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Properties

Label 16.0.22935441770...2641.4
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 89^{8}$
Root discriminant $78.98$
Ramified primes $17, 89$
Class number $28$ (GRH)
Class group $[28]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35320816, -19477920, 26393084, -12899464, 7955375, -2782084, 874948, -318070, 78408, -28424, 8536, -1564, 812, -34, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 44*x^14 - 34*x^13 + 812*x^12 - 1564*x^11 + 8536*x^10 - 28424*x^9 + 78408*x^8 - 318070*x^7 + 874948*x^6 - 2782084*x^5 + 7955375*x^4 - 12899464*x^3 + 26393084*x^2 - 19477920*x + 35320816)
 
gp: K = bnfinit(x^16 + 44*x^14 - 34*x^13 + 812*x^12 - 1564*x^11 + 8536*x^10 - 28424*x^9 + 78408*x^8 - 318070*x^7 + 874948*x^6 - 2782084*x^5 + 7955375*x^4 - 12899464*x^3 + 26393084*x^2 - 19477920*x + 35320816, 1)
 

Normalized defining polynomial

\( x^{16} + 44 x^{14} - 34 x^{13} + 812 x^{12} - 1564 x^{11} + 8536 x^{10} - 28424 x^{9} + 78408 x^{8} - 318070 x^{7} + 874948 x^{6} - 2782084 x^{5} + 7955375 x^{4} - 12899464 x^{3} + 26393084 x^{2} - 19477920 x + 35320816 \)

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:  \(2293544177031779368362112832641=17^{12}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.98$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{80} a^{14} - \frac{1}{80} a^{13} - \frac{1}{80} a^{12} + \frac{3}{80} a^{11} - \frac{1}{16} a^{10} + \frac{3}{80} a^{9} - \frac{7}{80} a^{8} - \frac{3}{16} a^{7} + \frac{1}{16} a^{6} - \frac{3}{16} a^{5} - \frac{7}{80} a^{4} + \frac{33}{80} a^{3} - \frac{3}{40} a^{2} + \frac{1}{20} a + \frac{1}{5}$, $\frac{1}{14790823729646957647812046164925509491036098880} a^{15} - \frac{88834337479518958951435116860010444556767509}{14790823729646957647812046164925509491036098880} a^{14} + \frac{306899246690843356275773647924818969330894757}{14790823729646957647812046164925509491036098880} a^{13} + \frac{131149246627422092752233965447040836965607881}{14790823729646957647812046164925509491036098880} a^{12} - \frac{308044956199322666175537628774042383730105589}{14790823729646957647812046164925509491036098880} a^{11} + \frac{666186914129029939186437153329340252119488573}{14790823729646957647812046164925509491036098880} a^{10} - \frac{201683911939459507508183209443281574284145421}{14790823729646957647812046164925509491036098880} a^{9} - \frac{477316617392193456296066031927181143048277539}{14790823729646957647812046164925509491036098880} a^{8} - \frac{383726382627195246290492301379346946841458741}{2958164745929391529562409232985101898207219776} a^{7} + \frac{338780733835038656419399994800561482312163879}{2958164745929391529562409232985101898207219776} a^{6} + \frac{1871984639992091157161216187161061828220900473}{14790823729646957647812046164925509491036098880} a^{5} + \frac{3131554699468752647320067855058204166572846799}{14790823729646957647812046164925509491036098880} a^{4} - \frac{43981177964879986472760787734516708615054371}{369770593241173941195301154123137737275902472} a^{3} - \frac{693973061471852256734270488249338440699605837}{3697705932411739411953011541231377372759024720} a^{2} - \frac{314289961182856974856462893247410489892982061}{924426483102934852988252885307844343189756180} a - \frac{339772835463275360436150610174809458742476403}{924426483102934852988252885307844343189756180}$

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

Class group and class number

$C_{28}$, which has order $28$ (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:  \( 90233389.6986 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.437257.1, 4.4.38915873.1, 4.0.25721.1, 8.0.11246687297.1, 8.0.1514445171352129.1, 8.0.89085010079537.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$