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Properties

Label 16.8.80706182379...3125.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 11^{2}\cdot 19^{2}\cdot 29^{9}\cdot 571^{2}$
Root discriminant $64.07$
Ramified primes $5, 11, 19, 29, 571$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1702

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![95216, -36008, -286748, -219878, 179631, 176175, -39872, -52783, 2731, 10784, 611, -1558, -13, 144, -16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 16*x^14 + 144*x^13 - 13*x^12 - 1558*x^11 + 611*x^10 + 10784*x^9 + 2731*x^8 - 52783*x^7 - 39872*x^6 + 176175*x^5 + 179631*x^4 - 219878*x^3 - 286748*x^2 - 36008*x + 95216)
 
gp: K = bnfinit(x^16 - 6*x^15 - 16*x^14 + 144*x^13 - 13*x^12 - 1558*x^11 + 611*x^10 + 10784*x^9 + 2731*x^8 - 52783*x^7 - 39872*x^6 + 176175*x^5 + 179631*x^4 - 219878*x^3 - 286748*x^2 - 36008*x + 95216, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 16 x^{14} + 144 x^{13} - 13 x^{12} - 1558 x^{11} + 611 x^{10} + 10784 x^{9} + 2731 x^{8} - 52783 x^{7} - 39872 x^{6} + 176175 x^{5} + 179631 x^{4} - 219878 x^{3} - 286748 x^{2} - 36008 x + 95216 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(80706182379542447070058203125=5^{8}\cdot 11^{2}\cdot 19^{2}\cdot 29^{9}\cdot 571^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.07$
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:  $5, 11, 19, 29, 571$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{11} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{1092} a^{14} + \frac{1}{91} a^{13} - \frac{1}{273} a^{12} - \frac{16}{91} a^{11} - \frac{5}{52} a^{10} + \frac{103}{273} a^{9} - \frac{37}{1092} a^{8} + \frac{163}{546} a^{7} - \frac{233}{1092} a^{6} + \frac{93}{364} a^{5} - \frac{29}{546} a^{4} - \frac{449}{1092} a^{3} - \frac{25}{364} a^{2} + \frac{79}{273} a + \frac{27}{91}$, $\frac{1}{750215587879322180501427603481280952} a^{15} + \frac{39329064614576171141474587005545}{125035931313220363416904600580213492} a^{14} - \frac{3537153712600721644105670166948971}{62517965656610181708452300290106746} a^{13} - \frac{2722596590834005368337309995858592}{31258982828305090854226150145053373} a^{12} - \frac{1123768867332653635237642270581283}{107173655411331740071632514783040136} a^{11} - \frac{102374631729309919689857405884467433}{375107793939661090250713801740640476} a^{10} + \frac{118591442288161280203578320276560559}{750215587879322180501427603481280952} a^{9} + \frac{13160089267555480623116642398424839}{62517965656610181708452300290106746} a^{8} + \frac{113122436259349284409201582099193085}{250071862626440726833809201160426984} a^{7} - \frac{171398087016026736034245756285321115}{750215587879322180501427603481280952} a^{6} + \frac{9059316434605193206603306792391687}{31258982828305090854226150145053373} a^{5} + \frac{241621943923930855000765355256649411}{750215587879322180501427603481280952} a^{4} + \frac{91488342334845387512021095167782579}{750215587879322180501427603481280952} a^{3} - \frac{75273840367395123661060340685946835}{375107793939661090250713801740640476} a^{2} - \frac{597706035581984879407404439008406}{2404537140638853142632780780388721} a - \frac{1895791361617651304340126413313853}{13396706926416467508954064347880017}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 268023448.09 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1702:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 104 conjugacy class representatives for t16n1702 are not computed
Character table for t16n1702 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.5702505625.3

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.4.2$x^{8} - 24389 x^{2} + 13438339$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
571Data not computed