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Properties

Label 16.8.24968628224...2241.2
Degree $16$
Signature $[8, 4]$
Discriminant $37^{12}\cdot 41^{14}$
Root discriminant $386.67$
Ramified primes $37, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11574258736, -34702486496, 36703872896, -8754113842, -3185922987, 1810848260, -113711176, -67215741, 10245030, -790746, 146765, 27466, -6102, 71, -32, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 32*x^14 + 71*x^13 - 6102*x^12 + 27466*x^11 + 146765*x^10 - 790746*x^9 + 10245030*x^8 - 67215741*x^7 - 113711176*x^6 + 1810848260*x^5 - 3185922987*x^4 - 8754113842*x^3 + 36703872896*x^2 - 34702486496*x - 11574258736)
 
gp: K = bnfinit(x^16 - 2*x^15 - 32*x^14 + 71*x^13 - 6102*x^12 + 27466*x^11 + 146765*x^10 - 790746*x^9 + 10245030*x^8 - 67215741*x^7 - 113711176*x^6 + 1810848260*x^5 - 3185922987*x^4 - 8754113842*x^3 + 36703872896*x^2 - 34702486496*x - 11574258736, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 32 x^{14} + 71 x^{13} - 6102 x^{12} + 27466 x^{11} + 146765 x^{10} - 790746 x^{9} + 10245030 x^{8} - 67215741 x^{7} - 113711176 x^{6} + 1810848260 x^{5} - 3185922987 x^{4} - 8754113842 x^{3} + 36703872896 x^{2} - 34702486496 x - 11574258736 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(249686282242731792647174886492450339042241=37^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $386.67$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{332} a^{14} - \frac{19}{83} a^{13} - \frac{23}{83} a^{12} - \frac{41}{332} a^{11} - \frac{13}{83} a^{10} + \frac{43}{166} a^{9} + \frac{53}{332} a^{8} + \frac{5}{83} a^{7} - \frac{53}{166} a^{6} + \frac{55}{332} a^{5} - \frac{5}{166} a^{4} - \frac{14}{83} a^{3} + \frac{37}{332} a^{2} - \frac{23}{83} a + \frac{6}{83}$, $\frac{1}{1483064288838206130627594112159997854574456613470133532539364721635224536} a^{15} - \frac{23219056825121797972492819629390362410301112313444546244833201855291}{370766072209551532656898528039999463643614153367533383134841180408806134} a^{14} - \frac{33386180853047183985400708265226899296762892370200832330607062248337334}{185383036104775766328449264019999731821807076683766691567420590204403067} a^{13} + \frac{1051913885432273829511346887279110502733908657063559770828824079031089}{6650512506000924352590108126278017285087249387758446334257240904193832} a^{12} - \frac{157224859773005432669181865606998633421340838632471212373783110359722759}{370766072209551532656898528039999463643614153367533383134841180408806134} a^{11} + \frac{190544682885073849816377163070469943381384846846185696264661428383269461}{741532144419103065313797056079998927287228306735066766269682360817612268} a^{10} + \frac{695087421490684871769798144416220894786887484931907343165079261706953041}{1483064288838206130627594112159997854574456613470133532539364721635224536} a^{9} - \frac{107490739363992415479575696365666150938454789176606943040395942685029799}{370766072209551532656898528039999463643614153367533383134841180408806134} a^{8} + \frac{98835769392525941602455697404580107939936735690683009350191334019581603}{741532144419103065313797056079998927287228306735066766269682360817612268} a^{7} - \frac{688701715465084765806736129637928728124166542241582803236676567993986041}{1483064288838206130627594112159997854574456613470133532539364721635224536} a^{6} + \frac{58974816832286818515715467146085533196147869946624504398056788523562873}{741532144419103065313797056079998927287228306735066766269682360817612268} a^{5} - \frac{48343884418718977991758978137357583223960309368575722415826151426669447}{185383036104775766328449264019999731821807076683766691567420590204403067} a^{4} + \frac{99169134644970124559536422573141838292245359967194166748479267350637789}{1483064288838206130627594112159997854574456613470133532539364721635224536} a^{3} + \frac{97500485482936061614864847199883362912963412727151906320113274656112753}{370766072209551532656898528039999463643614153367533383134841180408806134} a^{2} + \frac{35342522292758924196489668679595879713709905239040524442982679574081117}{185383036104775766328449264019999731821807076683766691567420590204403067} a - \frac{67630663731829412643752483307919189899176530636366317102186581617534700}{185383036104775766328449264019999731821807076683766691567420590204403067}$

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:  $11$
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:  \( 135707488387000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{1517}) \), \(\Q(\sqrt{37}) \), 4.4.68921.1, 4.4.94352849.1, \(\Q(\sqrt{37}, \sqrt{41})\), 8.8.8902460114416801.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
37Data not computed
$41$41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$