Properties

Label 16.0.499...233.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.997\times 10^{36}$
Root discriminant $196.64$
Ramified primes $53, 97$
Class number $176578$ (GRH)
Class group $[176578]$ (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 + 80*x^14 - 458*x^13 + 1725*x^12 - 15981*x^11 + 16603*x^10 - 96450*x^9 + 409565*x^8 + 2849968*x^7 + 5772337*x^6 + 44508608*x^5 - 27303927*x^4 + 143015618*x^3 + 481082447*x^2 - 1618135041*x + 1134515881)
 
gp: K = bnfinit(x^16 - 3*x^15 + 80*x^14 - 458*x^13 + 1725*x^12 - 15981*x^11 + 16603*x^10 - 96450*x^9 + 409565*x^8 + 2849968*x^7 + 5772337*x^6 + 44508608*x^5 - 27303927*x^4 + 143015618*x^3 + 481082447*x^2 - 1618135041*x + 1134515881, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1134515881, -1618135041, 481082447, 143015618, -27303927, 44508608, 5772337, 2849968, 409565, -96450, 16603, -15981, 1725, -458, 80, -3, 1]);
 

\( x^{16} - 3 x^{15} + 80 x^{14} - 458 x^{13} + 1725 x^{12} - 15981 x^{11} + 16603 x^{10} - 96450 x^{9} + 409565 x^{8} + 2849968 x^{7} + 5772337 x^{6} + 44508608 x^{5} - 27303927 x^{4} + 143015618 x^{3} + 481082447 x^{2} - 1618135041 x + 1134515881 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(4996656476111243051576134097472929233\)\(\medspace = 53^{4}\cdot 97^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $196.64$
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:  $53, 97$
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 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}{188296781055979671277839475368521177198730626515261587060903892890791} a^{15} - \frac{7302763546541464406210774713053622010680508652247158428852018095445}{188296781055979671277839475368521177198730626515261587060903892890791} a^{14} + \frac{76357544179845974456534162771212328590213832068923796979076641333804}{188296781055979671277839475368521177198730626515261587060903892890791} a^{13} - \frac{6048705588997714854380394353864849836439136841527009902829679863841}{188296781055979671277839475368521177198730626515261587060903892890791} a^{12} + \frac{15770634023829464934468079587113644681621716530836298155012516326283}{188296781055979671277839475368521177198730626515261587060903892890791} a^{11} + \frac{55242309962539013772958735157949835615707129917614309565414299038975}{188296781055979671277839475368521177198730626515261587060903892890791} a^{10} - \frac{12128626872387541397214642010966982867635854842054783161452350853254}{188296781055979671277839475368521177198730626515261587060903892890791} a^{9} + \frac{52230114084075422160427313103246041547222172543353884430485504641814}{188296781055979671277839475368521177198730626515261587060903892890791} a^{8} + \frac{51949889240139217541619456270719889047819539679176153881901624594570}{188296781055979671277839475368521177198730626515261587060903892890791} a^{7} - \frac{34096554471950092706814956234023632781383317405592465662694677147319}{188296781055979671277839475368521177198730626515261587060903892890791} a^{6} - \frac{25883808170792598073879288820146251105826507634600076611484693041954}{188296781055979671277839475368521177198730626515261587060903892890791} a^{5} + \frac{31370158614108503669678170955881990314020328078312151076788492668774}{188296781055979671277839475368521177198730626515261587060903892890791} a^{4} + \frac{57184635559554859576723472298518670335784450159835119563758429992464}{188296781055979671277839475368521177198730626515261587060903892890791} a^{3} + \frac{8330475039277387418569289475224832767918367165460092305474224191311}{188296781055979671277839475368521177198730626515261587060903892890791} a^{2} - \frac{55943985774264048711727189069968234281707034734007215637376001389129}{188296781055979671277839475368521177198730626515261587060903892890791} a + \frac{46518936607702393673729452625217135490148141526165262557026350590485}{188296781055979671277839475368521177198730626515261587060903892890791}$

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

Class group and class number

$C_{176578}$, which has order $176578$ (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:  $7$
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:  \( 1675810.87182 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 1675810.87182 \cdot 176578}{2\sqrt{4996656476111243051576134097472929233}}\approx 0.160779547514$ (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R $16$

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
$53$53.8.4.2$x^{8} - 148877 x^{2} + 142028658$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
53.8.0.1$x^{8} + x^{2} - x + 33$$1$$8$$0$$C_8$$[\ ]^{8}$
97Data not computed