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Properties

Label 16.8.99253389787...5625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 11^{4}\cdot 19^{2}\cdot 29^{8}\cdot 31^{2}$
Root discriminant $48.67$
Ramified primes $5, 11, 19, 29, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T860

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![54391, 176189, -226205, 82699, -40185, -66691, 66320, -41847, 19210, -1045, -2091, 1136, -250, 29, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 4*x^14 + 29*x^13 - 250*x^12 + 1136*x^11 - 2091*x^10 - 1045*x^9 + 19210*x^8 - 41847*x^7 + 66320*x^6 - 66691*x^5 - 40185*x^4 + 82699*x^3 - 226205*x^2 + 176189*x + 54391)
 
gp: K = bnfinit(x^16 - 6*x^15 + 4*x^14 + 29*x^13 - 250*x^12 + 1136*x^11 - 2091*x^10 - 1045*x^9 + 19210*x^8 - 41847*x^7 + 66320*x^6 - 66691*x^5 - 40185*x^4 + 82699*x^3 - 226205*x^2 + 176189*x + 54391, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 4 x^{14} + 29 x^{13} - 250 x^{12} + 1136 x^{11} - 2091 x^{10} - 1045 x^{9} + 19210 x^{8} - 41847 x^{7} + 66320 x^{6} - 66691 x^{5} - 40185 x^{4} + 82699 x^{3} - 226205 x^{2} + 176189 x + 54391 \)

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:  \(992533897870849042547265625=5^{8}\cdot 11^{4}\cdot 19^{2}\cdot 29^{8}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.67$
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:  $5, 11, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} - \frac{7}{29} a^{11} + \frac{2}{29} a^{10} - \frac{5}{29} a^{9} - \frac{10}{29} a^{8} - \frac{1}{29} a^{7} + \frac{2}{29} a^{5} - \frac{9}{29} a^{4} - \frac{3}{29} a^{3} + \frac{7}{29} a^{2} + \frac{2}{29} a - \frac{7}{29}$, $\frac{1}{29} a^{13} + \frac{11}{29} a^{11} + \frac{9}{29} a^{10} + \frac{13}{29} a^{9} - \frac{13}{29} a^{8} - \frac{7}{29} a^{7} + \frac{2}{29} a^{6} + \frac{5}{29} a^{5} - \frac{8}{29} a^{4} - \frac{14}{29} a^{3} - \frac{7}{29} a^{2} + \frac{7}{29} a + \frac{9}{29}$, $\frac{1}{56179409} a^{14} - \frac{128462}{56179409} a^{13} - \frac{503684}{56179409} a^{12} - \frac{18771587}{56179409} a^{11} + \frac{10909810}{56179409} a^{10} - \frac{156402}{5107219} a^{9} - \frac{5025}{12673} a^{8} + \frac{1272939}{5107219} a^{7} + \frac{2085811}{4321493} a^{6} + \frac{25372231}{56179409} a^{5} + \frac{17954174}{56179409} a^{4} - \frac{91518}{5107219} a^{3} - \frac{14973055}{56179409} a^{2} + \frac{730471}{56179409} a + \frac{17199640}{56179409}$, $\frac{1}{19203726171827503115055357198574201} a^{15} + \frac{139236934791572045399195051}{19203726171827503115055357198574201} a^{14} + \frac{22548199253478587653273358928}{91883857281471306770599795208489} a^{13} + \frac{295024390299712605876709596070179}{19203726171827503115055357198574201} a^{12} - \frac{26027108771917872456004659457714}{77747879238168028805892134407183} a^{11} + \frac{7172714478361568468816759051258607}{19203726171827503115055357198574201} a^{10} - \frac{174903964943035155578911360052601}{1745793288347954828641396108961291} a^{9} - \frac{29550132668195371530942272091499}{91883857281471306770599795208489} a^{8} + \frac{9388972580288589193360392147577668}{19203726171827503115055357198574201} a^{7} - \frac{860984143091709617716594567283764}{1745793288347954828641396108961291} a^{6} - \frac{2569050978491787552959547111662464}{19203726171827503115055357198574201} a^{5} - \frac{112390228147768387887198349888343}{834944616166413178915450312981487} a^{4} + \frac{207114971539423272744145066140153}{1477209705525192547311950553736477} a^{3} - \frac{468864774288500615080172775692}{37728342184336941286945691942189} a^{2} - \frac{7421307481126717560084849920594163}{19203726171827503115055357198574201} a - \frac{3345581636369726739615739600}{38484421186027060350812339075299}$

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:  \( 12551349.8585 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T860:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n860 are not computed
Character table for t16n860 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), 4.4.725.1 x2, 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$