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Properties

Label 16.16.1332256312...3281.1
Degree $16$
Signature $[16, 0]$
Discriminant $37^{8}\cdot 41^{14}$
Root discriminant $156.78$
Ramified primes $37, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2033869, -14116519, -14494532, 62563841, 101413382, 10490174, -36909197, -7678424, 5254594, 877563, -380372, -32724, 12853, 450, -190, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 190*x^14 + 450*x^13 + 12853*x^12 - 32724*x^11 - 380372*x^10 + 877563*x^9 + 5254594*x^8 - 7678424*x^7 - 36909197*x^6 + 10490174*x^5 + 101413382*x^4 + 62563841*x^3 - 14494532*x^2 - 14116519*x - 2033869)
 
gp: K = bnfinit(x^16 - 2*x^15 - 190*x^14 + 450*x^13 + 12853*x^12 - 32724*x^11 - 380372*x^10 + 877563*x^9 + 5254594*x^8 - 7678424*x^7 - 36909197*x^6 + 10490174*x^5 + 101413382*x^4 + 62563841*x^3 - 14494532*x^2 - 14116519*x - 2033869, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 190 x^{14} + 450 x^{13} + 12853 x^{12} - 32724 x^{11} - 380372 x^{10} + 877563 x^{9} + 5254594 x^{8} - 7678424 x^{7} - 36909197 x^{6} + 10490174 x^{5} + 101413382 x^{4} + 62563841 x^{3} - 14494532 x^{2} - 14116519 x - 2033869 \)

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:  \(133225631225242544609121034154723281=37^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.78$
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 Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{74} a^{12} + \frac{8}{37} a^{11} - \frac{5}{37} a^{10} - \frac{31}{74} a^{9} - \frac{2}{37} a^{8} - \frac{7}{37} a^{7} - \frac{11}{74} a^{6} + \frac{2}{37} a^{5} + \frac{16}{37} a^{4} + \frac{5}{74} a^{3} + \frac{16}{37} a^{2} + \frac{12}{37} a - \frac{11}{74}$, $\frac{1}{6142} a^{13} - \frac{35}{6142} a^{12} - \frac{80}{3071} a^{11} - \frac{853}{6142} a^{10} - \frac{1531}{6142} a^{9} + \frac{1242}{3071} a^{8} + \frac{1}{166} a^{7} + \frac{1823}{6142} a^{6} - \frac{900}{3071} a^{5} + \frac{2813}{6142} a^{4} - \frac{815}{6142} a^{3} + \frac{713}{3071} a^{2} - \frac{1901}{6142} a - \frac{327}{6142}$, $\frac{1}{21294314} a^{14} - \frac{326}{10647157} a^{13} - \frac{83311}{21294314} a^{12} - \frac{8960753}{21294314} a^{11} - \frac{1873371}{10647157} a^{10} - \frac{1640663}{21294314} a^{9} + \frac{796555}{21294314} a^{8} - \frac{622379}{10647157} a^{7} - \frac{3849987}{21294314} a^{6} + \frac{6541613}{21294314} a^{5} + \frac{3963295}{10647157} a^{4} - \frac{5958763}{21294314} a^{3} + \frac{6502601}{21294314} a^{2} + \frac{2022610}{10647157} a - \frac{1354657}{21294314}$, $\frac{1}{47502582838273303714485328854894745214362} a^{15} - \frac{493266419234192870791623554704434}{23751291419136651857242664427447372607181} a^{14} - \frac{1930319141239492737537456863224027660}{23751291419136651857242664427447372607181} a^{13} + \frac{317779756376915853148387343891815384827}{47502582838273303714485328854894745214362} a^{12} + \frac{9883539628027940356809952868734533735501}{23751291419136651857242664427447372607181} a^{11} + \frac{7390165179996593096577802254565357137722}{23751291419136651857242664427447372607181} a^{10} - \frac{9631559646364204318813386026632200216733}{47502582838273303714485328854894745214362} a^{9} + \frac{9271729661828238407592230938893960012717}{23751291419136651857242664427447372607181} a^{8} - \frac{7986505999282217879739263789272502347959}{23751291419136651857242664427447372607181} a^{7} + \frac{6095232379640480794556760157554355984955}{47502582838273303714485328854894745214362} a^{6} - \frac{10078149635814468798706313183639718726586}{23751291419136651857242664427447372607181} a^{5} + \frac{7484067199161588292309028313056165863388}{23751291419136651857242664427447372607181} a^{4} + \frac{6202320382004119451749165627417446016219}{47502582838273303714485328854894745214362} a^{3} - \frac{6669648947411123531893078774295234095862}{23751291419136651857242664427447372607181} a^{2} + \frac{6937891475963927942139915389946924731936}{23751291419136651857242664427447372607181} a + \frac{9409041084078340806846502585245142323423}{23751291419136651857242664427447372607181}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 2274016052920 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

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 $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{1517}) \), \(\Q(\sqrt{37}, \sqrt{41})\), 4.4.68921.1, 4.4.94352849.1, 8.8.8902460114416801.1, 8.8.266618600943089.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$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}$