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Properties

Label 16.0.35028437828...8881.3
Degree $16$
Signature $[0, 8]$
Discriminant $31^{4}\cdot 41^{14}$
Root discriminant $60.82$
Ramified primes $31, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![855625, -1271875, 648725, -68160, 89578, -219357, 150416, -31116, -10545, 5496, 468, -611, 30, 28, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625)
 
gp: K = bnfinit(x^16 - 5*x^15 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 5 x^{14} + 28 x^{13} + 30 x^{12} - 611 x^{11} + 468 x^{10} + 5496 x^{9} - 10545 x^{8} - 31116 x^{7} + 150416 x^{6} - 219357 x^{5} + 89578 x^{4} - 68160 x^{3} + 648725 x^{2} - 1271875 x + 855625 \)

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:  \(35028437828275611780530528881=31^{4}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.82$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{3}{10} a^{10} + \frac{2}{5} a^{8} + \frac{3}{10} a^{7} - \frac{2}{5} a^{6} - \frac{1}{10} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{3}{10} a$, $\frac{1}{2950} a^{14} + \frac{7}{590} a^{13} - \frac{57}{295} a^{12} - \frac{447}{2950} a^{11} + \frac{7}{118} a^{10} + \frac{632}{1475} a^{9} + \frac{403}{2950} a^{8} - \frac{1209}{2950} a^{7} + \frac{67}{295} a^{6} - \frac{141}{2950} a^{5} + \frac{51}{2950} a^{4} - \frac{396}{1475} a^{3} + \frac{1423}{2950} a^{2} - \frac{43}{590} a + \frac{18}{59}$, $\frac{1}{1699336988581362722042676899084372750} a^{15} - \frac{1641443928199552436576359409033}{33986739771627254440853537981687455} a^{14} - \frac{985509840524034484542310336676679}{339867397716272544408535379816874550} a^{13} + \frac{71916287198358290479530556274073089}{849668494290681361021338449542186375} a^{12} - \frac{28431058167444213635989942599486703}{169933698858136272204267689908437275} a^{11} - \frac{188647280539371688001005314556094961}{1699336988581362722042676899084372750} a^{10} + \frac{229761511375463730725237261061657219}{849668494290681361021338449542186375} a^{9} + \frac{363663410254603573396576634607917193}{849668494290681361021338449542186375} a^{8} + \frac{1959284315120523880966997387036611}{9185605343683041740771226481537150} a^{7} - \frac{389423872108008762127837627926718733}{849668494290681361021338449542186375} a^{6} - \frac{364378162425078093564454917463867132}{849668494290681361021338449542186375} a^{5} - \frac{654121507825309352486017853499169927}{1699336988581362722042676899084372750} a^{4} - \frac{199351371808864047367506776357573016}{849668494290681361021338449542186375} a^{3} - \frac{5407441881609128078772032230192747}{169933698858136272204267689908437275} a^{2} - \frac{8217856702196337485407600356740523}{67973479543254508881707075963374910} a + \frac{135453879066552793948402098406401}{367424213747321669630849059261486}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 3814663.53647 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.2136551.1, 4.4.68921.1, 4.2.52111.1, 8.0.194754273881.1, 8.4.187158857199641.1, 8.4.4564850175601.2

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$