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Properties

Label 16.0.96606406032...9033.9
Degree $16$
Signature $[0, 8]$
Discriminant $71^{12}\cdot 73^{9}$
Root discriminant $273.25$
Ramified primes $71, 73$
Class number $28$ (GRH)
Class group $[2, 14]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![85829254185005, -59659550358172, 1482211608009, 3826741307380, 104869687622, -96096389750, 76437998, 2204480747, 74993769, -29359470, -1814397, 275627, 29338, -1329, -189, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 189*x^14 - 1329*x^13 + 29338*x^12 + 275627*x^11 - 1814397*x^10 - 29359470*x^9 + 74993769*x^8 + 2204480747*x^7 + 76437998*x^6 - 96096389750*x^5 + 104869687622*x^4 + 3826741307380*x^3 + 1482211608009*x^2 - 59659550358172*x + 85829254185005)
 
gp: K = bnfinit(x^16 - 2*x^15 - 189*x^14 - 1329*x^13 + 29338*x^12 + 275627*x^11 - 1814397*x^10 - 29359470*x^9 + 74993769*x^8 + 2204480747*x^7 + 76437998*x^6 - 96096389750*x^5 + 104869687622*x^4 + 3826741307380*x^3 + 1482211608009*x^2 - 59659550358172*x + 85829254185005, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 189 x^{14} - 1329 x^{13} + 29338 x^{12} + 275627 x^{11} - 1814397 x^{10} - 29359470 x^{9} + 74993769 x^{8} + 2204480747 x^{7} + 76437998 x^{6} - 96096389750 x^{5} + 104869687622 x^{4} + 3826741307380 x^{3} + 1482211608009 x^{2} - 59659550358172 x + 85829254185005 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(966064060320802955505014160902035909033=71^{12}\cdot 73^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $273.25$
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:  $71, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{15} + \frac{59068208033483438874253289368743350304870691181880407863918024854870327373219939537536028680526992}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{14} + \frac{99421062381958095293859901861350365242369638255423589018681384314072915304750830185795826249604564}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{13} - \frac{25626374814348201064483352983478879435964656727589624685802586562240337093628684484436541695504458}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{12} + \frac{104412346131168549476616834437217210397325447971919237380664579825918815874957498025130996922956111}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{11} + \frac{135905034905979219602105316200560198803196785030588487321272952871748197605829450382374775383133696}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{10} - \frac{74154285767801187662712727947908538563012676948073592717730522122850727481105527567383208029182173}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{9} - \frac{47442765702562526699878039808567616083276771531510841488414782168117522144555138475135574503215202}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{8} + \frac{103978434826271345062856011078243568345337644467552039858150524841612138398160942868395801097861011}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{7} + \frac{124518636962993211319880553926993627821217647946892416996169730329234220089813707417436326482757086}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{6} - \frac{20145758840060236668743533932250634009456524289636855680410860876096238191121789474473843702036968}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{5} + \frac{38830070381879054606943862840762381295007863686046438216285952085351527266877162685811983583862518}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{4} + \frac{134112886849146443670150308784112076084036794182556498394475084321384372473490002315960563075090844}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{3} - \frac{99973353847593091851944971475164845127117863121108415721878919526637039244272983379796135106251524}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a^{2} + \frac{146852054231145666913561361484764945087330004234530093819516976700015886417567547502847753695370398}{296076058072799029442241208769846017021453099789211617880784276985611268095538624454340378188581435} a + \frac{12118785995154105769971483165080966752813784615250523569866579188596968982859981188298265864895387}{59215211614559805888448241753969203404290619957842323576156855397122253619107724890868075637716287}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{14}$, which has order $28$ (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:  $7$
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:  \( 378844007220 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-71}) \), 4.0.367993.1, 8.0.9885575907577.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ $16$ $16$ $16$

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
$71$71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
73Data not computed