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Properties

Label 16.16.1398629213...2672.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{59}\cdot 113^{6}\cdot 1039^{4}$
Root discriminant $430.63$
Ramified primes $2, 113, 1039$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T942

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29761106423144258, 0, -7218285677220744, 0, 392949677090456, 0, -8643065177076, 0, 94332572973, 0, -542445276, 0, 1596442, 0, -2120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2120*x^14 + 1596442*x^12 - 542445276*x^10 + 94332572973*x^8 - 8643065177076*x^6 + 392949677090456*x^4 - 7218285677220744*x^2 + 29761106423144258)
 
gp: K = bnfinit(x^16 - 2120*x^14 + 1596442*x^12 - 542445276*x^10 + 94332572973*x^8 - 8643065177076*x^6 + 392949677090456*x^4 - 7218285677220744*x^2 + 29761106423144258, 1)
 

Normalized defining polynomial

\( x^{16} - 2120 x^{14} + 1596442 x^{12} - 542445276 x^{10} + 94332572973 x^{8} - 8643065177076 x^{6} + 392949677090456 x^{4} - 7218285677220744 x^{2} + 29761106423144258 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1398629213800614161925350535491003636252672=2^{59}\cdot 113^{6}\cdot 1039^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $430.63$
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:  $2, 113, 1039$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1039} a^{10} - \frac{42}{1039} a^{8} - \frac{501}{1039} a^{6} - \frac{210}{1039} a^{2}$, $\frac{1}{1039} a^{11} - \frac{42}{1039} a^{9} - \frac{501}{1039} a^{7} - \frac{210}{1039} a^{3}$, $\frac{1}{2073759841} a^{12} - \frac{2120}{2073759841} a^{10} - \frac{28608650}{121985873} a^{8} - \frac{874305}{1995919} a^{6} + \frac{1013380128}{2073759841} a^{4} - \frac{3118}{17663} a^{2} - \frac{8}{17}$, $\frac{1}{2073759841} a^{13} - \frac{2120}{2073759841} a^{11} - \frac{28608650}{121985873} a^{9} - \frac{874305}{1995919} a^{7} + \frac{1013380128}{2073759841} a^{5} - \frac{3118}{17663} a^{3} - \frac{8}{17} a$, $\frac{1}{20208581579281235199653695558477191761474189281} a^{14} + \frac{211231502913332175998751945781139806}{1188740092898896188214923268145717162439658193} a^{12} + \frac{9355585584745699183867886510452374289949115}{20208581579281235199653695558477191761474189281} a^{10} - \frac{9114507604471598262022134749962210116349325}{19450030393918416939031468294973235574084879} a^{8} - \frac{8478973934041823547933220352285631606270748805}{20208581579281235199653695558477191761474189281} a^{6} + \frac{8655014605693413341745356312203362023360529}{19450030393918416939031468294973235574084879} a^{4} - \frac{4069927474214097035445257565339924549}{165663294300326359919182572546553745297} a^{2} + \frac{67897665582220423783569704826663912}{159444941578754918112783996676182623}$, $\frac{1}{20208581579281235199653695558477191761474189281} a^{15} + \frac{211231502913332175998751945781139806}{1188740092898896188214923268145717162439658193} a^{13} + \frac{9355585584745699183867886510452374289949115}{20208581579281235199653695558477191761474189281} a^{11} - \frac{9114507604471598262022134749962210116349325}{19450030393918416939031468294973235574084879} a^{9} - \frac{8478973934041823547933220352285631606270748805}{20208581579281235199653695558477191761474189281} a^{7} + \frac{8655014605693413341745356312203362023360529}{19450030393918416939031468294973235574084879} a^{5} - \frac{4069927474214097035445257565339924549}{165663294300326359919182572546553745297} a^{3} + \frac{67897665582220423783569704826663912}{159444941578754918112783996676182623} a$

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:  $15$
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:  \( 12384955387700000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T942:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n942
Character table for t16n942 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.26778533888.1

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
Arithmetically equvalently 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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $16$

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
$2$2.8.29.78$x^{8} + 4 x^{6} + 12 x^{4} + 16 x^{3} + 16 x^{2} + 16 x + 14$$8$$1$$29$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
2.8.30.35$x^{8} + 20 x^{6} + 14 x^{4} + 4 x^{2} + 23$$8$$1$$30$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
$113$113.4.3.2$x^{4} - 1017$$4$$1$$3$$C_4$$[\ ]_{4}$
113.4.3.2$x^{4} - 1017$$4$$1$$3$$C_4$$[\ ]_{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
1039Data not computed