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Properties

Label 16.8.29875440679...3521.2
Degree $16$
Signature $[8, 4]$
Discriminant $31^{12}\cdot 41^{14}$
Root discriminant $338.62$
Ramified primes $31, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25815211523, 14761359476, -16369099293, -3849727644, -361748214, 778367308, 384213103, 26256544, -9080099, -6058156, -772287, -55852, -17602, -154, 49, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 49*x^14 - 154*x^13 - 17602*x^12 - 55852*x^11 - 772287*x^10 - 6058156*x^9 - 9080099*x^8 + 26256544*x^7 + 384213103*x^6 + 778367308*x^5 - 361748214*x^4 - 3849727644*x^3 - 16369099293*x^2 + 14761359476*x + 25815211523)
 
gp: K = bnfinit(x^16 - 4*x^15 + 49*x^14 - 154*x^13 - 17602*x^12 - 55852*x^11 - 772287*x^10 - 6058156*x^9 - 9080099*x^8 + 26256544*x^7 + 384213103*x^6 + 778367308*x^5 - 361748214*x^4 - 3849727644*x^3 - 16369099293*x^2 + 14761359476*x + 25815211523, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 49 x^{14} - 154 x^{13} - 17602 x^{12} - 55852 x^{11} - 772287 x^{10} - 6058156 x^{9} - 9080099 x^{8} + 26256544 x^{7} + 384213103 x^{6} + 778367308 x^{5} - 361748214 x^{4} - 3849727644 x^{3} - 16369099293 x^{2} + 14761359476 x + 25815211523 \)

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:  \(29875440679295555535575643982688502833521=31^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $338.62$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{17158} a^{14} - \frac{404}{8579} a^{13} + \frac{843}{8579} a^{12} + \frac{1687}{8579} a^{11} - \frac{1627}{8579} a^{10} + \frac{1105}{17158} a^{9} - \frac{69}{746} a^{8} + \frac{8055}{17158} a^{7} - \frac{3918}{8579} a^{6} - \frac{8057}{17158} a^{5} - \frac{1767}{8579} a^{4} - \frac{4253}{17158} a^{3} + \frac{1786}{8579} a^{2} - \frac{6069}{17158} a - \frac{311}{746}$, $\frac{1}{126280329476680880373878061699718985544156339739805453309397353500486039901726} a^{15} - \frac{444310675339436600706590319272986972895399203681038306442565665096605860}{63140164738340440186939030849859492772078169869902726654698676750243019950863} a^{14} + \frac{15411004400304517121970633930086477383764491367410517676943589359269623571300}{63140164738340440186939030849859492772078169869902726654698676750243019950863} a^{13} - \frac{13314586569210176375724530528618527666283583426301571851794864487747173763944}{63140164738340440186939030849859492772078169869902726654698676750243019950863} a^{12} + \frac{9772995800207978763323616562265658207567166079049026660739615924984166735676}{63140164738340440186939030849859492772078169869902726654698676750243019950863} a^{11} - \frac{9523812103167721795953458292492227168857393799912182204416599868232601495285}{63140164738340440186939030849859492772078169869902726654698676750243019950863} a^{10} - \frac{29243240506130543837192374863733030254324106220451356517992049766039912310285}{126280329476680880373878061699718985544156339739805453309397353500486039901726} a^{9} - \frac{2010151090221087587833779679757910094571128771655664477304174113507198517855}{126280329476680880373878061699718985544156339739805453309397353500486039901726} a^{8} - \frac{10861091616569639599278038383176363030872463213892228594592421852118539097144}{63140164738340440186939030849859492772078169869902726654698676750243019950863} a^{7} - \frac{37000343307761047923542117806574090001386280271780613713220317752578795353227}{126280329476680880373878061699718985544156339739805453309397353500486039901726} a^{6} + \frac{53776523971897546732436292008560644143244536544330124831300058134039548828011}{126280329476680880373878061699718985544156339739805453309397353500486039901726} a^{5} - \frac{672477528638628460810409643579583364964496905113062240316421336249616994181}{126280329476680880373878061699718985544156339739805453309397353500486039901726} a^{4} + \frac{27083951232543276111934701818839320602729358715321611136095916575558678296447}{63140164738340440186939030849859492772078169869902726654698676750243019950863} a^{3} + \frac{49681907235818352174776914342149198285678760545790517485194302980083875246877}{126280329476680880373878061699718985544156339739805453309397353500486039901726} a^{2} - \frac{7404775192129572124007587948942545459537671797277299621246906899048234654945}{126280329476680880373878061699718985544156339739805453309397353500486039901726} a + \frac{659042046002223256677550277772272932819255496977837867122607228911703011967}{5490449107681777407559915726074738501919840858252411013452058847847219126162}$

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:  \( 122379871312000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$31$31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$