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Properties

Label 16.0.10107540821...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{4}\cdot 5^{8}\cdot 13^{8}\cdot 1249^{2}\cdot 1511^{2}$
Root discriminant $365.42$
Ramified primes $2, 3, 5, 13, 1249, 1511$
Class number $959830400$ (GRH)
Class group $[2, 2, 2, 2, 2, 29994700]$ (GRH)
Galois group 16T1547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25628389555519, -11384486485632, 6210498313016, -1784176112812, 572310574986, -114044064440, 27035672060, -3717247748, 728118680, -66371504, 11627920, -641420, 107194, -2936, 516, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 516*x^14 - 2936*x^13 + 107194*x^12 - 641420*x^11 + 11627920*x^10 - 66371504*x^9 + 728118680*x^8 - 3717247748*x^7 + 27035672060*x^6 - 114044064440*x^5 + 572310574986*x^4 - 1784176112812*x^3 + 6210498313016*x^2 - 11384486485632*x + 25628389555519)
 
gp: K = bnfinit(x^16 - 4*x^15 + 516*x^14 - 2936*x^13 + 107194*x^12 - 641420*x^11 + 11627920*x^10 - 66371504*x^9 + 728118680*x^8 - 3717247748*x^7 + 27035672060*x^6 - 114044064440*x^5 + 572310574986*x^4 - 1784176112812*x^3 + 6210498313016*x^2 - 11384486485632*x + 25628389555519, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 516 x^{14} - 2936 x^{13} + 107194 x^{12} - 641420 x^{11} + 11627920 x^{10} - 66371504 x^{9} + 728118680 x^{8} - 3717247748 x^{7} + 27035672060 x^{6} - 114044064440 x^{5} + 572310574986 x^{4} - 1784176112812 x^{3} + 6210498313016 x^{2} - 11384486485632 x + 25628389555519 \)

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:  \(101075408216203634510751650650521600000000=2^{40}\cdot 3^{4}\cdot 5^{8}\cdot 13^{8}\cdot 1249^{2}\cdot 1511^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $365.42$
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:  $2, 3, 5, 13, 1249, 1511$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{12} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{5}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{12}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{10} + \frac{1}{12} a^{9} + \frac{1}{3} a^{6} + \frac{5}{12} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{12} a + \frac{1}{3}$, $\frac{1}{1596} a^{14} - \frac{47}{1596} a^{13} - \frac{2}{133} a^{12} + \frac{113}{798} a^{11} + \frac{5}{1596} a^{10} + \frac{253}{1596} a^{9} + \frac{3}{19} a^{8} + \frac{199}{399} a^{7} + \frac{167}{532} a^{6} - \frac{29}{532} a^{5} - \frac{13}{133} a^{4} - \frac{269}{798} a^{3} + \frac{91}{228} a^{2} - \frac{463}{1596} a + \frac{52}{399}$, $\frac{1}{40612616226770842118312083237546302726572331428216274640536747877651007891794445388} a^{15} - \frac{4387900310286924076430336898267780916563717602945637116654299666634773794660469}{20306308113385421059156041618773151363286165714108137320268373938825503945897222694} a^{14} + \frac{355820203080679921721710243317060976516664596462834063830119723066082736016762797}{13537538742256947372770694412515434242190777142738758213512249292550335963931481796} a^{13} - \frac{75027094784452182879243699759682870367299113664197546371204970207489332551937147}{1933934106036706767538670630359347748884396734676965459073178470364333709133068828} a^{12} - \frac{3160622018014532709959376120144453319678265475124901350777692634351768556685978703}{13537538742256947372770694412515434242190777142738758213512249292550335963931481796} a^{11} - \frac{646716974321175698804511487173385514982810668393205429115894574692304726005823615}{6768769371128473686385347206257717121095388571369379106756124646275167981965740898} a^{10} + \frac{892642501887709250623846733711630856377529751470362739463936059334365107581184131}{13537538742256947372770694412515434242190777142738758213512249292550335963931481796} a^{9} + \frac{3145268636958628445673034999643380921840891793491408554054096124871961525776449709}{13537538742256947372770694412515434242190777142738758213512249292550335963931481796} a^{8} - \frac{8924357435080789297676517681754064634799218633205103959975474919249916624795975311}{40612616226770842118312083237546302726572331428216274640536747877651007891794445388} a^{7} + \frac{9085505919381014567571152375978389803873704892273202465536293204685259380571494079}{20306308113385421059156041618773151363286165714108137320268373938825503945897222694} a^{6} - \frac{3762166624718487435666994277176915123450885364505859713787401543016280881680886523}{13537538742256947372770694412515434242190777142738758213512249292550335963931481796} a^{5} + \frac{2592823479882778891675171238649359797465792729587471936233857490281010473864956603}{13537538742256947372770694412515434242190777142738758213512249292550335963931481796} a^{4} + \frac{2973849398837536546135151487511996840797943473261999391376409210888020085558882647}{40612616226770842118312083237546302726572331428216274640536747877651007891794445388} a^{3} - \frac{9576243711374483153222034475742334478921194157027272455952081258945003929258196467}{20306308113385421059156041618773151363286165714108137320268373938825503945897222694} a^{2} + \frac{12204959863170727528277538412173804695523609943162340054915940426688822520438761321}{40612616226770842118312083237546302726572331428216274640536747877651007891794445388} a + \frac{12880920928512594217650422192465650233283554952100978454595240518395777077704703391}{40612616226770842118312083237546302726572331428216274640536747877651007891794445388}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{29994700}$, which has order $959830400$ (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:  \( 113414.110194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 133 conjugacy class representatives for t16n1547 are not computed
Character table for t16n1547 is not computed

Intermediate fields

\(\Q(\sqrt{26}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{13})\), 8.8.421149081600.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1249Data not computed
1511Data not computed