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Properties

Label 16.0.58479365965...2064.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 17^{8}\cdot 47^{4}$
Root discriminant $47.09$
Ramified primes $2, 17, 47$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T394)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![142706, -289968, 355184, -367360, 307471, -202954, 115769, -64282, 37061, -20564, 9754, -3644, 945, -122, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 - 122*x^13 + 945*x^12 - 3644*x^11 + 9754*x^10 - 20564*x^9 + 37061*x^8 - 64282*x^7 + 115769*x^6 - 202954*x^5 + 307471*x^4 - 367360*x^3 + 355184*x^2 - 289968*x + 142706)
 
gp: K = bnfinit(x^16 - 2*x^15 + x^14 - 122*x^13 + 945*x^12 - 3644*x^11 + 9754*x^10 - 20564*x^9 + 37061*x^8 - 64282*x^7 + 115769*x^6 - 202954*x^5 + 307471*x^4 - 367360*x^3 + 355184*x^2 - 289968*x + 142706, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + x^{14} - 122 x^{13} + 945 x^{12} - 3644 x^{11} + 9754 x^{10} - 20564 x^{9} + 37061 x^{8} - 64282 x^{7} + 115769 x^{6} - 202954 x^{5} + 307471 x^{4} - 367360 x^{3} + 355184 x^{2} - 289968 x + 142706 \)

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:  \(584793659653495312365912064=2^{34}\cdot 17^{8}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.09$
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:  $2, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} + \frac{8}{17} a^{9} - \frac{3}{17} a^{8} + \frac{1}{17} a^{7} + \frac{7}{17} a^{6} + \frac{6}{17} a^{5} + \frac{8}{17} a^{4} + \frac{2}{17} a^{3} + \frac{8}{17} a^{2} - \frac{2}{17} a - \frac{4}{17}$, $\frac{1}{17} a^{11} + \frac{1}{17} a^{9} + \frac{8}{17} a^{8} - \frac{1}{17} a^{7} + \frac{1}{17} a^{6} - \frac{6}{17} a^{5} + \frac{6}{17} a^{4} - \frac{8}{17} a^{3} + \frac{2}{17} a^{2} - \frac{5}{17} a - \frac{2}{17}$, $\frac{1}{136} a^{12} - \frac{1}{68} a^{11} + \frac{3}{136} a^{10} - \frac{23}{68} a^{9} - \frac{5}{17} a^{8} + \frac{11}{68} a^{7} - \frac{11}{136} a^{6} - \frac{19}{68} a^{5} + \frac{47}{136} a^{4} + \frac{7}{17} a^{3} - \frac{11}{34} a^{2} - \frac{8}{17} a + \frac{15}{68}$, $\frac{1}{136} a^{13} - \frac{1}{136} a^{11} + \frac{13}{34} a^{9} - \frac{21}{68} a^{8} - \frac{63}{136} a^{7} - \frac{13}{34} a^{6} - \frac{61}{136} a^{5} + \frac{31}{68} a^{4} + \frac{3}{34} a^{3} + \frac{4}{17} a^{2} - \frac{21}{68} a + \frac{9}{34}$, $\frac{1}{641512} a^{14} + \frac{99}{160378} a^{13} - \frac{411}{160378} a^{12} + \frac{8741}{320756} a^{11} - \frac{13381}{641512} a^{10} + \frac{4503}{80189} a^{9} + \frac{10665}{641512} a^{8} - \frac{124537}{320756} a^{7} - \frac{31055}{160378} a^{6} - \frac{22033}{160378} a^{5} + \frac{320735}{641512} a^{4} + \frac{35262}{80189} a^{3} - \frac{32519}{320756} a^{2} - \frac{35233}{160378} a - \frac{8097}{320756}$, $\frac{1}{85138526526032085645373744} a^{15} + \frac{14925096753275805567}{85138526526032085645373744} a^{14} + \frac{1571322719039387813403}{803193646472000807975224} a^{13} - \frac{114965630100041358835751}{42569263263016042822686872} a^{12} + \frac{58075712990149390888617}{1979965733163536875473808} a^{11} + \frac{2073674656852757742917853}{85138526526032085645373744} a^{10} + \frac{231591718814945982889505}{956612657595866130846896} a^{9} + \frac{27117943541925499281797133}{85138526526032085645373744} a^{8} - \frac{10211468895540716636313743}{21284631631508021411343436} a^{7} - \frac{667897418897993959030693}{3040661661644003058763348} a^{6} - \frac{21448893085636706436291115}{85138526526032085645373744} a^{5} - \frac{2222132267045292641340907}{5008148619178357979139632} a^{4} - \frac{10056498939700326694660623}{42569263263016042822686872} a^{3} + \frac{361805638713559071767093}{42569263263016042822686872} a^{2} - \frac{19864399242505164898254397}{42569263263016042822686872} a + \frac{15936286722311113875855001}{42569263263016042822686872}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( -1 \) (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:  \( 9591102.68109 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T394):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 17 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.1088.2 x2, 4.0.2312.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.342102016.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$