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Properties

Label 16.0.64803441755...0625.8
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 61^{12}$
Root discriminant $72.98$
Ramified primes $5, 61$
Class number $72$ (GRH)
Class group $[2, 6, 6]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![791267920, 691165560, 365353240, 107817400, 39593521, 35047982, 17453050, 2060735, 341046, 216170, 46571, -4780, 2136, 135, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 10*x^14 + 135*x^13 + 2136*x^12 - 4780*x^11 + 46571*x^10 + 216170*x^9 + 341046*x^8 + 2060735*x^7 + 17453050*x^6 + 35047982*x^5 + 39593521*x^4 + 107817400*x^3 + 365353240*x^2 + 691165560*x + 791267920)
 
gp: K = bnfinit(x^16 - 2*x^15 + 10*x^14 + 135*x^13 + 2136*x^12 - 4780*x^11 + 46571*x^10 + 216170*x^9 + 341046*x^8 + 2060735*x^7 + 17453050*x^6 + 35047982*x^5 + 39593521*x^4 + 107817400*x^3 + 365353240*x^2 + 691165560*x + 791267920, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 10 x^{14} + 135 x^{13} + 2136 x^{12} - 4780 x^{11} + 46571 x^{10} + 216170 x^{9} + 341046 x^{8} + 2060735 x^{7} + 17453050 x^{6} + 35047982 x^{5} + 39593521 x^{4} + 107817400 x^{3} + 365353240 x^{2} + 691165560 x + 791267920 \)

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:  \(648034417553121620683837890625=5^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.98$
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:  $5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} - \frac{9}{19} a^{11} + \frac{5}{19} a^{10} - \frac{1}{19} a^{9} + \frac{8}{19} a^{8} + \frac{6}{19} a^{7} - \frac{5}{19} a^{6} - \frac{5}{19} a^{5} - \frac{6}{19} a^{4} + \frac{4}{19} a^{3} + \frac{5}{19} a^{2} - \frac{6}{19} a$, $\frac{1}{114} a^{13} - \frac{1}{57} a^{12} - \frac{10}{57} a^{11} - \frac{23}{114} a^{10} - \frac{28}{57} a^{9} + \frac{4}{19} a^{8} - \frac{13}{38} a^{7} + \frac{6}{19} a^{6} + \frac{9}{19} a^{5} - \frac{1}{6} a^{4} + \frac{26}{57} a^{3} - \frac{14}{57} a^{2} + \frac{5}{38} a + \frac{1}{3}$, $\frac{1}{12722480028} a^{14} + \frac{3845867}{1060206669} a^{13} - \frac{24056135}{2120413338} a^{12} + \frac{454579517}{4240826676} a^{11} - \frac{154379497}{706804446} a^{10} + \frac{43966787}{3180620007} a^{9} + \frac{1583246593}{4240826676} a^{8} + \frac{158303791}{353402223} a^{7} - \frac{419237135}{2120413338} a^{6} - \frac{1417478941}{12722480028} a^{5} - \frac{410933497}{3180620007} a^{4} + \frac{3030269759}{6361240014} a^{3} - \frac{4549159631}{12722480028} a^{2} - \frac{2341979699}{6361240014} a - \frac{74562500}{167401053}$, $\frac{1}{4302505944404911086567481713417201530683077531925024780392} a^{15} - \frac{39323322453319275259822195913753128211471358935}{2151252972202455543283740856708600765341538765962512390196} a^{14} - \frac{813155912559336654010259454884943760245087183114479275}{239028108022495060364860095189844529482393196218056932244} a^{13} - \frac{11888081657147875810516401984913303925389068820873400393}{478056216044990120729720190379689058964786392436113864488} a^{12} + \frac{17107265760745140094881136207429723387350220613447014819}{358542162033742590547290142784766794223589794327085398366} a^{11} + \frac{483829367553677715035738861517532033105943046166057982303}{1075626486101227771641870428354300382670769382981256195098} a^{10} + \frac{143314663090871225079142410151764054948635534027735046211}{4302505944404911086567481713417201530683077531925024780392} a^{9} + \frac{273093661834822544980614821252094426631543893795861115525}{717084324067485181094580285569533588447179588654170796732} a^{8} + \frac{292138699474735879117279478575304810640360560974289448349}{717084324067485181094580285569533588447179588654170796732} a^{7} + \frac{1584523189766961051040979803976143926184080878123188719959}{4302505944404911086567481713417201530683077531925024780392} a^{6} + \frac{27400265686936908473369053222515098690139380697469837393}{717084324067485181094580285569533588447179588654170796732} a^{5} + \frac{500803894627110799782824923113551762979759443594823464683}{2151252972202455543283740856708600765341538765962512390196} a^{4} + \frac{99000381859713365399874504048697415566468650903699637}{506594365289639831221886461016978868560352941472391944} a^{3} + \frac{16690901406107133709212569119688329832086277600132107327}{56611920321117251139045812018647388561619441209539799742} a^{2} - \frac{78063162936907078503358923079727573246364115516662347}{537813243050613885820935214177150191335384691490628097549} a - \frac{6385496605025256374151678695791215042010240183614685268}{28305960160558625569522906009323694280809720604769899871}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}$, which has order $72$ (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:  \( 7602239.42975 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{305}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 4.0.1525.1 x2, 4.0.18605.1 x2, 4.4.28372625.1, 4.4.28372625.2, 8.0.8653650625.2, 8.8.805005849390625.1, 8.0.805005849390625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$