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Properties

Label 16.4.10787325443...625.34
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, 16525, -18895, -15055, 16470, -4715, -4614, 8350, -6741, 2388, 103, -292, 69, 0, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 2*x^14 + 69*x^12 - 292*x^11 + 103*x^10 + 2388*x^9 - 6741*x^8 + 8350*x^7 - 4614*x^6 - 4715*x^5 + 16470*x^4 - 15055*x^3 - 18895*x^2 + 16525*x + 11125)
 
gp: K = bnfinit(x^16 - x^15 + 2*x^14 + 69*x^12 - 292*x^11 + 103*x^10 + 2388*x^9 - 6741*x^8 + 8350*x^7 - 4614*x^6 - 4715*x^5 + 16470*x^4 - 15055*x^3 - 18895*x^2 + 16525*x + 11125, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 2 x^{14} + 69 x^{12} - 292 x^{11} + 103 x^{10} + 2388 x^{9} - 6741 x^{8} + 8350 x^{7} - 4614 x^{6} - 4715 x^{5} + 16470 x^{4} - 15055 x^{3} - 18895 x^{2} + 16525 x + 11125 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{10} a^{6} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{3}{10} a^{8} + \frac{2}{5} a^{7} - \frac{3}{10} a^{6} - \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2}$, $\frac{1}{100} a^{14} + \frac{1}{100} a^{13} - \frac{1}{100} a^{12} - \frac{7}{100} a^{11} + \frac{1}{10} a^{10} + \frac{13}{100} a^{9} + \frac{29}{100} a^{8} + \frac{31}{100} a^{7} + \frac{13}{50} a^{6} - \frac{3}{100} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{3}{10} a^{2} - \frac{3}{20} a + \frac{1}{4}$, $\frac{1}{10340431427792696606429333374600} a^{15} + \frac{3622254284282442413445414593}{1292553928474087075803666671825} a^{14} + \frac{128505538356874755411980304101}{5170215713896348303214666687300} a^{13} + \frac{781025445386538634329494957}{1034043142779269660642933337460} a^{12} - \frac{877380279334440016918029696321}{10340431427792696606429333374600} a^{11} - \frac{28887202205198142543736019867}{333562304122345051820301076600} a^{10} + \frac{1144394389217132643199319135209}{5170215713896348303214666687300} a^{9} - \frac{125254334739875042443605798639}{272116616520860437011298246700} a^{8} - \frac{3089705211025934547072146533351}{10340431427792696606429333374600} a^{7} + \frac{856895921188304538848657773523}{2068086285558539321285866674920} a^{6} - \frac{3127022248050338596467628364079}{10340431427792696606429333374600} a^{5} - \frac{291920097681761194840365043029}{1034043142779269660642933337460} a^{4} + \frac{1035662529571916006065178799}{2473787422916913063739074970} a^{3} + \frac{964351505014870275171661141209}{2068086285558539321285866674920} a^{2} - \frac{339773978019075469439879358477}{1034043142779269660642933337460} a - \frac{70237971746431838443757335261}{413617257111707864257173334984}$

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:  \( 59279231.8251 \) (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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.2.1.1$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