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Properties

Label 16.0.23462097005...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{8}\cdot 5^{14}\cdot 41^{2}\cdot 563021^{2}$
Root discriminant $333.54$
Ramified primes $2, 3, 5, 41, 563021$
Class number $502970944$ (GRH)
Class group $[2, 2, 4, 31435684]$ (GRH)
Galois group 16T1162

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75558851901361, -1393015120376, 8329269294588, 656088145268, 247939978456, 60097040484, 5006084840, 1709712244, 173831769, 16709548, 4670180, -47388, 63806, -1764, 412, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 412*x^14 - 1764*x^13 + 63806*x^12 - 47388*x^11 + 4670180*x^10 + 16709548*x^9 + 173831769*x^8 + 1709712244*x^7 + 5006084840*x^6 + 60097040484*x^5 + 247939978456*x^4 + 656088145268*x^3 + 8329269294588*x^2 - 1393015120376*x + 75558851901361)
 
gp: K = bnfinit(x^16 - 8*x^15 + 412*x^14 - 1764*x^13 + 63806*x^12 - 47388*x^11 + 4670180*x^10 + 16709548*x^9 + 173831769*x^8 + 1709712244*x^7 + 5006084840*x^6 + 60097040484*x^5 + 247939978456*x^4 + 656088145268*x^3 + 8329269294588*x^2 - 1393015120376*x + 75558851901361, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 412 x^{14} - 1764 x^{13} + 63806 x^{12} - 47388 x^{11} + 4670180 x^{10} + 16709548 x^{9} + 173831769 x^{8} + 1709712244 x^{7} + 5006084840 x^{6} + 60097040484 x^{5} + 247939978456 x^{4} + 656088145268 x^{3} + 8329269294588 x^{2} - 1393015120376 x + 75558851901361 \)

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:  \(23462097005834430807696998400000000000000=2^{40}\cdot 3^{8}\cdot 5^{14}\cdot 41^{2}\cdot 563021^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $333.54$
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:  $2, 3, 5, 41, 563021$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
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}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{15} + \frac{2828713803569571153646555023008252210477894458125350426668406204066679321714262339432308519}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{14} + \frac{3448832706706279014098178373223418232189929131710783506386936094652005027574006279109365202}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{13} - \frac{169045081672973516745935565080059082870354980323162004732776339515787851848091604062955919}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{12} + \frac{3849590442171742719744911835455199806779517807771156360092192191745854840460000111404599587}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{11} - \frac{1838507937965566656488867691785830087142583335211135376583663440132943929453710690585032934}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{10} + \frac{735061268128091951020686293144428153491559517175178332720018204382192582430572350385256434}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{9} + \frac{3023213224220881185274979552693337595010994614032309363360697751022080198429505884432478528}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{8} - \frac{2714834320495003175968119064505049682689983817161008647666731607841325499968135313935053574}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{7} - \frac{3887891738118650174362369044193015152977435917625615888161842264246095913408986012596743531}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{6} + \frac{450145096785918196081310437612231985824058913413200663442732832969045402667636537939281523}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{5} + \frac{3221474705452628082821438675004186246986688710972031420788982796315264494818439159642275837}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{4} + \frac{910279621394114822522267333498261725818178953942346695582043983918130773500423892004010473}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{3} - \frac{2319573295161477018978486747016682354131530082850056619882031702047417410978507932525889199}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a^{2} + \frac{1070530460921324098555938968322796596933555989538372782477454324980741821362938111228268670}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141} a - \frac{3341981283786330103549814050176211116680382159232058299961873032519866156362224111717479517}{8494936724122784322334003380833564417280170164953981588457035365937919517618394761892421141}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{31435684}$, which has order $502970944$ (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:  \( 114709.055882 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1162:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1162
Character table for t16n1162 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.414720000000.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
563021Data not computed