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Properties

Label 16.0.40892419155...0273.1
Degree $16$
Signature $[0, 8]$
Discriminant $71^{8}\cdot 97^{15}$
Root discriminant $614.08$
Ramified primes $71, 97$
Class number $1546273474$ (GRH)
Class group $[1546273474]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24670268999409811, -10799903260538834, 4135020510780571, -684237862730000, 176031116223380, -16001130613788, 3561203748047, -188524940501, 40240148282, -1198723539, 263030578, -3890525, 957458, -5140, 1701, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 1701*x^14 - 5140*x^13 + 957458*x^12 - 3890525*x^11 + 263030578*x^10 - 1198723539*x^9 + 40240148282*x^8 - 188524940501*x^7 + 3561203748047*x^6 - 16001130613788*x^5 + 176031116223380*x^4 - 684237862730000*x^3 + 4135020510780571*x^2 - 10799903260538834*x + 24670268999409811)
 
gp: K = bnfinit(x^16 - x^15 + 1701*x^14 - 5140*x^13 + 957458*x^12 - 3890525*x^11 + 263030578*x^10 - 1198723539*x^9 + 40240148282*x^8 - 188524940501*x^7 + 3561203748047*x^6 - 16001130613788*x^5 + 176031116223380*x^4 - 684237862730000*x^3 + 4135020510780571*x^2 - 10799903260538834*x + 24670268999409811, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 1701 x^{14} - 5140 x^{13} + 957458 x^{12} - 3890525 x^{11} + 263030578 x^{10} - 1198723539 x^{9} + 40240148282 x^{8} - 188524940501 x^{7} + 3561203748047 x^{6} - 16001130613788 x^{5} + 176031116223380 x^{4} - 684237862730000 x^{3} + 4135020510780571 x^{2} - 10799903260538834 x + 24670268999409811 \)

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:  \(408924191550659033534214062789206959759730273=71^{8}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $614.08$
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:  $71, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(6887=71\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{6887}(1,·)$, $\chi_{6887}(70,·)$, $\chi_{6887}(3407,·)$, $\chi_{6887}(212,·)$, $\chi_{6887}(283,·)$, $\chi_{6887}(5537,·)$, $\chi_{6887}(4900,·)$, $\chi_{6887}(3622,·)$, $\chi_{6887}(5608,·)$, $\chi_{6887}(1066,·)$, $\chi_{6887}(4332,·)$, $\chi_{6887}(2413,·)$, $\chi_{6887}(3054,·)$, $\chi_{6887}(5750,·)$, $\chi_{6887}(6036,·)$, $\chi_{6887}(1918,·)$$\rbrace$
This is 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}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{15} - \frac{5056216651590323541498942028676246839830147649603548183584128008768074604093007384613026976932720108702705481547}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{14} - \frac{7140789221634286067202070848395049495818016158460060381456032050956004740813200932286876470870448615644673344799}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{13} + \frac{6836947103678099242679076509849316240083299780146364974365293778455254053728855694325512298987923508627784018511}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{12} - \frac{356301416984502183299478689409758408807238437263772119668984979676962763142649833956969645011152507937073290540}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{11} + \frac{3515910028923024122246882131767833390613112276828762021805248812900996810513960007431333632722118122022993620431}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{10} + \frac{2841110762675669655568896026156517667042832573773480529301856216218644729725573653991383028752695847917279657995}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{9} + \frac{6723686285283713883538634107786158924848362873123678665787465511024982252360787891523748830784870791194987288198}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{8} - \frac{2252363744699512632114723216619998568820996911825201764298673797057333407741804692252768607639535132062617447011}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{7} - \frac{498487236713468458405002902830988452022926629122225545357405466558235102880944777464298887431274309036904997049}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{6} + \frac{7680066119746322808382071741285840826438354200833906405734215167402648218503242029178981361880916510065005845915}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{5} - \frac{5217466425380783864919124412624458204867794598918164480973714086703480841154236002890214950969653843760746839021}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{4} - \frac{6625792927599340730342521037458628688022073606905050971793069534744230409926961457976262959524621316245098246172}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{3} + \frac{1147524009058334605433721258424072157853077051216906794380978529175284291617129523763001451421197804997595634893}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a^{2} + \frac{2737030275328453506712240051859055179480486845552199999126069970750026156601404105001618514008087133185484010945}{16994595042283242800126150357421429122564271825399459753592999457798508909785969262687199426182898503305092002199} a - \frac{53458883304497970441054492368373462937452059612214410686546004907439485080263312507787344304734404564084789822}{150394646391887104425895135906384328518267892260172210208787605821225742564477604094576986072415031002699929223}$

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

Class group and class number

$C_{1546273474}$, which has order $1546273474$ (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:  \( 1675810.87182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

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.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
71Data not computed
97Data not computed