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Properties

Label 16.16.2140705834...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 5^{8}\cdot 13^{6}\cdot 41^{4}\cdot 311^{4}$
Root discriminant $248.69$
Ramified primes $2, 5, 13, 41, 311$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times C_4).C_2^4$ (as 16T205)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12924181690872944, 5400237968704032, -2292621554536112, -518503792322688, 142907859799296, 15875325749312, -3734495952480, -211911186816, 46041170792, 1378342360, -291746664, -4501752, 968618, 6976, -1584, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 1584*x^14 + 6976*x^13 + 968618*x^12 - 4501752*x^11 - 291746664*x^10 + 1378342360*x^9 + 46041170792*x^8 - 211911186816*x^7 - 3734495952480*x^6 + 15875325749312*x^5 + 142907859799296*x^4 - 518503792322688*x^3 - 2292621554536112*x^2 + 5400237968704032*x + 12924181690872944)
 
gp: K = bnfinit(x^16 - 4*x^15 - 1584*x^14 + 6976*x^13 + 968618*x^12 - 4501752*x^11 - 291746664*x^10 + 1378342360*x^9 + 46041170792*x^8 - 211911186816*x^7 - 3734495952480*x^6 + 15875325749312*x^5 + 142907859799296*x^4 - 518503792322688*x^3 - 2292621554536112*x^2 + 5400237968704032*x + 12924181690872944, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 1584 x^{14} + 6976 x^{13} + 968618 x^{12} - 4501752 x^{11} - 291746664 x^{10} + 1378342360 x^{9} + 46041170792 x^{8} - 211911186816 x^{7} - 3734495952480 x^{6} + 15875325749312 x^{5} + 142907859799296 x^{4} - 518503792322688 x^{3} - 2292621554536112 x^{2} + 5400237968704032 x + 12924181690872944 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(214070583463493396843509212774400000000=2^{32}\cdot 5^{8}\cdot 13^{6}\cdot 41^{4}\cdot 311^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $248.69$
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:  $2, 5, 13, 41, 311$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{32184783017053472737977800426684053845444471190649713274962133434058601757523148593425048182368021398696} a^{15} - \frac{1587793900266541604917609664538703046553891640491124792349744569316287520270442881986953500817196311113}{32184783017053472737977800426684053845444471190649713274962133434058601757523148593425048182368021398696} a^{14} - \frac{606755465998478615327230122814511275127773786594641283677246397063409327069287418314512911966620100447}{32184783017053472737977800426684053845444471190649713274962133434058601757523148593425048182368021398696} a^{13} - \frac{53375604372507249715805526147277079246963657721226213159650459040277299693409967750987527837359667015}{8046195754263368184494450106671013461361117797662428318740533358514650439380787148356262045592005349674} a^{12} + \frac{1604648116924833009172570124435189873781939319105430208395616440220001825705362553396674023677429129147}{16092391508526736368988900213342026922722235595324856637481066717029300878761574296712524091184010699348} a^{11} + \frac{1880693381819284682782876630806036622256446093573342001242808774980338871604642009673491569389458784115}{16092391508526736368988900213342026922722235595324856637481066717029300878761574296712524091184010699348} a^{10} + \frac{483583144790926038837240670105207437693791761984789192123067449623510061859025730816097358249877362299}{8046195754263368184494450106671013461361117797662428318740533358514650439380787148356262045592005349674} a^{9} - \frac{85690063167127520314431157107680599408843962975036224493403021650091894894722799364949620245455264299}{16092391508526736368988900213342026922722235595324856637481066717029300878761574296712524091184010699348} a^{8} + \frac{701223355076217880601991544234224463964294579450655599292919196119147567867656152384229240933557627594}{4023097877131684092247225053335506730680558898831214159370266679257325219690393574178131022796002674837} a^{7} + \frac{152091003867722891994670389942442316336696421437189634499124315125627356201054794038824791900482253945}{8046195754263368184494450106671013461361117797662428318740533358514650439380787148356262045592005349674} a^{6} - \frac{1156699650949095871222176492288286319242498371058648482901716376135756630112972726097455150399594049801}{8046195754263368184494450106671013461361117797662428318740533358514650439380787148356262045592005349674} a^{5} + \frac{331642705673850191406072699440419129113482055490341904085987597932140121910892743312516104854343806715}{4023097877131684092247225053335506730680558898831214159370266679257325219690393574178131022796002674837} a^{4} + \frac{1476797794592092814066652388147015178333172462946727079705840796484190142039968797461879771498458845126}{4023097877131684092247225053335506730680558898831214159370266679257325219690393574178131022796002674837} a^{3} - \frac{1067382358791808044292024279863106430778538936116212618160373886719984036353208464373866101863645511530}{4023097877131684092247225053335506730680558898831214159370266679257325219690393574178131022796002674837} a^{2} - \frac{486108335752790751947191402609578409428617857739002870780720067532982226652667935242237873704742617659}{4023097877131684092247225053335506730680558898831214159370266679257325219690393574178131022796002674837} a - \frac{1773268328627738951119202260521417432737636316840399946793731101113480503172869735219713391933369410534}{4023097877131684092247225053335506730680558898831214159370266679257325219690393574178131022796002674837}$

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:  $15$
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:  \( 43404899056800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).C_2^4$ (as 16T205):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $(C_2\times C_4).C_2^4$
Character table for $(C_2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.432640000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
5Data not computed
$13$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.6.3$x^{8} + 65 x^{4} + 1352$$4$$2$$6$$C_8$$[\ ]_{4}^{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$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
311Data not computed