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Properties

Label 16.16.3668342962...9169.1
Degree $16$
Signature $[16, 0]$
Discriminant $53^{14}\cdot 149^{14}$
Root discriminant $2572.05$
Ramified primes $53, 149$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25609000866480631, -5002354276372688, 10306247757699384, 1776491162706154, -1057291387991512, -153750994966120, 26357066282862, 5365653798128, 94978012812, -28799442787, -1379884912, 46612678, 3533194, -16435, -3216, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 3216*x^14 - 16435*x^13 + 3533194*x^12 + 46612678*x^11 - 1379884912*x^10 - 28799442787*x^9 + 94978012812*x^8 + 5365653798128*x^7 + 26357066282862*x^6 - 153750994966120*x^5 - 1057291387991512*x^4 + 1776491162706154*x^3 + 10306247757699384*x^2 - 5002354276372688*x - 25609000866480631)
 
gp: K = bnfinit(x^16 - 5*x^15 - 3216*x^14 - 16435*x^13 + 3533194*x^12 + 46612678*x^11 - 1379884912*x^10 - 28799442787*x^9 + 94978012812*x^8 + 5365653798128*x^7 + 26357066282862*x^6 - 153750994966120*x^5 - 1057291387991512*x^4 + 1776491162706154*x^3 + 10306247757699384*x^2 - 5002354276372688*x - 25609000866480631, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 3216 x^{14} - 16435 x^{13} + 3533194 x^{12} + 46612678 x^{11} - 1379884912 x^{10} - 28799442787 x^{9} + 94978012812 x^{8} + 5365653798128 x^{7} + 26357066282862 x^{6} - 153750994966120 x^{5} - 1057291387991512 x^{4} + 1776491162706154 x^{3} + 10306247757699384 x^{2} - 5002354276372688 x - 25609000866480631 \)

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:  \(3668342962637889127310596327197122046055622675946369169=53^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2572.05$
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:  $53, 149$
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}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{15} + \frac{824328722483575322861716747166241607183379127503077367099064500406215756433138328546478298508978740825145618055705258}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{14} - \frac{392615572325462765577319902502226723730927223237625854171787034275985906296388340381674924923031978134125712779930532}{923924034257157101109646096366213899174831301263565194820262314174950212905670042940807818556695124537505349655010763} a^{13} + \frac{1457406455528521690497558435187144059138766341272266397166606095129837024340534905198965896957530117260189551871730558}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{12} + \frac{1619973941866461556463307013294556015112654500514703113946717206210827422544086008151302107833081658109632026078805483}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{11} + \frac{892446052918262637516665634799298114767165883988940253829138184249779013696178083204381459230783447124694186367167990}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{10} + \frac{2188641019179195578134602486369821127402696664601183407263697431741443649887597611021426085880520584929918590477805923}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{9} - \frac{2058821962061926349180092170621966332262829943423667615492379233797323011715443901000215729923759287923051669284496726}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{8} + \frac{1375338194390439722655535033876234469721479869441942150262927015699625568164876952194358524446771243693685234026862504}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{7} + \frac{134913955125007503620536068181204825858688396193541559099079721722866452605992153600986964221004568597126774049523199}{380439308223535276927501333797852782013165829932056256690696247013214793549393547093273807640992110103678673387357373} a^{6} - \frac{371569630133052156627963613210316011762338547087150694552850831344971229746032275757727102520878001858356753819500758}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{5} - \frac{2001400185506118480825733557082283437289530694094715466760442846532504369996927176353874547911858551582393639167627660}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{4} + \frac{1858036568206824766356174903232394172576117457041557092283352717137763212613378762787421698320105096547350971288019146}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{3} - \frac{2374203587826538940828666977791551059921592432894432831631928275771796024289676674854256415590666348932539340497255604}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341} a^{2} + \frac{28174665639393146607833840778494781336862616185149783102266199139293264347214707956510088066460527899139378876031774}{57234232210620351396172767031535374285166540786238551891520674329421694604776020359165086105281998865155198651195357} a + \frac{454747187730606279896538008327694570051580109759163842435838715229881782778967990070790393169041127146437997424356981}{6467468239800099707767522674563497294223819108844956363741836199224651490339690300585654729896865871762537447585075341}$

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:  \( 8573309611210000000000 \) (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

\(\Q(\sqrt{7897}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{53}, \sqrt{149})\), 8.8.242534110929108256632529.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$149$149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$