545, -11950, -5715, 4525, 34, -747, 205, -29, -85, 24, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 + 24*x^13 - 85*x^12 - 29*x^11 + 205*x^10 - 747*x^9 + 34*x^8 + 4525*x^7 - 5715*x^6 - 11950*x^5 + 13545*x^4 + 16850*x^3 - 14825*x^2 - 15500*x + 11125)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 + 24*x^13 - 85*x^12 - 29*x^11 + 205*x^10 - 747*x^9 + 34*x^8 + 4525*x^7 - 5715*x^6 - 11950*x^5 + 13545*x^4 + 16850*x^3 - 14825*x^2 - 15500*x + 11125, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} + 24 x^{13} - 85 x^{12} - 29 x^{11} + 205 x^{10} - 747 x^{9} + 34 x^{8} + 4525 x^{7} - 5715 x^{6} - 11950 x^{5} + 13545 x^{4} + 16850 x^{3} - 14825 x^{2} - 15500 x + 11125 \)

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:  \(1078732544346879404306640625=5^{10}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.93$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{25} a^{12} - \frac{2}{25} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{2}{5} a^{7} - \frac{6}{25} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{9} + \frac{2}{25} a^{8} - \frac{1}{25} a^{7} + \frac{3}{25} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{125} a^{14} - \frac{1}{125} a^{13} - \frac{1}{125} a^{12} + \frac{11}{125} a^{9} + \frac{4}{125} a^{8} + \frac{59}{125} a^{7} - \frac{9}{25} a^{6} + \frac{6}{25} a^{4} + \frac{9}{25} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{316983096835108955661417625} a^{15} - \frac{151677689348437888309759}{316983096835108955661417625} a^{14} + \frac{2177606928077225838670432}{316983096835108955661417625} a^{13} + \frac{4666775569971386910990698}{316983096835108955661417625} a^{12} + \frac{2325755627951751189425499}{63396619367021791132283525} a^{11} + \frac{15627551772677604518561876}{316983096835108955661417625} a^{10} - \frac{2180411403464057247224799}{316983096835108955661417625} a^{9} + \frac{10208674052081251410208317}{316983096835108955661417625} a^{8} + \frac{73853118525120096084902983}{316983096835108955661417625} a^{7} + \frac{22443395087858244620019224}{63396619367021791132283525} a^{6} - \frac{20027887712620742192947874}{63396619367021791132283525} a^{5} - \frac{11863366956285976568339179}{63396619367021791132283525} a^{4} + \frac{7383947632973457820608538}{63396619367021791132283525} a^{3} - \frac{956764431008293996586393}{2535864774680871645291341} a^{2} + \frac{235135241596182426402852}{12679323873404358226456705} a + \frac{534731600286097328039663}{2535864774680871645291341}$

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:  $9$
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:  \( 51846373.8921 \) (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, 8.4.65037750625.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed