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Properties

Label 16.8.68470160614...3824.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{64}\cdot 3^{4}\cdot 7^{6}\cdot 79^{4}$
Root discriminant $130.23$
Ramified primes $2, 3, 7, 79$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T467)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12419943838, -36835875024, 39182469104, -16401426768, -1449183488, 1510348640, -162023704, -12091152, 11342580, -1009872, -46000, 16064, -6516, -16, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 - 16*x^13 - 6516*x^12 + 16064*x^11 - 46000*x^10 - 1009872*x^9 + 11342580*x^8 - 12091152*x^7 - 162023704*x^6 + 1510348640*x^5 - 1449183488*x^4 - 16401426768*x^3 + 39182469104*x^2 - 36835875024*x + 12419943838)
 
gp: K = bnfinit(x^16 + 40*x^14 - 16*x^13 - 6516*x^12 + 16064*x^11 - 46000*x^10 - 1009872*x^9 + 11342580*x^8 - 12091152*x^7 - 162023704*x^6 + 1510348640*x^5 - 1449183488*x^4 - 16401426768*x^3 + 39182469104*x^2 - 36835875024*x + 12419943838, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} - 16 x^{13} - 6516 x^{12} + 16064 x^{11} - 46000 x^{10} - 1009872 x^{9} + 11342580 x^{8} - 12091152 x^{7} - 162023704 x^{6} + 1510348640 x^{5} - 1449183488 x^{4} - 16401426768 x^{3} + 39182469104 x^{2} - 36835875024 x + 12419943838 \)

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:  \(6847016061481864788618324763213824=2^{64}\cdot 3^{4}\cdot 7^{6}\cdot 79^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.23$
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:  $2, 3, 7, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} + \frac{7}{17} a^{12} + \frac{6}{17} a^{11} - \frac{2}{17} a^{10} + \frac{5}{17} a^{9} - \frac{1}{17} a^{8} + \frac{4}{17} a^{7} - \frac{1}{17} a^{6} + \frac{2}{17} a^{5} - \frac{7}{17} a^{4} - \frac{5}{17} a^{3} - \frac{3}{17} a^{2} - \frac{1}{17} a - \frac{5}{17}$, $\frac{1}{85} a^{14} + \frac{2}{85} a^{13} + \frac{1}{17} a^{12} + \frac{36}{85} a^{11} - \frac{19}{85} a^{10} + \frac{42}{85} a^{9} - \frac{5}{17} a^{8} + \frac{6}{17} a^{7} - \frac{2}{17} a^{6} - \frac{1}{5} a^{5} - \frac{38}{85} a^{4} + \frac{22}{85} a^{3} - \frac{37}{85} a^{2} + \frac{42}{85}$, $\frac{1}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{15} + \frac{603048852927263226890258179274641688974649926964894012514231764073372407}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{14} - \frac{5085808006459941703474076808428810507258802387020678797877820494337219912}{303538982877890545643578501734587164060527746526914793329551312732297699783} a^{13} + \frac{554563347802295219844469272106081559744864132473369919076444674457126305841}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{12} + \frac{719716415310852086301301807630372971132817352484132904729755937444483186686}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{11} + \frac{326872514611243012341488189094348747535187652458873164601218524192068790742}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{10} + \frac{97240984667540914445553252008907120061375016422586828984001893210400979546}{303538982877890545643578501734587164060527746526914793329551312732297699783} a^{9} - \frac{99013560641083300311742500097216353804303203619585713467335584103157473360}{303538982877890545643578501734587164060527746526914793329551312732297699783} a^{8} + \frac{68851017627334884425106043843295829832797030407466254274797231700820396136}{303538982877890545643578501734587164060527746526914793329551312732297699783} a^{7} + \frac{455172705531058099302911454607138193498773671125750149958694329673184135438}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{6} - \frac{409059374971489302563262238489396520868325547660061290140039782064078217503}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{5} + \frac{60327297237931233012353154409954301841638797204534257388250778420014778982}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{4} + \frac{59232080542945970108025560383486043576513672606203019890935609251490647713}{1517694914389452728217892508672935820302638732634573966647756563661488498915} a^{3} - \frac{398419916820043840110681519586984721181057373312299293107442161798209555}{7403389826290013308379963456941150342939701134802799837306129578836529263} a^{2} - \frac{22411564260522384177136865661745363864423145159218902609045602368044104339}{89276171434673689895170147568996224723684631331445527449868033156558146995} a - \frac{130886223509529031131035227255022560817249272287297160565564185585704801395}{303538982877890545643578501734587164060527746526914793329551312732297699783}$

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:  $11$
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:  \( 94167996561.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T467):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7168.1, \(\Q(\zeta_{16})^+\), 4.4.14336.1, 8.8.3288334336.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
79.4.2.2$x^{4} - 79 x^{2} + 18723$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$