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Properties

Label 16.0.85587369363...3933.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{10}\cdot 53^{9}$
Root discriminant $41.76$
Ramified primes $11, 53$
Class number $2$ (GRH)
Class group $[2]$ (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![21101, 26465, 40434, 35362, 16697, 8765, 3093, -3539, -2463, -205, 149, 54, 26, 17, 2, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 2*x^14 + 17*x^13 + 26*x^12 + 54*x^11 + 149*x^10 - 205*x^9 - 2463*x^8 - 3539*x^7 + 3093*x^6 + 8765*x^5 + 16697*x^4 + 35362*x^3 + 40434*x^2 + 26465*x + 21101)
 
gp: K = bnfinit(x^16 - 6*x^15 + 2*x^14 + 17*x^13 + 26*x^12 + 54*x^11 + 149*x^10 - 205*x^9 - 2463*x^8 - 3539*x^7 + 3093*x^6 + 8765*x^5 + 16697*x^4 + 35362*x^3 + 40434*x^2 + 26465*x + 21101, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 2 x^{14} + 17 x^{13} + 26 x^{12} + 54 x^{11} + 149 x^{10} - 205 x^{9} - 2463 x^{8} - 3539 x^{7} + 3093 x^{6} + 8765 x^{5} + 16697 x^{4} + 35362 x^{3} + 40434 x^{2} + 26465 x + 21101 \)

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:  \(85587369363492766398473933=11^{10}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.76$
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:  $11, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{666125457320129864759926287675717} a^{15} + \frac{11105229153162585437990096538514}{666125457320129864759926287675717} a^{14} + \frac{13417951659943963700831272187964}{222041819106709954919975429225239} a^{13} + \frac{14110176646444591543750511535038}{666125457320129864759926287675717} a^{12} + \frac{63374417216609682696990294152431}{666125457320129864759926287675717} a^{11} - \frac{9707779290003497583859704886850}{666125457320129864759926287675717} a^{10} + \frac{38315555815505067951168187638829}{222041819106709954919975429225239} a^{9} + \frac{26955730552280587191165174765206}{666125457320129864759926287675717} a^{8} + \frac{28537716679722423747679574947034}{95160779617161409251418041096531} a^{7} - \frac{100353092429669473400609548264072}{222041819106709954919975429225239} a^{6} - \frac{85731251835315909745838663904045}{222041819106709954919975429225239} a^{5} + \frac{132040077865203180351680427718469}{666125457320129864759926287675717} a^{4} + \frac{273555062529166027577450817515704}{666125457320129864759926287675717} a^{3} - \frac{30858921735080623172921328393394}{666125457320129864759926287675717} a^{2} + \frac{2213687907064311431478003792389}{95160779617161409251418041096531} a + \frac{250068269941830647249278364038132}{666125457320129864759926287675717}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 863511.366972 \) (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

\(\Q(\sqrt{-11}) \), 4.0.6413.1, 8.0.2179708157.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$53$53.4.3.3$x^{4} + 106$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$