Properties

Label 16.0.6158959248447369.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 7^{4}\cdot 13^{6}$
Root discriminant $9.70$
Ramified primes $3, 7, 13$
Class number $1$
Class group Trivial
Galois Group $C_4.C_2^2:D_4$ (as 16T211)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 7, -15, 20, -15, 1, 6, 7, -24, 23, -3, -10, 6, 2, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*x^2 - 3*x + 1)
gp: K = bnfinit(x^16 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 3 x^{15} \) \(\mathstrut +\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 6 x^{13} \) \(\mathstrut -\mathstrut 10 x^{12} \) \(\mathstrut -\mathstrut 3 x^{11} \) \(\mathstrut +\mathstrut 23 x^{10} \) \(\mathstrut -\mathstrut 24 x^{9} \) \(\mathstrut +\mathstrut 7 x^{8} \) \(\mathstrut +\mathstrut 6 x^{7} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut -\mathstrut 15 x^{5} \) \(\mathstrut +\mathstrut 20 x^{4} \) \(\mathstrut -\mathstrut 15 x^{3} \) \(\mathstrut +\mathstrut 7 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6158959248447369=3^{12}\cdot 7^{4}\cdot 13^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.70$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{238663} a^{15} + \frac{107485}{238663} a^{14} - \frac{89485}{238663} a^{13} + \frac{32552}{238663} a^{12} - \frac{88877}{238663} a^{11} - \frac{495}{14039} a^{10} + \frac{21273}{238663} a^{9} - \frac{38003}{238663} a^{8} + \frac{89451}{238663} a^{7} - \frac{107187}{238663} a^{6} - \frac{98593}{238663} a^{5} + \frac{27453}{238663} a^{4} + \frac{38752}{238663} a^{3} - \frac{10378}{238663} a^{2} + \frac{405}{238663} a + \frac{95971}{238663}$

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:  \( -\frac{8884}{1717} a^{15} + \frac{19678}{1717} a^{14} - \frac{1713}{1717} a^{13} - \frac{56036}{1717} a^{12} + \frac{44856}{1717} a^{11} + \frac{3878}{101} a^{10} - \frac{155389}{1717} a^{9} + \frac{85641}{1717} a^{8} + \frac{15313}{1717} a^{7} - \frac{45251}{1717} a^{6} - \frac{46225}{1717} a^{5} + \frac{98399}{1717} a^{4} - \frac{96684}{1717} a^{3} + \frac{50196}{1717} a^{2} - \frac{16358}{1717} a + \frac{11194}{1717} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{1083357}{238663} a^{15} - \frac{2391837}{238663} a^{14} + \frac{232129}{238663} a^{13} + \frac{6797422}{238663} a^{12} - \frac{5524607}{238663} a^{11} - \frac{463280}{14039} a^{10} + \frac{19092569}{238663} a^{9} - \frac{10756428}{238663} a^{8} - \frac{1982806}{238663} a^{7} + \frac{5901129}{238663} a^{6} + \frac{5788894}{238663} a^{5} - \frac{12239163}{238663} a^{4} + \frac{11691273}{238663} a^{3} - \frac{6108917}{238663} a^{2} + \frac{2006295}{238663} a - \frac{1487988}{238663} \),  \( \frac{573866}{238663} a^{15} - \frac{1181329}{238663} a^{14} + \frac{2711}{238663} a^{13} + \frac{3435641}{238663} a^{12} - \frac{2398697}{238663} a^{11} - \frac{237207}{14039} a^{10} + \frac{9069499}{238663} a^{9} - \frac{5093907}{238663} a^{8} - \frac{421115}{238663} a^{7} + \frac{2742667}{238663} a^{6} + \frac{2814839}{238663} a^{5} - \frac{5767907}{238663} a^{4} + \frac{5803467}{238663} a^{3} - \frac{3087465}{238663} a^{2} + \frac{1151283}{238663} a - \frac{811972}{238663} \),  \( \frac{676016}{238663} a^{15} - \frac{1518557}{238663} a^{14} + \frac{141524}{238663} a^{13} + \frac{4285514}{238663} a^{12} - \frac{3398379}{238663} a^{11} - \frac{303174}{14039} a^{10} + \frac{11705127}{238663} a^{9} - \frac{6201314}{238663} a^{8} - \frac{930409}{238663} a^{7} + \frac{2494405}{238663} a^{6} + \frac{3834478}{238663} a^{5} - \frac{7166185}{238663} a^{4} + \frac{7049064}{238663} a^{3} - \frac{4252434}{238663} a^{2} + \frac{1710660}{238663} a - \frac{934373}{238663} \),  \( \frac{699650}{238663} a^{15} - \frac{1547039}{238663} a^{14} + \frac{284540}{238663} a^{13} + \frac{4169970}{238663} a^{12} - \frac{3682997}{238663} a^{11} - \frac{251361}{14039} a^{10} + \frac{12085594}{238663} a^{9} - \frac{8184651}{238663} a^{8} + \frac{509649}{238663} a^{7} + \frac{3599244}{238663} a^{6} + \frac{3038396}{238663} a^{5} - \frac{7505343}{238663} a^{4} + \frac{8595879}{238663} a^{3} - \frac{4896511}{238663} a^{2} + \frac{1974573}{238663} a - \frac{1008911}{238663} \),  \( \frac{590026}{238663} a^{15} - \frac{1451706}{238663} a^{14} + \frac{461554}{238663} a^{13} + \frac{3701372}{238663} a^{12} - \frac{4047724}{238663} a^{11} - \frac{206099}{14039} a^{10} + \frac{11791752}{238663} a^{9} - \frac{8722433}{238663} a^{8} + \frac{2580}{238663} a^{7} + \frac{4248779}{238663} a^{6} + \frac{2150158}{238663} a^{5} - \frac{7711248}{238663} a^{4} + \frac{8409368}{238663} a^{3} - \frac{4686497}{238663} a^{2} + \frac{1490845}{238663} a - \frac{752786}{238663} \),  \( \frac{1837616}{238663} a^{15} - \frac{4206996}{238663} a^{14} + \frac{693892}{238663} a^{13} + \frac{11514862}{238663} a^{12} - \frac{10195581}{238663} a^{11} - \frac{749099}{14039} a^{10} + \frac{33211903}{238663} a^{9} - \frac{20504099}{238663} a^{8} - \frac{1597782}{238663} a^{7} + \frac{9812891}{238663} a^{6} + \frac{8760770}{238663} a^{5} - \frac{21276573}{238663} a^{4} + \frac{21702277}{238663} a^{3} - \frac{12106320}{238663} a^{2} + \frac{4379180}{238663} a - \frac{2656040}{238663} \),  \( \frac{2403860}{238663} a^{15} - \frac{5308916}{238663} a^{14} + \frac{652419}{238663} a^{13} + \frac{14810016}{238663} a^{12} - \frac{12260715}{238663} a^{11} - \frac{975868}{14039} a^{10} + \frac{41713447}{238663} a^{9} - \frac{24960033}{238663} a^{8} - \frac{1715565}{238663} a^{7} + \frac{12114097}{238663} a^{6} + \frac{11740331}{238663} a^{5} - \frac{26029143}{238663} a^{4} + \frac{27602794}{238663} a^{3} - \frac{15090122}{238663} a^{2} + \frac{5546172}{238663} a - \frac{3218216}{238663} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 33.7844971841 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_4.C_2^2:D_4$ (as 16T211):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.117.1, 4.0.2457.2, 8.0.8719893.2, 8.0.1601613.1, 8.0.6036849.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$