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Properties

Label 16.0.37801841960...3808.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 17^{15}\cdot 67^{4}$
Root discriminant $81.49$
Ramified primes $2, 17, 67$
Class number $71296$ (GRH)
Class group $[2, 2, 2, 2, 2, 2228]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![342569057, 0, 506184129, 0, 224207594, 0, 44966581, 0, 4752843, 0, 279905, 0, 9146, 0, 153, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 153*x^14 + 9146*x^12 + 279905*x^10 + 4752843*x^8 + 44966581*x^6 + 224207594*x^4 + 506184129*x^2 + 342569057)
 
gp: K = bnfinit(x^16 + 153*x^14 + 9146*x^12 + 279905*x^10 + 4752843*x^8 + 44966581*x^6 + 224207594*x^4 + 506184129*x^2 + 342569057, 1)
 

Normalized defining polynomial

\( x^{16} + 153 x^{14} + 9146 x^{12} + 279905 x^{10} + 4752843 x^{8} + 44966581 x^{6} + 224207594 x^{4} + 506184129 x^{2} + 342569057 \)

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:  \(3780184196000221150082102263808=2^{16}\cdot 17^{15}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.49$
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:  $2, 17, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{67} a^{8} - \frac{4}{67} a^{6} - \frac{8}{67} a^{4} + \frac{29}{67} a^{2}$, $\frac{1}{67} a^{9} - \frac{4}{67} a^{7} - \frac{8}{67} a^{5} + \frac{29}{67} a^{3}$, $\frac{1}{67} a^{10} - \frac{24}{67} a^{6} - \frac{3}{67} a^{4} - \frac{18}{67} a^{2}$, $\frac{1}{67} a^{11} - \frac{24}{67} a^{7} - \frac{3}{67} a^{5} - \frac{18}{67} a^{3}$, $\frac{1}{4489} a^{12} + \frac{19}{4489} a^{10} - \frac{33}{4489} a^{8} + \frac{1118}{4489} a^{6} + \frac{1002}{4489} a^{4} + \frac{20}{67} a^{2}$, $\frac{1}{4489} a^{13} + \frac{19}{4489} a^{11} - \frac{33}{4489} a^{9} + \frac{1118}{4489} a^{7} + \frac{1002}{4489} a^{5} + \frac{20}{67} a^{3}$, $\frac{1}{130416608931012743341} a^{14} + \frac{575419048554243}{130416608931012743341} a^{12} - \frac{47876653383732872}{130416608931012743341} a^{10} + \frac{582653506663821181}{130416608931012743341} a^{8} + \frac{61926698279447664576}{130416608931012743341} a^{6} - \frac{823037454631139947}{1946516551209145423} a^{4} + \frac{11145186654142587}{29052485838942469} a^{2} - \frac{37987555083107}{433619191626007}$, $\frac{1}{130416608931012743341} a^{15} + \frac{575419048554243}{130416608931012743341} a^{13} - \frac{47876653383732872}{130416608931012743341} a^{11} + \frac{582653506663821181}{130416608931012743341} a^{9} + \frac{61926698279447664576}{130416608931012743341} a^{7} - \frac{823037454631139947}{1946516551209145423} a^{5} + \frac{11145186654142587}{29052485838942469} a^{3} - \frac{37987555083107}{433619191626007} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2228}$, which has order $71296$ (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:  \( -1 \) (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
17Data not computed
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$