Properties

Label 16.8.309...833.1
Degree $16$
Signature $[8, 4]$
Discriminant $3.090\times 10^{36}$
Root discriminant $190.82$
Ramified primes $47, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (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 - 114*x^14 + 900*x^13 - 2931*x^12 - 21510*x^11 + 319146*x^10 - 2247134*x^9 + 8400619*x^8 - 16164554*x^7 + 84204306*x^6 - 557059493*x^5 + 1493038571*x^4 - 895327284*x^3 - 2914304292*x^2 + 5483733054*x - 2806619339)
 
gp: K = bnfinit(x^16 - 3*x^15 - 114*x^14 + 900*x^13 - 2931*x^12 - 21510*x^11 + 319146*x^10 - 2247134*x^9 + 8400619*x^8 - 16164554*x^7 + 84204306*x^6 - 557059493*x^5 + 1493038571*x^4 - 895327284*x^3 - 2914304292*x^2 + 5483733054*x - 2806619339, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2806619339, 5483733054, -2914304292, -895327284, 1493038571, -557059493, 84204306, -16164554, 8400619, -2247134, 319146, -21510, -2931, 900, -114, -3, 1]);
 

\( x^{16} - 3 x^{15} - 114 x^{14} + 900 x^{13} - 2931 x^{12} - 21510 x^{11} + 319146 x^{10} - 2247134 x^{9} + 8400619 x^{8} - 16164554 x^{7} + 84204306 x^{6} - 557059493 x^{5} + 1493038571 x^{4} - 895327284 x^{3} - 2914304292 x^{2} + 5483733054 x - 2806619339 \)

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:  \(3090063795858197568077038853384324833\)\(\medspace = 47^{4}\cdot 97^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $190.82$
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:  $47, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1413628494353550092930460545782743559529573267555705944244895543358} a^{15} + \frac{292541661997411536372524920245085678396818197533126464495602954473}{1413628494353550092930460545782743559529573267555705944244895543358} a^{14} - \frac{31896029930867427902575104900047022454869447242432877856243778959}{706814247176775046465230272891371779764786633777852972122447771679} a^{13} + \frac{42310524275195302446695940035478873137836726103941440139339943389}{706814247176775046465230272891371779764786633777852972122447771679} a^{12} - \frac{181304477570533562630126693493633887002092135440340861176190135759}{706814247176775046465230272891371779764786633777852972122447771679} a^{11} + \frac{32688027978818429379655457624260867897300205196498149522857855091}{706814247176775046465230272891371779764786633777852972122447771679} a^{10} - \frac{417198160492527674839065151350943745339434455033855589020164300807}{1413628494353550092930460545782743559529573267555705944244895543358} a^{9} + \frac{115248458891151647591761820844041440508906961922495235420249283981}{706814247176775046465230272891371779764786633777852972122447771679} a^{8} + \frac{64133034322848074121314995966503865627174152358681431224189741307}{706814247176775046465230272891371779764786633777852972122447771679} a^{7} - \frac{236667989701052621246424447532258138434101862115526458200038952805}{706814247176775046465230272891371779764786633777852972122447771679} a^{6} + \frac{159975978705040958644970643724045357073901421225884317997638255929}{706814247176775046465230272891371779764786633777852972122447771679} a^{5} - \frac{254621623772012110161863828651973520202326936003406068587741858879}{706814247176775046465230272891371779764786633777852972122447771679} a^{4} - \frac{330159508787301176637481323147781842644031424404251701245548906966}{706814247176775046465230272891371779764786633777852972122447771679} a^{3} + \frac{132992987761909691362467447888681752189198678935904189630440971196}{706814247176775046465230272891371779764786633777852972122447771679} a^{2} + \frac{46454438672975709966034008795432768443918672802885132151624766952}{706814247176775046465230272891371779764786633777852972122447771679} a - \frac{503093205977229575602456864745924709194645955827393236366547523375}{1413628494353550092930460545782743559529573267555705944244895543358}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 459261691864 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 459261691864 \cdot 4}{2\sqrt{3090063795858197568077038853384324833}}\approx 0.208480736646374$ (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$47$47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97Data not computed