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Properties

Label 16.8.42905375783...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 29^{6}\cdot 41^{4}\cdot 79^{4}$
Root discriminant $168.67$
Ramified primes $2, 5, 29, 41, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2376465001, 0, -31979344, 0, -52272376, 0, -10578, 0, 367554, 0, 1084, 0, -1039, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 1039*x^12 + 1084*x^10 + 367554*x^8 - 10578*x^6 - 52272376*x^4 - 31979344*x^2 + 2376465001)
 
gp: K = bnfinit(x^16 - 2*x^14 - 1039*x^12 + 1084*x^10 + 367554*x^8 - 10578*x^6 - 52272376*x^4 - 31979344*x^2 + 2376465001, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{14} - 1039 x^{12} + 1084 x^{10} + 367554 x^{8} - 10578 x^{6} - 52272376 x^{4} - 31979344 x^{2} + 2376465001 \)

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:  \(429053757833296459233761689600000000=2^{24}\cdot 5^{8}\cdot 29^{6}\cdot 41^{4}\cdot 79^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.67$
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:  $2, 5, 29, 41, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{10} a^{6} + \frac{1}{5} a^{2} - \frac{1}{10}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{7} + \frac{1}{5} a^{3} - \frac{1}{10} a$, $\frac{1}{410} a^{10} - \frac{1}{205} a^{8} + \frac{191}{410} a^{6} + \frac{91}{205} a^{4} - \frac{93}{410} a^{2} + \frac{1}{10}$, $\frac{1}{410} a^{11} - \frac{1}{205} a^{9} + \frac{191}{410} a^{7} + \frac{91}{205} a^{5} - \frac{93}{410} a^{3} + \frac{1}{10} a$, $\frac{1}{33620} a^{12} - \frac{1}{820} a^{11} + \frac{39}{33620} a^{10} - \frac{39}{820} a^{9} - \frac{1121}{33620} a^{8} - \frac{15}{82} a^{7} - \frac{11257}{33620} a^{6} - \frac{91}{410} a^{5} - \frac{6571}{33620} a^{4} + \frac{11}{820} a^{3} - \frac{98}{205} a^{2} - \frac{1}{2} a + \frac{1}{20}$, $\frac{1}{33620} a^{13} - \frac{1}{16810} a^{11} + \frac{321}{16810} a^{9} - \frac{1}{20} a^{8} + \frac{12851}{33620} a^{7} - \frac{9}{20} a^{6} - \frac{14033}{33620} a^{5} - \frac{217}{820} a^{3} - \frac{1}{10} a^{2} + \frac{9}{20} a - \frac{9}{20}$, $\frac{1}{72706223778462820} a^{14} - \frac{44681244329}{72706223778462820} a^{12} - \frac{1}{820} a^{11} + \frac{16893943177113}{14541244755692564} a^{10} + \frac{1}{410} a^{9} + \frac{905391893186617}{18176555944615705} a^{8} + \frac{219}{820} a^{7} - \frac{237417764747556}{519330169846163} a^{6} - \frac{91}{410} a^{5} + \frac{28647582693473}{443330632795505} a^{4} + \frac{93}{820} a^{3} + \frac{3947175889549}{43251769053220} a^{2} - \frac{1}{20} a - \frac{662320637}{36376592980}$, $\frac{1}{72706223778462820} a^{15} - \frac{44681244329}{72706223778462820} a^{13} - \frac{1049102668384}{18176555944615705} a^{11} - \frac{1}{820} a^{10} + \frac{163588636941529}{72706223778462820} a^{9} + \frac{1}{410} a^{8} + \frac{373825967570569}{1038660339692326} a^{7} + \frac{219}{820} a^{6} - \frac{27900076761041}{177332253118202} a^{5} - \frac{91}{410} a^{4} + \frac{226369127379}{2162588452661} a^{3} + \frac{93}{820} a^{2} + \frac{17525975853}{36376592980} a - \frac{1}{20}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 507241225343 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n839 are not computed
Character table for t16n839 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.31600.1, 4.4.916400.1, 4.4.725.1, 8.8.839788960000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$79$79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.8.4.1$x^{8} + 37446 x^{4} - 493039 x^{2} + 350550729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$