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Properties

Label 16.8.30187924998...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 11^{2}\cdot 29^{6}$
Root discriminant $52.18$
Ramified primes $2, 5, 11, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![138911, 5368, -243180, -29872, 143971, 67876, -23864, -32992, -12127, -532, 3384, 1696, 155, -84, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 84*x^13 + 155*x^12 + 1696*x^11 + 3384*x^10 - 532*x^9 - 12127*x^8 - 32992*x^7 - 23864*x^6 + 67876*x^5 + 143971*x^4 - 29872*x^3 - 243180*x^2 + 5368*x + 138911)
 
gp: K = bnfinit(x^16 - 32*x^14 - 84*x^13 + 155*x^12 + 1696*x^11 + 3384*x^10 - 532*x^9 - 12127*x^8 - 32992*x^7 - 23864*x^6 + 67876*x^5 + 143971*x^4 - 29872*x^3 - 243180*x^2 + 5368*x + 138911, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} - 84 x^{13} + 155 x^{12} + 1696 x^{11} + 3384 x^{10} - 532 x^{9} - 12127 x^{8} - 32992 x^{7} - 23864 x^{6} + 67876 x^{5} + 143971 x^{4} - 29872 x^{3} - 243180 x^{2} + 5368 x + 138911 \)

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:  \(3018792499821936640000000000=2^{32}\cdot 5^{10}\cdot 11^{2}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.18$
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:  $2, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{295} a^{14} - \frac{141}{295} a^{13} - \frac{5}{59} a^{12} - \frac{14}{59} a^{11} + \frac{13}{59} a^{10} - \frac{134}{295} a^{9} - \frac{147}{295} a^{8} - \frac{9}{295} a^{7} - \frac{29}{59} a^{6} - \frac{61}{295} a^{5} - \frac{48}{295} a^{4} - \frac{127}{295} a^{3} + \frac{28}{59} a^{2} - \frac{74}{295} a + \frac{89}{295}$, $\frac{1}{106169799287255739150930280239213964825} a^{15} + \frac{118967255216030962340165589324170552}{106169799287255739150930280239213964825} a^{14} + \frac{29313130346716124250781302828383729567}{106169799287255739150930280239213964825} a^{13} + \frac{638930718048262922070352437402073556}{21233959857451147830186056047842792965} a^{12} + \frac{4210482018462656604207788579382457628}{21233959857451147830186056047842792965} a^{11} + \frac{17523874544652832390467215153965344426}{106169799287255739150930280239213964825} a^{10} - \frac{24089779674010951112222542838807993564}{106169799287255739150930280239213964825} a^{9} + \frac{5667319791670856770225573535431160897}{21233959857451147830186056047842792965} a^{8} - \frac{252300783875012743375369663983223459}{798269167573351422187445716084315525} a^{7} + \frac{7251625408209927187900232525599158609}{106169799287255739150930280239213964825} a^{6} - \frac{40727208817847926621583568218385902746}{106169799287255739150930280239213964825} a^{5} - \frac{49422318230471826766273571798642824586}{106169799287255739150930280239213964825} a^{4} + \frac{51387066288585004037595689329942642989}{106169799287255739150930280239213964825} a^{3} - \frac{10561330286028422673577540094315050534}{106169799287255739150930280239213964825} a^{2} + \frac{6747119581692676261954722400660186752}{106169799287255739150930280239213964825} a + \frac{25367265114593845762643153825740620317}{106169799287255739150930280239213964825}$

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:  $11$
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:  \( 26780763.3724 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), 4.4.725.1, 4.4.46400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.2152960000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$