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Properties

Label 16.0.11623374807...1313.3
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 67^{8}$
Root discriminant $116.57$
Ramified primes $17, 67$
Class number $128$ (GRH)
Class group $[8, 16]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![991543, -337425, -673612, 508030, 302225, -165231, -175249, 38860, 69939, -37834, 10451, -1341, 543, -212, 21, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 21*x^14 - 212*x^13 + 543*x^12 - 1341*x^11 + 10451*x^10 - 37834*x^9 + 69939*x^8 + 38860*x^7 - 175249*x^6 - 165231*x^5 + 302225*x^4 + 508030*x^3 - 673612*x^2 - 337425*x + 991543)
 
gp: K = bnfinit(x^16 - 2*x^15 + 21*x^14 - 212*x^13 + 543*x^12 - 1341*x^11 + 10451*x^10 - 37834*x^9 + 69939*x^8 + 38860*x^7 - 175249*x^6 - 165231*x^5 + 302225*x^4 + 508030*x^3 - 673612*x^2 - 337425*x + 991543, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 21 x^{14} - 212 x^{13} + 543 x^{12} - 1341 x^{11} + 10451 x^{10} - 37834 x^{9} + 69939 x^{8} + 38860 x^{7} - 175249 x^{6} - 165231 x^{5} + 302225 x^{4} + 508030 x^{3} - 673612 x^{2} - 337425 x + 991543 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1162337480711184271576898233831313=17^{15}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.57$
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:  $17, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{72128804715585720627007708328519859269837502400293} a^{15} + \frac{6425371464541739499408806606214937469489014278254}{72128804715585720627007708328519859269837502400293} a^{14} + \frac{28982578362709376484252104380010370458490907169071}{72128804715585720627007708328519859269837502400293} a^{13} + \frac{31890281535358707737526828781466095185771359921555}{72128804715585720627007708328519859269837502400293} a^{12} - \frac{29274764735618793091446889982697929072853190879166}{72128804715585720627007708328519859269837502400293} a^{11} + \frac{10971833124969249243184610087251537772334280266241}{72128804715585720627007708328519859269837502400293} a^{10} + \frac{16512170832438680706881906446805691746141893554203}{72128804715585720627007708328519859269837502400293} a^{9} - \frac{23806671006716839448776737570263457356830593699544}{72128804715585720627007708328519859269837502400293} a^{8} - \frac{181610777465481254585715902428676533997038464926}{1472016422767055523000157312826935903466071477557} a^{7} - \frac{30845572755570335451366157185161418400554003883846}{72128804715585720627007708328519859269837502400293} a^{6} + \frac{170206169456548575526326981360501431628644003641}{3434704986456462887000367063262850441420833447633} a^{5} - \frac{8390944220721972731134864749944081562646062438671}{24042934905195240209002569442839953089945834133431} a^{4} - \frac{664976931709241646242135165428248288213032734241}{1759239139404529771390431910451703884630182985373} a^{3} + \frac{270918040716846658769648446340869429818504337587}{2185721355017749109909324494803632099085984921221} a^{2} + \frac{7778321129840997604030910452416174199192609233676}{24042934905195240209002569442839953089945834133431} a - \frac{2230977459891449027497816048622221433631004765028}{10304114959369388661001101189788551324262500342899}$

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

Class group and class number

$C_{8}\times C_{16}$, which has order $128$ (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:  $7$
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:  \( 141428679.325 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1139}) \), 4.0.22054457.1, 8.0.8268784250602433.3

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
17Data not computed
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$