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Properties

Label 16.0.86380562306...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{12}$
Root discriminant $41.79$
Ramified primes $5, 29$
Class number $72$ (GRH)
Class group $[6, 12]$ (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![279841, 109503, -5819, -33488, -29020, 7953, 9868, 5196, 1233, -1196, 308, 49, -36, 16, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + 16*x^13 - 36*x^12 + 49*x^11 + 308*x^10 - 1196*x^9 + 1233*x^8 + 5196*x^7 + 9868*x^6 + 7953*x^5 - 29020*x^4 - 33488*x^3 - 5819*x^2 + 109503*x + 279841)
 
gp: K = bnfinit(x^16 - x^15 - 3*x^14 + 16*x^13 - 36*x^12 + 49*x^11 + 308*x^10 - 1196*x^9 + 1233*x^8 + 5196*x^7 + 9868*x^6 + 7953*x^5 - 29020*x^4 - 33488*x^3 - 5819*x^2 + 109503*x + 279841, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 3 x^{14} + 16 x^{13} - 36 x^{12} + 49 x^{11} + 308 x^{10} - 1196 x^{9} + 1233 x^{8} + 5196 x^{7} + 9868 x^{6} + 7953 x^{5} - 29020 x^{4} - 33488 x^{3} - 5819 x^{2} + 109503 x + 279841 \)

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:  \(86380562306022715087890625=5^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.79$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{30} a^{12} + \frac{1}{5} a^{11} - \frac{1}{3} a^{10} - \frac{1}{6} a^{9} - \frac{7}{15} a^{8} + \frac{3}{10} a^{6} + \frac{1}{3} a^{5} + \frac{1}{5} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{7}{15} a + \frac{1}{30}$, $\frac{1}{794113362390} a^{13} + \frac{3350826131}{264704454130} a^{12} + \frac{110488244731}{397056681195} a^{11} - \frac{52237280605}{158822672478} a^{10} + \frac{261866113081}{794113362390} a^{9} - \frac{36159685968}{132352227065} a^{8} + \frac{2200166893}{264704454130} a^{7} - \frac{16717935859}{34526667930} a^{6} + \frac{53950083801}{132352227065} a^{5} - \frac{89116133203}{794113362390} a^{4} + \frac{23165527127}{158822672478} a^{3} - \frac{130318358032}{397056681195} a^{2} - \frac{136395242177}{794113362390} a + \frac{4355001143}{11508889310}$, $\frac{1}{18264607334970} a^{14} - \frac{1}{18264607334970} a^{13} - \frac{241859715173}{18264607334970} a^{12} - \frac{8464342306891}{18264607334970} a^{11} - \frac{8832826167419}{18264607334970} a^{10} + \frac{6785617808293}{18264607334970} a^{9} - \frac{5541405781717}{18264607334970} a^{8} - \frac{210920299981}{794113362390} a^{7} - \frac{5502108115963}{18264607334970} a^{6} + \frac{1610987437723}{18264607334970} a^{5} - \frac{2846770435351}{18264607334970} a^{4} + \frac{3666864325831}{18264607334970} a^{3} - \frac{8932452719851}{18264607334970} a^{2} + \frac{69847832939}{794113362390} a + \frac{8976480761}{34526667930}$, $\frac{1}{2100429843521550} a^{15} + \frac{11}{1050214921760775} a^{14} + \frac{503}{2100429843521550} a^{13} + \frac{2090652852919}{140028656234770} a^{12} - \frac{265028329305068}{1050214921760775} a^{11} + \frac{535271113303591}{2100429843521550} a^{10} + \frac{223789121370517}{700143281173850} a^{9} - \frac{2194225192163}{45661518337425} a^{8} + \frac{968343783609499}{2100429843521550} a^{7} - \frac{321403185993277}{2100429843521550} a^{6} + \frac{25876649816017}{350071640586925} a^{5} + \frac{883302742147649}{2100429843521550} a^{4} - \frac{534010716395173}{2100429843521550} a^{3} - \frac{15150805801127}{45661518337425} a^{2} + \frac{4404272819}{264704454130} a - \frac{2277173166}{28772223275}$

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

Class group and class number

$C_{6}\times C_{12}$, 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:  \( \frac{127016319}{70014328117385} a^{15} + \frac{12161132}{14002865623477} a^{14} - \frac{381048957}{70014328117385} a^{13} + \frac{2032261104}{70014328117385} a^{12} - \frac{4572587484}{70014328117385} a^{11} + \frac{6223799631}{70014328117385} a^{10} + \frac{47462879623}{70014328117385} a^{9} - \frac{6604848588}{3044101222495} a^{8} + \frac{156611121327}{70014328117385} a^{7} + \frac{659976793524}{70014328117385} a^{6} + \frac{1253397035892}{70014328117385} a^{5} + \frac{8484149735551}{70014328117385} a^{4} - \frac{737202715476}{14002865623477} a^{3} - \frac{184935760464}{3044101222495} a^{2} - \frac{1397179509}{132352227065} a + \frac{1143146871}{5754444655} \) (order $10$)
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:  \( 249792.59056 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.609725.1 x2, 4.0.121945.1 x2, \(\Q(\zeta_{5})\), 4.0.105125.2, 8.0.371764575625.5, 8.0.11051265625.1, 8.8.9294114390625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$