Properties

Label 16.8.927...113.1
Degree $16$
Signature $[8, 4]$
Discriminant $9.271\times 10^{33}$
Root discriminant $132.72$
Ramified primes $11, 97$
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 - x^15 - 45*x^14 + x^13 - 708*x^12 + 1115*x^11 + 16242*x^10 + 36896*x^9 + 185901*x^8 - 544814*x^7 - 1337386*x^6 - 7688656*x^5 - 9683921*x^4 + 17611964*x^3 + 39000939*x^2 + 15688356*x - 1027001)
 
gp: K = bnfinit(x^16 - x^15 - 45*x^14 + x^13 - 708*x^12 + 1115*x^11 + 16242*x^10 + 36896*x^9 + 185901*x^8 - 544814*x^7 - 1337386*x^6 - 7688656*x^5 - 9683921*x^4 + 17611964*x^3 + 39000939*x^2 + 15688356*x - 1027001, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1027001, 15688356, 39000939, 17611964, -9683921, -7688656, -1337386, -544814, 185901, 36896, 16242, 1115, -708, 1, -45, -1, 1]);
 

\( x^{16} - x^{15} - 45 x^{14} + x^{13} - 708 x^{12} + 1115 x^{11} + 16242 x^{10} + 36896 x^{9} + 185901 x^{8} - 544814 x^{7} - 1337386 x^{6} - 7688656 x^{5} - 9683921 x^{4} + 17611964 x^{3} + 39000939 x^{2} + 15688356 x - 1027001 \)

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:  \(9271430660151733401059603251196113\)\(\medspace = 11^{4}\cdot 97^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $132.72$
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:  $11, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{113} a^{14} - \frac{27}{113} a^{13} - \frac{25}{113} a^{12} - \frac{32}{113} a^{11} - \frac{2}{113} a^{10} + \frac{52}{113} a^{9} - \frac{18}{113} a^{8} - \frac{21}{113} a^{7} - \frac{44}{113} a^{6} - \frac{56}{113} a^{5} + \frac{19}{113} a^{4} + \frac{46}{113} a^{3} + \frac{37}{113} a^{2} - \frac{6}{113} a + \frac{14}{113}$, $\frac{1}{1367770287506511467794409571837555967549964280890409361571} a^{15} + \frac{1811171302251735847137316559970432361394629894114070127}{1367770287506511467794409571837555967549964280890409361571} a^{14} - \frac{23631826414167383680163269889427712180071344017046380607}{1367770287506511467794409571837555967549964280890409361571} a^{13} - \frac{104485003065795585873841895947160521849532907403088445546}{1367770287506511467794409571837555967549964280890409361571} a^{12} - \frac{62617152451030581958471684593345091239149825497725728161}{1367770287506511467794409571837555967549964280890409361571} a^{11} + \frac{30955717104849242975953133053357117022926603976502971740}{1367770287506511467794409571837555967549964280890409361571} a^{10} + \frac{3963596069145596602960035390914911915110632885729314998}{12104161836340809449508049308296955465043931689295658067} a^{9} - \frac{483891163127067186267039012573378694326159623484059278467}{1367770287506511467794409571837555967549964280890409361571} a^{8} + \frac{450879100389647412202881957368849594701403798163685285155}{1367770287506511467794409571837555967549964280890409361571} a^{7} - \frac{32002532234455841162711024195156416529479756936011383039}{1367770287506511467794409571837555967549964280890409361571} a^{6} + \frac{367366804873688252549333604460700719564011523262908933329}{1367770287506511467794409571837555967549964280890409361571} a^{5} + \frac{262623239516783972031584875959999281446375893366777030942}{1367770287506511467794409571837555967549964280890409361571} a^{4} + \frac{548227997753212479891941902174554863391144676669375928372}{1367770287506511467794409571837555967549964280890409361571} a^{3} - \frac{226244737646774248547593491914924404874887183986982243841}{1367770287506511467794409571837555967549964280890409361571} a^{2} - \frac{174500399496401998791750089352225734404069129752978941560}{1367770287506511467794409571837555967549964280890409361571} a + \frac{509380960472968466757356843372033118673836963855903381818}{1367770287506511467794409571837555967549964280890409361571}$

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:  \( 41336464901.6 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 41336464901.6 \cdot 2}{2\sqrt{9271430660151733401059603251196113}}\approx 0.171285024608$ (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ R $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$11$11.8.4.2$x^{8} - 1331 x^{2} + 29282$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
97Data not computed