↑

Properties

Label 16.8.10998771473...7801.1
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 73^{14}$
Root discriminant $1005.97$
Ramified primes $37, 73$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6139708248207609, 3450107191595130, -165474707995824, -246602526605672, 51666615448253, -1698734028116, 53744166666, -95757847887, 9389893780, 456652722, -85402450, 661327, 252311, -628, -701, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 701*x^14 - 628*x^13 + 252311*x^12 + 661327*x^11 - 85402450*x^10 + 456652722*x^9 + 9389893780*x^8 - 95757847887*x^7 + 53744166666*x^6 - 1698734028116*x^5 + 51666615448253*x^4 - 246602526605672*x^3 - 165474707995824*x^2 + 3450107191595130*x - 6139708248207609)
 
gp: K = bnfinit(x^16 - 2*x^15 - 701*x^14 - 628*x^13 + 252311*x^12 + 661327*x^11 - 85402450*x^10 + 456652722*x^9 + 9389893780*x^8 - 95757847887*x^7 + 53744166666*x^6 - 1698734028116*x^5 + 51666615448253*x^4 - 246602526605672*x^3 - 165474707995824*x^2 + 3450107191595130*x - 6139708248207609, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 701 x^{14} - 628 x^{13} + 252311 x^{12} + 661327 x^{11} - 85402450 x^{10} + 456652722 x^{9} + 9389893780 x^{8} - 95757847887 x^{7} + 53744166666 x^{6} - 1698734028116 x^{5} + 51666615448253 x^{4} - 246602526605672 x^{3} - 165474707995824 x^{2} + 3450107191595130 x - 6139708248207609 \)

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:  \(1099877147396767625299704166328746389549375427801=37^{14}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1005.97$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{999} a^{12} + \frac{88}{999} a^{11} + \frac{19}{999} a^{10} + \frac{142}{999} a^{9} + \frac{10}{999} a^{8} + \frac{3}{37} a^{7} - \frac{230}{999} a^{6} + \frac{131}{999} a^{5} - \frac{236}{999} a^{4} + \frac{125}{999} a^{3} - \frac{32}{999} a^{2} - \frac{11}{37} a - \frac{40}{111}$, $\frac{1}{2997} a^{13} - \frac{1}{2997} a^{12} - \frac{487}{2997} a^{11} + \frac{116}{2997} a^{10} - \frac{307}{2997} a^{9} - \frac{476}{2997} a^{8} - \frac{446}{2997} a^{7} - \frac{5}{333} a^{6} + \frac{253}{999} a^{5} - \frac{394}{999} a^{4} + \frac{499}{999} a^{3} - \frac{1445}{2997} a^{2} + \frac{11}{333} a + \frac{119}{333}$, $\frac{1}{122877} a^{14} + \frac{28}{122877} a^{12} - \frac{18899}{122877} a^{11} - \frac{8369}{122877} a^{10} + \frac{2518}{40959} a^{9} + \frac{79}{2997} a^{8} - \frac{653}{122877} a^{7} - \frac{16678}{40959} a^{6} - \frac{14239}{40959} a^{5} - \frac{12182}{40959} a^{4} + \frac{11605}{122877} a^{3} - \frac{16859}{122877} a^{2} + \frac{529}{13653} a - \frac{4870}{13653}$, $\frac{1}{946331272639503020455422573272912405787548646733265646154466727626330585355863383660508966151637197} a^{15} + \frac{210162620806495275476300852864758992389750313622046643727271930746311671248191046816330903345}{315443757546501006818474191090970801929182882244421882051488909208776861785287794553502988717212399} a^{14} - \frac{40527783491278950052301559756099129410461070314279979293261887633948592003813703278453837759481}{946331272639503020455422573272912405787548646733265646154466727626330585355863383660508966151637197} a^{13} + \frac{329583918229293968978125178241714022616172526498034102697472334958021851059129803409240188872320}{946331272639503020455422573272912405787548646733265646154466727626330585355863383660508966151637197} a^{12} + \frac{124847371296965561268824277683746485897643966478587632852160813453671250217705345696104337477326432}{946331272639503020455422573272912405787548646733265646154466727626330585355863383660508966151637197} a^{11} + \frac{44200315587811722341694411202864630982515371796726476988823973723220188484269709255006926993236185}{315443757546501006818474191090970801929182882244421882051488909208776861785287794553502988717212399} a^{10} + \frac{100804584686354625507501608009638307731860839402983001634530778008271541562448273195583164823337330}{946331272639503020455422573272912405787548646733265646154466727626330585355863383660508966151637197} a^{9} - \frac{14097527063626378081014111752120589438909284355179069804686778413502042657255404035601460190032711}{946331272639503020455422573272912405787548646733265646154466727626330585355863383660508966151637197} a^{8} + \frac{48894538050581385923958302519643470714292908407100905403564625028947247592708633721021465111779066}{315443757546501006818474191090970801929182882244421882051488909208776861785287794553502988717212399} a^{7} - \frac{5582636784671810797075696723214161306180509641551552443159037594658858296001085610041224005565786}{35049306394055667424274910121218977992131431360491320227943212134308540198365310505944776524134711} a^{6} - \frac{38230665387644184155376449626231426915248604461570491157867069911427241535045839997743547618532584}{105147919182167002272824730363656933976394294081473960683829636402925620595095931517834329572404133} a^{5} + \frac{451755841616488882251864754268489123285402657047060004346861334680160148195518767994717469641271599}{946331272639503020455422573272912405787548646733265646154466727626330585355863383660508966151637197} a^{4} - \frac{252724902381369875822000327114086466539265185235013849578241904047321502121091609462624314239915175}{946331272639503020455422573272912405787548646733265646154466727626330585355863383660508966151637197} a^{3} - \frac{155547854457266560520347009306839315351377724157709787211901189033218539816087112958854167185990546}{315443757546501006818474191090970801929182882244421882051488909208776861785287794553502988717212399} a^{2} + \frac{6801096791708648273327453724541793899189185840886677020920503681984416609756458757677175145457187}{105147919182167002272824730363656933976394294081473960683829636402925620595095931517834329572404133} a - \frac{433893585268786998058518634453000834965868621996618122520099581989019693357655559437558717805575}{35049306394055667424274910121218977992131431360491320227943212134308540198365310505944776524134711}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 6913299719910000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{2701}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{37}, \sqrt{73})\), 8.8.388282220975269366201.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$73$73.8.7.2$x^{8} - 1825$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.4$x^{8} - 1140625$$8$$1$$7$$C_8$$[\ ]_{8}$