Properties

Label 16.8.344...761.1
Degree $16$
Signature $[8, 4]$
Discriminant $3.442\times 10^{39}$
Root discriminant $295.84$
Ramified primes $19, 41$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 183*x^14 + 679*x^13 + 9311*x^12 + 43276*x^11 + 97932*x^10 - 10403552*x^9 - 8440783*x^8 + 567915747*x^7 - 854442249*x^6 - 9305286587*x^5 + 21874828800*x^4 + 74315288244*x^3 - 109960290119*x^2 - 166367096959*x + 190692533519)
 
gp: K = bnfinit(x^16 - 6*x^15 - 183*x^14 + 679*x^13 + 9311*x^12 + 43276*x^11 + 97932*x^10 - 10403552*x^9 - 8440783*x^8 + 567915747*x^7 - 854442249*x^6 - 9305286587*x^5 + 21874828800*x^4 + 74315288244*x^3 - 109960290119*x^2 - 166367096959*x + 190692533519, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![190692533519, -166367096959, -109960290119, 74315288244, 21874828800, -9305286587, -854442249, 567915747, -8440783, -10403552, 97932, 43276, 9311, 679, -183, -6, 1]);
 

\( x^{16} - 6 x^{15} - 183 x^{14} + 679 x^{13} + 9311 x^{12} + 43276 x^{11} + 97932 x^{10} - 10403552 x^{9} - 8440783 x^{8} + 567915747 x^{7} - 854442249 x^{6} - 9305286587 x^{5} + 21874828800 x^{4} + 74315288244 x^{3} - 109960290119 x^{2} - 166367096959 x + 190692533519 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3441922301185419076541988280060619730761\)\(\medspace = 19^{12}\cdot 41^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $295.84$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $19, 41$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{15} + \frac{839730111389703137193523712487454649022288448626280858307926423769280010028049860267}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{14} + \frac{573842811815714074343479266532249882053966426128663302244118294735685605170326190392}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{13} - \frac{874951229695962959659421958793199983966952858006164363071638100497397098539488008264}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{12} + \frac{1570459037813203446064015116820793253250543884104288494120865034260292432024477653500}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{11} - \frac{1786379462853367356193091281107103993805184071332911702260707148507019311295049599699}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{10} - \frac{562896220281047098994227796645284431987070855240194725289059683350563782263100037960}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{9} - \frac{1572131516132636513768735198586823891791764437826488540584879717320925018754625698421}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{8} + \frac{1092109915456630513665558199876471665428150101265866521549653072334206444413542521950}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{7} + \frac{233761340311541926600925727125704067520593794844182226402649610188986609603622564120}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{6} - \frac{618661085629801407095671893841756001636432283979284113081261390865146617307031771168}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{5} + \frac{1563542090519971318099988550875043357389405470138738145962760352088809448266182931890}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{4} - \frac{526002215440108142090615597415695934067221734186735974507219910443430696295316302293}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{3} + \frac{899148935554510467767678820398813009398970794778858759552699595332631922002624289168}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a^{2} + \frac{677964049839088094006029601284053136281912230463937120173037107128529766493648510521}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619} a - \frac{466385134460623996455385900783029917628749275235256921207459933629119259216895117617}{3707363878925230448788135063794017870082761160674528898609152666677261773889075282619}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 48469270476700 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 48469270476700 \cdot 1}{2\sqrt{3441922301185419076541988280060619730761}}\approx 0.164814491571640$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.25380571726445801.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
19Data not computed
41Data not computed