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Properties

Label 16.0.17612050268...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $18.42$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 10, -8, -12, 72, -52, -60, 187, -60, -52, 72, -12, -8, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 8*x^13 - 12*x^12 + 72*x^11 - 52*x^10 - 60*x^9 + 187*x^8 - 60*x^7 - 52*x^6 + 72*x^5 - 12*x^4 - 8*x^3 + 10*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 8*x^13 - 12*x^12 + 72*x^11 - 52*x^10 - 60*x^9 + 187*x^8 - 60*x^7 - 52*x^6 + 72*x^5 - 12*x^4 - 8*x^3 + 10*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 12 x^{12} + 72 x^{11} - 52 x^{10} - 60 x^{9} + 187 x^{8} - 60 x^{7} - 52 x^{6} + 72 x^{5} - 12 x^{4} - 8 x^{3} + 10 x^{2} - 4 x + 1 \)

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:  \(176120502681600000000=2^{36}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.42$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{2}{9} a^{6} - \frac{4}{9} a^{5} - \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} - \frac{4}{9} a - \frac{1}{9}$, $\frac{1}{21033} a^{14} + \frac{802}{21033} a^{13} - \frac{928}{21033} a^{12} - \frac{938}{21033} a^{11} - \frac{866}{7011} a^{10} + \frac{974}{21033} a^{9} + \frac{262}{2337} a^{8} + \frac{8881}{21033} a^{7} - \frac{3109}{7011} a^{6} - \frac{6037}{21033} a^{5} + \frac{3029}{7011} a^{4} - \frac{7949}{21033} a^{3} - \frac{7939}{21033} a^{2} - \frac{6209}{21033} a + \frac{4675}{21033}$, $\frac{1}{63099} a^{15} + \frac{880}{63099} a^{13} + \frac{8}{3321} a^{12} - \frac{499}{63099} a^{11} - \frac{4708}{63099} a^{10} - \frac{569}{63099} a^{9} - \frac{542}{3321} a^{8} + \frac{5309}{63099} a^{7} + \frac{21524}{63099} a^{6} - \frac{28895}{63099} a^{5} - \frac{18305}{63099} a^{4} - \frac{19862}{63099} a^{3} + \frac{22925}{63099} a^{2} + \frac{9172}{21033} a + \frac{22568}{63099}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{8972}{21033} a^{15} - \frac{1174}{779} a^{14} + \frac{3992}{1107} a^{13} - \frac{40868}{21033} a^{12} - \frac{114776}{21033} a^{11} + \frac{578536}{21033} a^{10} - \frac{204124}{21033} a^{9} - \frac{558517}{21033} a^{8} + \frac{1322764}{21033} a^{7} + \frac{33112}{21033} a^{6} - \frac{238312}{21033} a^{5} + \frac{390692}{21033} a^{4} + \frac{5144}{21033} a^{3} - \frac{26726}{21033} a^{2} + \frac{24140}{7011} a - \frac{3575}{21033} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10573.7624663 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), 8.0.3317760000.8, 8.0.132710400.1 x2, 8.0.1474560000.1 x2

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 8 siblings: 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$