Properties

Label 16.0.5960322509765625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 61^{2}$
Root discriminant $9.68$
Ramified primes $3, 5, 61$
Class number $1$
Class group Trivial
Galois Group $C_2\times C_2\wr C_4$ (as 16T261)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 6, -50, 160, -300, 374, -315, 159, -10, -56, 45, -10, -10, 11, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*x^2 + 1)
gp: K = bnfinit(x^16 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*x^2 + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 5 x^{15} \) \(\mathstrut +\mathstrut 11 x^{14} \) \(\mathstrut -\mathstrut 10 x^{13} \) \(\mathstrut -\mathstrut 10 x^{12} \) \(\mathstrut +\mathstrut 45 x^{11} \) \(\mathstrut -\mathstrut 56 x^{10} \) \(\mathstrut -\mathstrut 10 x^{9} \) \(\mathstrut +\mathstrut 159 x^{8} \) \(\mathstrut -\mathstrut 315 x^{7} \) \(\mathstrut +\mathstrut 374 x^{6} \) \(\mathstrut -\mathstrut 300 x^{5} \) \(\mathstrut +\mathstrut 160 x^{4} \) \(\mathstrut -\mathstrut 50 x^{3} \) \(\mathstrut +\mathstrut 6 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(5960322509765625=3^{8}\cdot 5^{12}\cdot 61^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.68$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 61$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{673721} a^{15} - \frac{249772}{673721} a^{14} + \frac{259698}{673721} a^{13} - \frac{153659}{673721} a^{12} - \frac{243043}{673721} a^{11} - \frac{162237}{673721} a^{10} - \frac{174543}{673721} a^{9} - \frac{56997}{673721} a^{8} + \frac{245128}{673721} a^{7} + \frac{184105}{673721} a^{6} + \frac{126252}{673721} a^{5} - \frac{72179}{673721} a^{4} - \frac{167786}{673721} a^{3} - \frac{61551}{673721} a^{2} - \frac{230876}{673721} a + \frac{78060}{673721}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( -\frac{74212}{673721} a^{15} + \frac{667512}{673721} a^{14} - \frac{1592492}{673721} a^{13} + \frac{1287504}{673721} a^{12} + \frac{1869667}{673721} a^{11} - \frac{6872957}{673721} a^{10} + \frac{7636101}{673721} a^{9} + \frac{3609531}{673721} a^{8} - \frac{24552371}{673721} a^{7} + \frac{41358601}{673721} a^{6} - \frac{41746179}{673721} a^{5} + \frac{26067396}{673721} a^{4} - \frac{8061542}{673721} a^{3} - \frac{679289}{673721} a^{2} + \frac{1044682}{673721} a - \frac{335562}{673721} \) (order $30$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{218781}{673721} a^{15} - \frac{1205064}{673721} a^{14} + \frac{2769929}{673721} a^{13} - \frac{2360384}{673721} a^{12} - \frac{3129263}{673721} a^{11} + \frac{12071045}{673721} a^{10} - \frac{13660223}{673721} a^{9} - \frac{5348436}{673721} a^{8} + \frac{43601791}{673721} a^{7} - \frac{77211895}{673721} a^{6} + \frac{81171774}{673721} a^{5} - \frac{54618681}{673721} a^{4} + \frac{20858891}{673721} a^{3} - \frac{1875146}{673721} a^{2} - \frac{397623}{673721} a - \frac{108769}{673721} \),  \( \frac{482684}{673721} a^{15} - \frac{1943703}{673721} a^{14} + \frac{3582498}{673721} a^{13} - \frac{2164471}{673721} a^{12} - \frac{5340613}{673721} a^{11} + \frac{15578189}{673721} a^{10} - \frac{14450503}{673721} a^{9} - \frac{11596170}{673721} a^{8} + \frac{59095259}{673721} a^{7} - \frac{100993951}{673721} a^{6} + \frac{109551278}{673721} a^{5} - \frac{81034604}{673721} a^{4} + \frac{39459604}{673721} a^{3} - \frac{10713762}{673721} a^{2} + \frac{713147}{673721} a + \frac{466115}{673721} \),  \( \frac{230251}{673721} a^{15} - \frac{754491}{673721} a^{14} + \frac{964285}{673721} a^{13} + \frac{319906}{673721} a^{12} - \frac{2974975}{673721} a^{11} + \frac{4619126}{673721} a^{10} - \frac{568922}{673721} a^{9} - \frac{9636982}{673721} a^{8} + \frac{20201983}{673721} a^{7} - \frac{23071479}{673721} a^{6} + \frac{18126011}{673721} a^{5} - \frac{10043116}{673721} a^{4} + \frac{5678785}{673721} a^{3} - \frac{3826671}{673721} a^{2} + \frac{2546792}{673721} a - \frac{135778}{673721} \),  \( \frac{454175}{673721} a^{15} - \frac{2424725}{673721} a^{14} + \frac{5393448}{673721} a^{13} - \frac{4728866}{673721} a^{12} - \frac{5648211}{673721} a^{11} + \frac{23109088}{673721} a^{10} - \frac{27308121}{673721} a^{9} - \frac{8315144}{673721} a^{8} + \frac{82829275}{673721} a^{7} - \frac{153907044}{673721} a^{6} + \frac{169885482}{673721} a^{5} - \frac{121924408}{673721} a^{4} + \frac{52252277}{673721} a^{3} - \frac{8978345}{673721} a^{2} - \frac{1518302}{673721} a - \frac{319683}{673721} \),  \( \frac{429524}{673721} a^{15} - \frac{1757651}{673721} a^{14} + \frac{3253829}{673721} a^{13} - \frac{1845435}{673721} a^{12} - \frac{5122350}{673721} a^{11} + \frac{14447146}{673721} a^{10} - \frac{12882793}{673721} a^{9} - \frac{12032708}{673721} a^{8} + \frac{55160035}{673721} a^{7} - \frac{90764969}{673721} a^{6} + \frac{94107977}{673721} a^{5} - \frac{64670755}{673721} a^{4} + \frac{28117788}{673721} a^{3} - \frac{6209452}{673721} a^{2} + \frac{232129}{673721} a - \frac{429567}{673721} \),  \( \frac{107622}{673721} a^{15} - \frac{168005}{673721} a^{14} - \frac{97529}{673721} a^{13} + \frac{740489}{673721} a^{12} - \frac{903363}{673721} a^{11} - \frac{116978}{673721} a^{10} + \frac{2717060}{673721} a^{9} - \frac{3943755}{673721} a^{8} + \frac{1619861}{673721} a^{7} + \frac{2982305}{673721} a^{6} - \frac{5502152}{673721} a^{5} + \frac{4670839}{673721} a^{4} - \frac{1068371}{673721} a^{3} - \frac{216850}{673721} a^{2} - \frac{506392}{673721} a + \frac{346171}{673721} \),  \( \frac{750447}{673721} a^{15} - \frac{3381232}{673721} a^{14} + \frac{6353662}{673721} a^{13} - \frac{3565260}{673721} a^{12} - \frac{10573195}{673721} a^{11} + \frac{28843137}{673721} a^{10} - \frac{24687857}{673721} a^{9} - \frac{26303930}{673721} a^{8} + \frac{109912015}{673721} a^{7} - \frac{172740553}{673721} a^{6} + \frac{168480664}{673721} a^{5} - \frac{106467252}{673721} a^{4} + \frac{39661771}{673721} a^{3} - \frac{7188747}{673721} a^{2} + \frac{1301719}{673721} a - \frac{821851}{673721} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 161.821613678 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times C_2\wr C_4$ (as 16T261):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 128
The 26 conjugacy class representatives for $C_2\times C_2\wr C_4$
Character table for $C_2\times C_2\wr C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.4.77203125.1, 8.4.77203125.2, \(\Q(\zeta_{15})\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$