Properties

Label 16.0.3345005787800625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{4}\cdot 13^{8}$
Root discriminant $9.34$
Ramified primes $3, 5, 13$
Class number $1$
Class group Trivial
Galois Group $D_8:C_2$ (as 16T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 10, -20, 28, -20, 1, 21, -28, 12, 6, -16, 21, -19, 12, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*x^2 - 4*x + 1)
gp: K = bnfinit(x^16 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 5 x^{15} \) \(\mathstrut +\mathstrut 12 x^{14} \) \(\mathstrut -\mathstrut 19 x^{13} \) \(\mathstrut +\mathstrut 21 x^{12} \) \(\mathstrut -\mathstrut 16 x^{11} \) \(\mathstrut +\mathstrut 6 x^{10} \) \(\mathstrut +\mathstrut 12 x^{9} \) \(\mathstrut -\mathstrut 28 x^{8} \) \(\mathstrut +\mathstrut 21 x^{7} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut -\mathstrut 20 x^{5} \) \(\mathstrut +\mathstrut 28 x^{4} \) \(\mathstrut -\mathstrut 20 x^{3} \) \(\mathstrut +\mathstrut 10 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3345005787800625=3^{8}\cdot 5^{4}\cdot 13^{8}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.34$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4201259} a^{15} - \frac{844160}{4201259} a^{14} + \frac{1138268}{4201259} a^{13} - \frac{476410}{4201259} a^{12} - \frac{1634204}{4201259} a^{11} + \frac{273623}{4201259} a^{10} + \frac{795002}{4201259} a^{9} - \frac{1897}{4201259} a^{8} + \frac{682328}{4201259} a^{7} + \frac{2016081}{4201259} a^{6} - \frac{1049503}{4201259} a^{5} - \frac{1487939}{4201259} a^{4} + \frac{743343}{4201259} a^{3} - \frac{867204}{4201259} a^{2} + \frac{2016916}{4201259} a - \frac{107421}{4201259}$

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:  \( -\frac{8947}{8033} a^{15} + \frac{40788}{8033} a^{14} - \frac{87386}{8033} a^{13} + \frac{122437}{8033} a^{12} - \frac{114059}{8033} a^{11} + \frac{64231}{8033} a^{10} + \frac{1220}{8033} a^{9} - \frac{121765}{8033} a^{8} + \frac{194988}{8033} a^{7} - \frac{72428}{8033} a^{6} - \frac{87250}{8033} a^{5} + \frac{166072}{8033} a^{4} - \frac{161088}{8033} a^{3} + \frac{64577}{8033} a^{2} - \frac{24285}{8033} a + \frac{11501}{8033} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{796875}{4201259} a^{15} - \frac{1213956}{4201259} a^{14} - \frac{2908118}{4201259} a^{13} + \frac{11752044}{4201259} a^{12} - \frac{21469083}{4201259} a^{11} + \frac{23193579}{4201259} a^{10} - \frac{15833414}{4201259} a^{9} + \frac{9183883}{4201259} a^{8} + \frac{15788997}{4201259} a^{7} - \frac{43108574}{4201259} a^{6} + \frac{26127264}{4201259} a^{5} + \frac{7333168}{4201259} a^{4} - \frac{30267134}{4201259} a^{3} + \frac{32911705}{4201259} a^{2} - \frac{12108058}{4201259} a + \frac{3744009}{4201259} \),  \( \frac{1962229}{4201259} a^{15} - \frac{9047969}{4201259} a^{14} + \frac{18754684}{4201259} a^{13} - \frac{24384095}{4201259} a^{12} + \frac{20878732}{4201259} a^{11} - \frac{9914533}{4201259} a^{10} - \frac{1902350}{4201259} a^{9} + \frac{25174615}{4201259} a^{8} - \frac{40649152}{4201259} a^{7} + \frac{6299933}{4201259} a^{6} + \frac{28920728}{4201259} a^{5} - \frac{28919017}{4201259} a^{4} + \frac{24494445}{4201259} a^{3} - \frac{8502428}{4201259} a^{2} + \frac{2068879}{4201259} a - \frac{3236120}{4201259} \),  \( \frac{3231593}{4201259} a^{15} - \frac{11650482}{4201259} a^{14} + \frac{19187251}{4201259} a^{13} - \frac{20263098}{4201259} a^{12} + \frac{9588762}{4201259} a^{11} + \frac{3390968}{4201259} a^{10} - \frac{14200458}{4201259} a^{9} + \frac{41317550}{4201259} a^{8} - \frac{32800164}{4201259} a^{7} - \frac{20568620}{4201259} a^{6} + \frac{38421018}{4201259} a^{5} - \frac{34519996}{4201259} a^{4} + \frac{15573192}{4201259} a^{3} + \frac{12043755}{4201259} a^{2} - \frac{2591707}{4201259} a + \frac{4878258}{4201259} \),  \( \frac{4648396}{4201259} a^{15} - \frac{18262619}{4201259} a^{14} + \frac{33827233}{4201259} a^{13} - \frac{41914424}{4201259} a^{12} + \frac{32910217}{4201259} a^{11} - \frac{14701024}{4201259} a^{10} - \frac{6318493}{4201259} a^{9} + \frac{55051796}{4201259} a^{8} - \frac{64352529}{4201259} a^{7} - \frac{3934792}{4201259} a^{6} + \frac{42617461}{4201259} a^{5} - \frac{47419252}{4201259} a^{4} + \frac{43968314}{4201259} a^{3} - \frac{12198061}{4201259} a^{2} + \frac{8116329}{4201259} a - \frac{3110789}{4201259} \),  \( \frac{173680}{4201259} a^{15} - \frac{2373477}{4201259} a^{14} + \frac{8345254}{4201259} a^{13} - \frac{15897831}{4201259} a^{12} + \frac{21111097}{4201259} a^{11} - \frac{18604204}{4201259} a^{10} + \frac{9972843}{4201259} a^{9} + \frac{2428501}{4201259} a^{8} - \frac{23393127}{4201259} a^{7} + \frac{28425538}{4201259} a^{6} - \frac{6059325}{4201259} a^{5} - \frac{14206948}{4201259} a^{4} + \frac{28531983}{4201259} a^{3} - \frac{26063124}{4201259} a^{2} + \frac{5397978}{4201259} a - \frac{3289320}{4201259} \),  \( \frac{1440473}{4201259} a^{15} - \frac{6691533}{4201259} a^{14} + \frac{14769575}{4201259} a^{13} - \frac{22096870}{4201259} a^{12} + \frac{22704388}{4201259} a^{11} - \frac{15575701}{4201259} a^{10} + \frac{3938985}{4201259} a^{9} + \frac{19247364}{4201259} a^{8} - \frac{34689460}{4201259} a^{7} + \frac{19371376}{4201259} a^{6} + \frac{4504900}{4201259} a^{5} - \frac{25864966}{4201259} a^{4} + \frac{32652499}{4201259} a^{3} - \frac{19407763}{4201259} a^{2} + \frac{12203739}{4201259} a - \frac{479904}{4201259} \),  \( \frac{1130760}{4201259} a^{15} - \frac{3713023}{4201259} a^{14} + \frac{6015181}{4201259} a^{13} - \frac{7338843}{4201259} a^{12} + \frac{6048556}{4201259} a^{11} - \frac{3976834}{4201259} a^{10} + \frac{469513}{4201259} a^{9} + \frac{10194147}{4201259} a^{8} - \frac{7804811}{4201259} a^{7} - \frac{4413315}{4201259} a^{6} + \frac{2019968}{4201259} a^{5} - \frac{6906874}{4201259} a^{4} + \frac{13447586}{4201259} a^{3} - \frac{4738145}{4201259} a^{2} + \frac{7091787}{4201259} a - \frac{569752}{4201259} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 22.9306095521 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_8:C_2$ (as 16T47):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), 4.0.117.1 x2, 4.2.507.1 x2, \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.2313441.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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$5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$