Properties

Label 16.8.584...953.1
Degree $16$
Signature $[8, 4]$
Discriminant $5.848\times 10^{35}$
Root discriminant $171.96$
Ramified primes $31, 97$
Class number $2$ (GRH)
Class group $[2]$ (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 - 3*x^15 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317)
 
gp: K = bnfinit(x^16 - 3*x^15 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23651317, 48133287, -115827035, 37514732, 39257376, -16023466, -3412593, 1864157, -257892, -80542, 47837, 412, -1476, 124, -17, -3, 1]);
 

\( x^{16} - 3 x^{15} - 17 x^{14} + 124 x^{13} - 1476 x^{12} + 412 x^{11} + 47837 x^{10} - 80542 x^{9} - 257892 x^{8} + 1864157 x^{7} - 3412593 x^{6} - 16023466 x^{5} + 39257376 x^{4} + 37514732 x^{3} - 115827035 x^{2} + 48133287 x + 23651317 \)

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:  \(584820771442796870588072252861681953\)\(\medspace = 31^{4}\cdot 97^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $171.96$
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:  $31, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{701} a^{14} + \frac{180}{701} a^{13} + \frac{213}{701} a^{12} - \frac{254}{701} a^{11} - \frac{281}{701} a^{10} + \frac{250}{701} a^{9} - \frac{349}{701} a^{8} - \frac{338}{701} a^{7} - \frac{85}{701} a^{6} - \frac{127}{701} a^{5} - \frac{49}{701} a^{4} + \frac{182}{701} a^{3} - \frac{64}{701} a^{2} - \frac{107}{701} a + \frac{197}{701}$, $\frac{1}{43638531836760902866245007194858323058531192289429213699} a^{15} - \frac{9705123810870083678679579527899292362345288502410775}{43638531836760902866245007194858323058531192289429213699} a^{14} - \frac{4743742443212269731838508595024388461880370585202200076}{43638531836760902866245007194858323058531192289429213699} a^{13} + \frac{8692766789351644211185769528182471326855851360649824148}{43638531836760902866245007194858323058531192289429213699} a^{12} - \frac{2990404522384147464396695401234314021168755347896873684}{43638531836760902866245007194858323058531192289429213699} a^{11} - \frac{11616802905815476811305158506276305578841063680893894612}{43638531836760902866245007194858323058531192289429213699} a^{10} - \frac{4248830760280568807042794447845344039083439571951001442}{43638531836760902866245007194858323058531192289429213699} a^{9} - \frac{18453289294389069913900174682315602714200284239211266566}{43638531836760902866245007194858323058531192289429213699} a^{8} - \frac{1616307856249763571905504719358653765786841443164281758}{43638531836760902866245007194858323058531192289429213699} a^{7} + \frac{7694793832270209079254058862531406155101027767772870921}{43638531836760902866245007194858323058531192289429213699} a^{6} - \frac{19727578777337907664165132841727281204551381398620184789}{43638531836760902866245007194858323058531192289429213699} a^{5} + \frac{16734218272123297065237452544601565107603084555761393557}{43638531836760902866245007194858323058531192289429213699} a^{4} + \frac{859049354192456182245452906330712757771345790523533331}{43638531836760902866245007194858323058531192289429213699} a^{3} + \frac{6722364924496777507696589404829993412116652778815800331}{43638531836760902866245007194858323058531192289429213699} a^{2} - \frac{11126329639360638379744026465694524707579232014994990680}{43638531836760902866245007194858323058531192289429213699} a - \frac{4350796053946759562402461221175690459960356497633923488}{43638531836760902866245007194858323058531192289429213699}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 321228696477 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 321228696477 \cdot 2}{2\sqrt{584820771442796870588072252861681953}}\approx 0.167595510021600$ (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

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ R $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$31$31.8.4.2$x^{8} - 59582 x^{2} + 15699857$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$
97Data not computed