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Properties

Label 16.16.6184124834...4401.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{12}\cdot 61^{12}$
Root discriminant $149.44$
Ramified primes $13, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![105543, -6612084, 55262142, -66676452, -87126427, 62582502, 28230493, -20231211, -2022752, 2467270, -119494, -107804, 11835, 1528, -221, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 221*x^14 + 1528*x^13 + 11835*x^12 - 107804*x^11 - 119494*x^10 + 2467270*x^9 - 2022752*x^8 - 20231211*x^7 + 28230493*x^6 + 62582502*x^5 - 87126427*x^4 - 66676452*x^3 + 55262142*x^2 - 6612084*x + 105543)
 
gp: K = bnfinit(x^16 - 5*x^15 - 221*x^14 + 1528*x^13 + 11835*x^12 - 107804*x^11 - 119494*x^10 + 2467270*x^9 - 2022752*x^8 - 20231211*x^7 + 28230493*x^6 + 62582502*x^5 - 87126427*x^4 - 66676452*x^3 + 55262142*x^2 - 6612084*x + 105543, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 221 x^{14} + 1528 x^{13} + 11835 x^{12} - 107804 x^{11} - 119494 x^{10} + 2467270 x^{9} - 2022752 x^{8} - 20231211 x^{7} + 28230493 x^{6} + 62582502 x^{5} - 87126427 x^{4} - 66676452 x^{3} + 55262142 x^{2} - 6612084 x + 105543 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(61841248347955294332328765542314401=13^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.44$
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:  $13, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{36} a^{10} + \frac{1}{36} a^{9} + \frac{1}{18} a^{8} + \frac{1}{36} a^{7} - \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{5}{36} a^{4} - \frac{2}{9} a^{3} + \frac{13}{36} a^{2} + \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{72} a^{11} + \frac{1}{72} a^{9} - \frac{1}{72} a^{8} - \frac{17}{72} a^{7} + \frac{5}{18} a^{6} + \frac{1}{72} a^{5} + \frac{23}{72} a^{4} + \frac{7}{24} a^{3} + \frac{29}{72} a^{2} - \frac{5}{24} a + \frac{1}{8}$, $\frac{1}{144} a^{12} - \frac{1}{144} a^{11} + \frac{1}{144} a^{10} - \frac{1}{72} a^{9} + \frac{1}{18} a^{8} + \frac{13}{144} a^{7} - \frac{67}{144} a^{6} + \frac{35}{72} a^{5} - \frac{25}{72} a^{4} + \frac{7}{18} a^{3} - \frac{5}{36} a^{2} - \frac{1}{2} a + \frac{7}{16}$, $\frac{1}{288} a^{13} - \frac{1}{288} a^{10} + \frac{1}{48} a^{9} + \frac{7}{96} a^{8} + \frac{5}{16} a^{7} - \frac{47}{96} a^{6} - \frac{31}{72} a^{5} - \frac{23}{48} a^{4} - \frac{3}{8} a^{3} - \frac{23}{72} a^{2} - \frac{1}{32} a - \frac{9}{32}$, $\frac{1}{3456} a^{14} - \frac{1}{3456} a^{13} + \frac{7}{3456} a^{11} - \frac{41}{3456} a^{10} + \frac{167}{3456} a^{9} - \frac{419}{3456} a^{8} + \frac{1121}{3456} a^{7} - \frac{485}{1152} a^{6} - \frac{257}{576} a^{5} + \frac{227}{1728} a^{4} - \frac{169}{432} a^{3} + \frac{49}{128} a^{2} - \frac{1}{3} a + \frac{43}{128}$, $\frac{1}{695120802506074437474985894584061915723391616768} a^{15} - \frac{2551657948233846172463119972439998514871199}{28963366771086434894791078941002579821807984032} a^{14} - \frac{221171860300581091457241960543911001334902841}{695120802506074437474985894584061915723391616768} a^{13} + \frac{842562153374741896701593630270921561714990887}{695120802506074437474985894584061915723391616768} a^{12} - \frac{1599615521942669671844504227143672234271162157}{347560401253037218737492947292030957861695808384} a^{11} + \frac{125467364249016819476542920243178675280599057}{115853467084345739579164315764010319287231936128} a^{10} - \frac{672119165025473276776743016053615991651459261}{19308911180724289929860719294001719881205322688} a^{9} - \frac{5023874654639356692194599290005901702471377485}{38617822361448579859721438588003439762410645376} a^{8} - \frac{126442806452369764520420510287876952322045891467}{347560401253037218737492947292030957861695808384} a^{7} - \frac{22790072154868755159855971395135982274325301407}{231706934168691479158328631528020638574463872256} a^{6} - \frac{1820843000024717349332148050910208589798177765}{43445050156629652342186618411503869732711976048} a^{5} - \frac{172757795616495762695224828696282638145257489689}{347560401253037218737492947292030957861695808384} a^{4} - \frac{305786884753918863428182568200061642162975586461}{695120802506074437474985894584061915723391616768} a^{3} + \frac{111570778830624427676623559074936153614077568721}{231706934168691479158328631528020638574463872256} a^{2} - \frac{19655975161022525420808295573206982938316587551}{77235644722897159719442877176006879524821290752} a - \frac{9291292036187648941363997427439756124226227085}{25745214907632386573147625725335626508273763584}$

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:  $15$
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:  \( 2287406525140 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{793}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{61})\), 4.4.48373.1 x2, 4.4.10309.1 x2, 4.4.498677257.1, 4.4.498677257.2, 8.8.395451064801.1, 8.8.248679006649044049.2, 8.8.248679006649044049.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$