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Properties

Label 16.0.61841248347...401.29
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{12}$
Root discriminant $149.44$
Ramified primes $13, 61$
Class number $103680$ (GRH)
Class group $[2, 2, 2, 36, 360]$ (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![190380537, -79315929, 63196200, -26219943, 21565485, -9535563, 2597153, -831663, 167858, -71199, 18125, -1998, 994, -72, 37, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 37*x^14 - 72*x^13 + 994*x^12 - 1998*x^11 + 18125*x^10 - 71199*x^9 + 167858*x^8 - 831663*x^7 + 2597153*x^6 - 9535563*x^5 + 21565485*x^4 - 26219943*x^3 + 63196200*x^2 - 79315929*x + 190380537)
 
gp: K = bnfinit(x^16 + 37*x^14 - 72*x^13 + 994*x^12 - 1998*x^11 + 18125*x^10 - 71199*x^9 + 167858*x^8 - 831663*x^7 + 2597153*x^6 - 9535563*x^5 + 21565485*x^4 - 26219943*x^3 + 63196200*x^2 - 79315929*x + 190380537, 1)
 

Normalized defining polynomial

\( x^{16} + 37 x^{14} - 72 x^{13} + 994 x^{12} - 1998 x^{11} + 18125 x^{10} - 71199 x^{9} + 167858 x^{8} - 831663 x^{7} + 2597153 x^{6} - 9535563 x^{5} + 21565485 x^{4} - 26219943 x^{3} + 63196200 x^{2} - 79315929 x + 190380537 \)

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:  \(61841248347955294332328765542314401=13^{12}\cdot 61^{12}\)
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 a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{54} a^{8} - \frac{1}{27} a^{7} - \frac{1}{27} a^{6} + \frac{2}{27} a^{5} + \frac{1}{54} a^{4} - \frac{1}{27} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{162} a^{9} + \frac{1}{27} a^{7} - \frac{5}{54} a^{5} - \frac{1}{162} a^{3} - \frac{1}{2} a^{2} + \frac{1}{18} a$, $\frac{1}{162} a^{10} - \frac{1}{27} a^{7} - \frac{1}{54} a^{6} + \frac{2}{27} a^{5} - \frac{7}{162} a^{4} + \frac{7}{54} a^{3} + \frac{7}{18} a^{2} - \frac{1}{3} a$, $\frac{1}{486} a^{11} + \frac{1}{486} a^{10} + \frac{1}{162} a^{8} - \frac{1}{18} a^{7} - \frac{1}{54} a^{6} + \frac{41}{486} a^{5} + \frac{77}{486} a^{4} - \frac{7}{81} a^{3} - \frac{7}{27} a^{2} - \frac{1}{18} a - \frac{1}{2}$, $\frac{1}{25272} a^{12} - \frac{23}{25272} a^{11} - \frac{7}{2808} a^{10} - \frac{11}{8424} a^{9} + \frac{1}{156} a^{8} - \frac{43}{936} a^{7} - \frac{47}{12636} a^{6} - \frac{121}{1944} a^{5} - \frac{595}{4212} a^{4} + \frac{43}{2808} a^{3} + \frac{175}{936} a^{2} - \frac{17}{156} a - \frac{49}{104}$, $\frac{1}{25272} a^{13} - \frac{5}{6318} a^{11} + \frac{1}{972} a^{10} + \frac{1}{936} a^{9} + \frac{23}{8424} a^{8} + \frac{815}{25272} a^{7} - \frac{17}{936} a^{6} - \frac{3193}{25272} a^{5} - \frac{3515}{25272} a^{4} + \frac{503}{4212} a^{3} + \frac{797}{2808} a^{2} + \frac{125}{936} a + \frac{17}{104}$, $\frac{1}{75816} a^{14} + \frac{1}{75816} a^{13} - \frac{1}{75816} a^{12} - \frac{67}{75816} a^{11} + \frac{1}{486} a^{10} - \frac{73}{25272} a^{9} + \frac{187}{37908} a^{8} - \frac{3451}{75816} a^{7} + \frac{1727}{37908} a^{6} + \frac{421}{5832} a^{5} - \frac{1265}{25272} a^{4} - \frac{77}{1053} a^{3} + \frac{977}{2808} a^{2} - \frac{5}{39} a - \frac{5}{52}$, $\frac{1}{24826784159406326624199249793213608} a^{15} - \frac{7366531733898744712433156398}{3103348019925790828024906224151701} a^{14} - \frac{180228377158293961095548598505}{24826784159406326624199249793213608} a^{13} - \frac{38674207767608322782645854657}{6206696039851581656049812448303402} a^{12} + \frac{4021412810661488114093705193487}{8275594719802108874733083264404536} a^{11} + \frac{9947055996208078978427102801651}{8275594719802108874733083264404536} a^{10} - \frac{35653948198811314766844670271135}{12413392079703163312099624896606804} a^{9} - \frac{7416096795594041481033990555227}{3103348019925790828024906224151701} a^{8} + \frac{50147164011567578563077006038771}{954876313823320254776894222815908} a^{7} - \frac{48411950508026204188690095970277}{6206696039851581656049812448303402} a^{6} - \frac{122146973343519950653503172864793}{8275594719802108874733083264404536} a^{5} + \frac{18391365784172014286577578476168}{114938815552807067704626156450063} a^{4} - \frac{48596015794649248011152988543929}{306503508140818847212336417200168} a^{3} - \frac{10015361718156588809199298791403}{76625877035204711803084104300042} a^{2} + \frac{20307239948757491609100496013759}{102167836046939615737445472400056} a - \frac{5037076162184265505595133860591}{11351981782993290637493941377784}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{36}\times C_{360}$, which has order $103680$ (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:  \( 308589653.463 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{793}) \), \(\Q(\sqrt{13}, \sqrt{61})\), 4.0.2950753.1 x2, 4.0.38359789.1 x2, 4.0.2197.1, 4.0.8175037.2, 8.0.1471473412124521.6, 8.0.66831229951369.5, 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ 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}$