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Properties

Label 16.8.24292037884...1449.2
Degree $16$
Signature $[8, 4]$
Discriminant $61^{2}\cdot 97^{14}$
Root discriminant $91.54$
Ramified primes $61, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2797, 1723, 4616, 5492, -20056, 12597, 27691, -30580, -3836, 13816, -2505, -2215, 660, 144, -56, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 56*x^14 + 144*x^13 + 660*x^12 - 2215*x^11 - 2505*x^10 + 13816*x^9 - 3836*x^8 - 30580*x^7 + 27691*x^6 + 12597*x^5 - 20056*x^4 + 5492*x^3 + 4616*x^2 + 1723*x + 2797)
 
gp: K = bnfinit(x^16 - x^15 - 56*x^14 + 144*x^13 + 660*x^12 - 2215*x^11 - 2505*x^10 + 13816*x^9 - 3836*x^8 - 30580*x^7 + 27691*x^6 + 12597*x^5 - 20056*x^4 + 5492*x^3 + 4616*x^2 + 1723*x + 2797, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 56 x^{14} + 144 x^{13} + 660 x^{12} - 2215 x^{11} - 2505 x^{10} + 13816 x^{9} - 3836 x^{8} - 30580 x^{7} + 27691 x^{6} + 12597 x^{5} - 20056 x^{4} + 5492 x^{3} + 4616 x^{2} + 1723 x + 2797 \)

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:  \(24292037884309209334711647701449=61^{2}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.54$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{10} - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a$, $\frac{1}{1164} a^{12} - \frac{89}{1164} a^{11} + \frac{119}{1164} a^{10} - \frac{26}{291} a^{9} + \frac{37}{582} a^{8} - \frac{37}{291} a^{7} + \frac{35}{291} a^{6} + \frac{20}{97} a^{5} + \frac{19}{97} a^{4} + \frac{53}{194} a^{3} + \frac{109}{388} a^{2} - \frac{89}{388} a - \frac{307}{1164}$, $\frac{1}{1164} a^{13} + \frac{55}{1164} a^{11} + \frac{11}{1164} a^{10} + \frac{65}{582} a^{9} + \frac{3}{97} a^{8} - \frac{19}{97} a^{7} - \frac{26}{291} a^{6} + \frac{9}{194} a^{5} - \frac{57}{194} a^{4} + \frac{37}{388} a^{3} - \frac{22}{97} a^{2} - \frac{499}{1164} a + \frac{31}{1164}$, $\frac{1}{1164} a^{14} - \frac{41}{1164} a^{11} - \frac{13}{1164} a^{10} - \frac{16}{291} a^{9} - \frac{56}{291} a^{8} - \frac{28}{291} a^{7} - \frac{20}{291} a^{6} - \frac{13}{97} a^{5} + \frac{125}{388} a^{4} + \frac{24}{97} a^{3} + \frac{35}{291} a^{2} + \frac{457}{1164} a - \frac{575}{1164}$, $\frac{1}{133994889993210667464} a^{15} - \frac{4687410355078807}{11166240832767555622} a^{14} - \frac{5912631117806833}{16749361249151333433} a^{13} - \frac{8915093487288761}{66997444996605333732} a^{12} - \frac{2462232870429555103}{66997444996605333732} a^{11} + \frac{449222711966939035}{44664963331070222488} a^{10} - \frac{337543608242272323}{5583120416383777811} a^{9} - \frac{2051600563298417281}{16749361249151333433} a^{8} - \frac{611336880046820112}{5583120416383777811} a^{7} - \frac{173688523591527155}{5583120416383777811} a^{6} + \frac{11076766529731995037}{44664963331070222488} a^{5} + \frac{936354585942979239}{11166240832767555622} a^{4} - \frac{7855550962720421918}{16749361249151333433} a^{3} - \frac{32140310522599243699}{66997444996605333732} a^{2} + \frac{7902436610154140889}{22332481665535111244} a - \frac{1229450225027936107}{44664963331070222488}$

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:  $11$
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:  \( 5218188605.98 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

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 t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$