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Properties

Label 16.8.12001634735...0192.6
Degree $16$
Signature $[8, 4]$
Discriminant $2^{59}\cdot 113^{6}$
Root discriminant $75.85$
Ramified primes $2, 113$
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![408608, 0, -5720512, 0, -3726288, 0, -330864, 0, 96382, 0, 13360, 0, -172, 0, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 44*x^14 - 172*x^12 + 13360*x^10 + 96382*x^8 - 330864*x^6 - 3726288*x^4 - 5720512*x^2 + 408608)
 
gp: K = bnfinit(x^16 - 44*x^14 - 172*x^12 + 13360*x^10 + 96382*x^8 - 330864*x^6 - 3726288*x^4 - 5720512*x^2 + 408608, 1)
 

Normalized defining polynomial

\( x^{16} - 44 x^{14} - 172 x^{12} + 13360 x^{10} + 96382 x^{8} - 330864 x^{6} - 3726288 x^{4} - 5720512 x^{2} + 408608 \)

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:  \(1200163473568415164592335880192=2^{59}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.85$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{15368} a^{12} + \frac{164}{1921} a^{10} - \frac{417}{1921} a^{8} + \frac{201}{1921} a^{6} + \frac{279}{7684} a^{4} - \frac{1}{17}$, $\frac{1}{30736} a^{13} + \frac{82}{1921} a^{11} - \frac{417}{3842} a^{9} - \frac{860}{1921} a^{7} + \frac{279}{15368} a^{5} - \frac{1}{2} a^{3} - \frac{1}{34} a$, $\frac{1}{49069758370707370256} a^{14} - \frac{531419770393955}{24534879185353685128} a^{12} - \frac{883080507017030333}{12267439592676842564} a^{10} - \frac{1329605104592043257}{6133719796338421282} a^{8} + \frac{278933456334193257}{791447715656570488} a^{6} + \frac{836121623051651785}{12267439592676842564} a^{4} - \frac{11025068847821189}{27140353081143457} a^{2} - \frac{7325560218933298}{27140353081143457}$, $\frac{1}{98139516741414740512} a^{15} - \frac{531419770393955}{49069758370707370256} a^{13} + \frac{545944847788045077}{6133719796338421282} a^{11} + \frac{434313698394291846}{3066859898169210641} a^{9} - \frac{512514259322377231}{1582895431313140976} a^{7} + \frac{836121623051651785}{24534879185353685128} a^{5} - \frac{49190490776785835}{108561412324573828} a^{3} + \frac{19814792862210159}{54280706162286914} a$

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:  $11$
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:  \( 1903740080.04 \) (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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $16$

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$2.8.30.48$x^{8} + 8 x^{7} + 16 x^{6} + 16 x^{4} + 30$$8$$1$$30$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
2.8.29.71$x^{8} + 20 x^{6} + 28 x^{4} + 14$$8$$1$$29$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
113Data not computed