Properties

Label 16.0.3091162921500672.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 193^{3}$
Root discriminant $9.29$
Ramified primes $2, 3, 193$
Class number $1$
Class group Trivial
Galois Group 16T1202

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 20, -46, 77, -104, 127, -142, 138, -118, 92, -62, 35, -18, 8, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*x + 1)
gp: K = bnfinit(x^16 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 2 x^{15} \) \(\mathstrut +\mathstrut 8 x^{14} \) \(\mathstrut -\mathstrut 18 x^{13} \) \(\mathstrut +\mathstrut 35 x^{12} \) \(\mathstrut -\mathstrut 62 x^{11} \) \(\mathstrut +\mathstrut 92 x^{10} \) \(\mathstrut -\mathstrut 118 x^{9} \) \(\mathstrut +\mathstrut 138 x^{8} \) \(\mathstrut -\mathstrut 142 x^{7} \) \(\mathstrut +\mathstrut 127 x^{6} \) \(\mathstrut -\mathstrut 104 x^{5} \) \(\mathstrut +\mathstrut 77 x^{4} \) \(\mathstrut -\mathstrut 46 x^{3} \) \(\mathstrut +\mathstrut 20 x^{2} \) \(\mathstrut -\mathstrut 6 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3091162921500672=2^{16}\cdot 3^{8}\cdot 193^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.29$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} + \frac{8}{19} a^{13} - \frac{2}{19} a^{12} - \frac{7}{19} a^{11} - \frac{3}{19} a^{9} + \frac{9}{19} a^{8} + \frac{4}{19} a^{7} - \frac{1}{19} a^{6} - \frac{8}{19} a^{5} - \frac{6}{19} a^{4} + \frac{8}{19} a^{3} + \frac{8}{19} a^{2} - \frac{2}{19} a + \frac{9}{19}$

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:  \( \frac{6292}{19} a^{15} - 420 a^{14} + \frac{44503}{19} a^{13} - \frac{80699}{19} a^{12} + \frac{161213}{19} a^{11} - 14327 a^{10} + \frac{379820}{19} a^{9} - \frac{464770}{19} a^{8} + \frac{528516}{19} a^{7} - \frac{507094}{19} a^{6} + \frac{428407}{19} a^{5} - \frac{341220}{19} a^{4} + \frac{235054}{19} a^{3} - \frac{117624}{19} a^{2} + \frac{39894}{19} a - \frac{8599}{19} \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{11035}{19} a^{15} - 736 a^{14} + \frac{78039}{19} a^{13} - \frac{141447}{19} a^{12} + \frac{282615}{19} a^{11} - 25111 a^{10} + \frac{665696}{19} a^{9} - \frac{814433}{19} a^{8} + \frac{926139}{19} a^{7} - \frac{888417}{19} a^{6} + \frac{750494}{19} a^{5} - \frac{597678}{19} a^{4} + \frac{411679}{19} a^{3} - \frac{205878}{19} a^{2} + \frac{69757}{19} a - \frac{15027}{19} \),  \( 418 a^{15} - 530 a^{14} + 2956 a^{13} - 5360 a^{12} + 10706 a^{11} - 18078 a^{10} + 25221 a^{9} - 30858 a^{8} + 35091 a^{7} - 33662 a^{6} + 28437 a^{5} - 22648 a^{4} + 15600 a^{3} - 7802 a^{2} + 2645 a - 570 \),  \( \frac{4694}{19} a^{15} - 312 a^{14} + \frac{33182}{19} a^{13} - \frac{60023}{19} a^{12} + \frac{120035}{19} a^{11} - 10657 a^{10} + \frac{282489}{19} a^{9} - \frac{345468}{19} a^{8} + \frac{392791}{19} a^{7} - \frac{376619}{19} a^{6} + \frac{318052}{19} a^{5} - \frac{253238}{19} a^{4} + \frac{174352}{19} a^{3} - \frac{87088}{19} a^{2} + \frac{29448}{19} a - \frac{6337}{19} \),  \( \frac{9936}{19} a^{15} - 662 a^{14} + \frac{70254}{19} a^{13} - \frac{127279}{19} a^{12} + \frac{254322}{19} a^{11} - 22598 a^{10} + \frac{598940}{19} a^{9} - \frac{732839}{19} a^{8} + \frac{833317}{19} a^{7} - \frac{799329}{19} a^{6} + \frac{675306}{19} a^{5} - \frac{537846}{19} a^{4} + \frac{370454}{19} a^{3} - \frac{185277}{19} a^{2} + \frac{62835}{19} a - \frac{13556}{19} \),  \( \frac{11321}{19} a^{15} - 754 a^{14} + \frac{80042}{19} a^{13} - \frac{144983}{19} a^{12} + \frac{289714}{19} a^{11} - 25741 a^{10} + \frac{682204}{19} a^{9} - \frac{834678}{19} a^{8} + \frac{949038}{19} a^{7} - \frac{910268}{19} a^{6} + \frac{768916}{19} a^{5} - \frac{612352}{19} a^{4} + \frac{421719}{19} a^{3} - \frac{210848}{19} a^{2} + \frac{71446}{19} a - \frac{15398}{19} \),  \( \frac{212}{19} a^{15} - 15 a^{14} + \frac{1506}{19} a^{13} - \frac{2818}{19} a^{12} + \frac{5546}{19} a^{11} - 496 a^{10} + \frac{13196}{19} a^{9} - \frac{16104}{19} a^{8} + \frac{18347}{19} a^{7} - \frac{17635}{19} a^{6} + \frac{14872}{19} a^{5} - \frac{11817}{19} a^{4} + \frac{8175}{19} a^{3} - \frac{4061}{19} a^{2} + \frac{1343}{19} a - \frac{277}{19} \),  \( \frac{1635}{19} a^{15} - 108 a^{14} + \frac{11541}{19} a^{13} - \frac{20826}{19} a^{12} + \frac{41641}{19} a^{11} - 3698 a^{10} + \frac{97866}{19} a^{9} - \frac{119710}{19} a^{8} + \frac{136025}{19} a^{7} - \frac{130322}{19} a^{6} + \frac{110002}{19} a^{5} - \frac{87577}{19} a^{4} + \frac{60219}{19} a^{3} - \frac{30031}{19} a^{2} + \frac{10125}{19} a - \frac{2176}{19} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 43.8698041698 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1202:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1024
The 106 conjugacy class representatives for t16n1202 are not computed
Character table for t16n1202 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 8.0.4002048.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$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}$
$193$$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.4.3.1$x^{4} - 193$$4$$1$$3$$C_4$$[\ ]_{4}$