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Properties

Label 16.0.13881687657...0409.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 89^{7}$
Root discriminant $43.04$
Ramified primes $11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5184, 9120, 12448, -21928, 15976, -4130, 4549, -4703, 1446, -597, 359, -134, 91, -17, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 - 17*x^13 + 91*x^12 - 134*x^11 + 359*x^10 - 597*x^9 + 1446*x^8 - 4703*x^7 + 4549*x^6 - 4130*x^5 + 15976*x^4 - 21928*x^3 + 12448*x^2 + 9120*x + 5184)
 
gp: K = bnfinit(x^16 - 3*x^15 + 2*x^14 - 17*x^13 + 91*x^12 - 134*x^11 + 359*x^10 - 597*x^9 + 1446*x^8 - 4703*x^7 + 4549*x^6 - 4130*x^5 + 15976*x^4 - 21928*x^3 + 12448*x^2 + 9120*x + 5184, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 2 x^{14} - 17 x^{13} + 91 x^{12} - 134 x^{11} + 359 x^{10} - 597 x^{9} + 1446 x^{8} - 4703 x^{7} + 4549 x^{6} - 4130 x^{5} + 15976 x^{4} - 21928 x^{3} + 12448 x^{2} + 9120 x + 5184 \)

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:  \(138816876576378001590580409=11^{12}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.04$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} + \frac{3}{16} a^{4} - \frac{7}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{5}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{128} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{128} a^{10} - \frac{1}{32} a^{9} + \frac{1}{16} a^{8} - \frac{21}{128} a^{7} + \frac{3}{16} a^{5} + \frac{21}{128} a^{4} + \frac{3}{16} a^{3} + \frac{9}{32} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{512} a^{14} - \frac{1}{512} a^{13} - \frac{1}{64} a^{12} + \frac{11}{512} a^{11} + \frac{1}{512} a^{10} + \frac{5}{128} a^{9} - \frac{29}{512} a^{8} + \frac{97}{512} a^{7} + \frac{7}{32} a^{6} + \frac{5}{512} a^{5} - \frac{17}{512} a^{4} - \frac{3}{128} a^{3} - \frac{13}{128} a^{2} - \frac{15}{32} a + \frac{15}{32}$, $\frac{1}{34439493135970307816448} a^{15} - \frac{2639998730609669715}{11479831045323435938816} a^{14} - \frac{28123892317900417837}{8609873283992576954112} a^{13} - \frac{708338735940600117029}{34439493135970307816448} a^{12} - \frac{141888910778899220759}{34439493135970307816448} a^{11} + \frac{3996629645256611981}{1076234160499072119264} a^{10} - \frac{905887465118565405517}{34439493135970307816448} a^{9} - \frac{976147592281068149645}{11479831045323435938816} a^{8} - \frac{265772536047794457847}{2869957761330858984704} a^{7} + \frac{4441181453149382292229}{34439493135970307816448} a^{6} + \frac{1861721211009676333495}{34439493135970307816448} a^{5} - \frac{859026920105512205869}{4304936641996288477056} a^{4} - \frac{3009588544329072889253}{8609873283992576954112} a^{3} + \frac{57838730266746087427}{1076234160499072119264} a^{2} + \frac{334673962987775445895}{2152468320998144238528} a + \frac{76445477352510212891}{179372360083178686544}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $7$
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:  \( 72547314.059 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.10769.1, 8.0.10321451129.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$11$11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$89$89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$