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Properties

Label 16.0.22850732699...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 7^{4}\cdot 29^{6}$
Root discriminant $38.45$
Ramified primes $2, 5, 7, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22961, -53248, 38363, 7550, -31805, 20588, 1639, -10030, 4909, 530, -1369, 446, 30, -60, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 22*x^14 - 60*x^13 + 30*x^12 + 446*x^11 - 1369*x^10 + 530*x^9 + 4909*x^8 - 10030*x^7 + 1639*x^6 + 20588*x^5 - 31805*x^4 + 7550*x^3 + 38363*x^2 - 53248*x + 22961)
 
gp: K = bnfinit(x^16 - 6*x^15 + 22*x^14 - 60*x^13 + 30*x^12 + 446*x^11 - 1369*x^10 + 530*x^9 + 4909*x^8 - 10030*x^7 + 1639*x^6 + 20588*x^5 - 31805*x^4 + 7550*x^3 + 38363*x^2 - 53248*x + 22961, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 22 x^{14} - 60 x^{13} + 30 x^{12} + 446 x^{11} - 1369 x^{10} + 530 x^{9} + 4909 x^{8} - 10030 x^{7} + 1639 x^{6} + 20588 x^{5} - 31805 x^{4} + 7550 x^{3} + 38363 x^{2} - 53248 x + 22961 \)

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:  \(22850732699536000000000000=2^{16}\cdot 5^{12}\cdot 7^{4}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.45$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3106436190834034162194761} a^{15} + \frac{1013213033942036037605459}{3106436190834034162194761} a^{14} - \frac{1188946073775142215477987}{3106436190834034162194761} a^{13} + \frac{1431225405543938684255940}{3106436190834034162194761} a^{12} - \frac{31218798078796532412812}{282403290075821287472251} a^{11} + \frac{631862812082720838889869}{3106436190834034162194761} a^{10} + \frac{643812042690332988090753}{3106436190834034162194761} a^{9} - \frac{465045450957048317885807}{3106436190834034162194761} a^{8} + \frac{70523538624753211063137}{3106436190834034162194761} a^{7} + \frac{1465578038820902180852992}{3106436190834034162194761} a^{6} + \frac{279014458737802989305463}{3106436190834034162194761} a^{5} + \frac{1088846863167777876387462}{3106436190834034162194761} a^{4} + \frac{506002622113720228845714}{3106436190834034162194761} a^{3} + \frac{161340661405166973866930}{3106436190834034162194761} a^{2} + \frac{596366211208389475620075}{3106436190834034162194761} a + \frac{236603805494367801525041}{3106436190834034162194761}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -\frac{424501107524281844986400}{3106436190834034162194761} a^{15} + \frac{1871550127183543276002769}{3106436190834034162194761} a^{14} - \frac{6358768585064147249001154}{3106436190834034162194761} a^{13} + \frac{15345810383500545972389676}{3106436190834034162194761} a^{12} + \frac{1064334551853291287927352}{282403290075821287472251} a^{11} - \frac{170748079859090975025349289}{3106436190834034162194761} a^{10} + \frac{309341379682598575688028602}{3106436190834034162194761} a^{9} + \frac{267851670037191046472749286}{3106436190834034162194761} a^{8} - \frac{1658082956445268855476814676}{3106436190834034162194761} a^{7} + \frac{1617347462048324628888406053}{3106436190834034162194761} a^{6} + \frac{1881839999397729006681381526}{3106436190834034162194761} a^{5} - \frac{5744370200241753394016353199}{3106436190834034162194761} a^{4} + \frac{4351434503252633359404339174}{3106436190834034162194761} a^{3} + \frac{3728687844182948623528377049}{3106436190834034162194761} a^{2} - \frac{10350165622257288651718868356}{3106436190834034162194761} a + \frac{6114007365094243440549572311}{3106436190834034162194761} \) (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:  \( 2131120.31788 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

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 $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.11600.1, \(\Q(\zeta_{5})\), 4.4.58000.1, 8.0.3364000000.5

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$