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Properties

Label 16.0.26873856000...000.12
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}$
Root discriminant $16.38$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![541, 790, -72, -986, -433, 604, 506, -368, -120, 2, 158, -128, 82, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 82*x^12 - 128*x^11 + 158*x^10 + 2*x^9 - 120*x^8 - 368*x^7 + 506*x^6 + 604*x^5 - 433*x^4 - 986*x^3 - 72*x^2 + 790*x + 541)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 82*x^12 - 128*x^11 + 158*x^10 + 2*x^9 - 120*x^8 - 368*x^7 + 506*x^6 + 604*x^5 - 433*x^4 - 986*x^3 - 72*x^2 + 790*x + 541, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 82 x^{12} - 128 x^{11} + 158 x^{10} + 2 x^{9} - 120 x^{8} - 368 x^{7} + 506 x^{6} + 604 x^{5} - 433 x^{4} - 986 x^{3} - 72 x^{2} + 790 x + 541 \)

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:  \(26873856000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.38$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{29} a^{10} - \frac{5}{29} a^{9} + \frac{9}{29} a^{8} - \frac{6}{29} a^{7} - \frac{5}{29} a^{6} - \frac{14}{29} a^{5} + \frac{4}{29} a^{4} - \frac{1}{29} a^{3} - \frac{11}{29} a^{2} - \frac{1}{29} a + \frac{4}{29}$, $\frac{1}{29} a^{11} + \frac{13}{29} a^{9} + \frac{10}{29} a^{8} - \frac{6}{29} a^{7} - \frac{10}{29} a^{6} - \frac{8}{29} a^{5} - \frac{10}{29} a^{4} + \frac{13}{29} a^{3} + \frac{2}{29} a^{2} - \frac{1}{29} a - \frac{9}{29}$, $\frac{1}{29} a^{12} - \frac{12}{29} a^{9} - \frac{7}{29} a^{8} + \frac{10}{29} a^{7} - \frac{1}{29} a^{6} - \frac{2}{29} a^{5} - \frac{10}{29} a^{4} - \frac{14}{29} a^{3} - \frac{3}{29} a^{2} + \frac{4}{29} a + \frac{6}{29}$, $\frac{1}{551} a^{13} + \frac{3}{551} a^{12} + \frac{3}{551} a^{11} - \frac{2}{551} a^{10} + \frac{91}{551} a^{9} + \frac{225}{551} a^{8} - \frac{223}{551} a^{7} + \frac{2}{551} a^{6} + \frac{110}{551} a^{5} - \frac{237}{551} a^{4} - \frac{16}{551} a^{3} + \frac{94}{551} a^{2} - \frac{169}{551} a + \frac{2}{551}$, $\frac{1}{539429} a^{14} - \frac{7}{539429} a^{13} + \frac{8979}{539429} a^{12} + \frac{2020}{539429} a^{11} + \frac{5260}{539429} a^{10} + \frac{113125}{539429} a^{9} - \frac{181586}{539429} a^{8} - \frac{227516}{539429} a^{7} + \frac{266052}{539429} a^{6} + \frac{206}{1691} a^{5} - \frac{41897}{539429} a^{4} + \frac{163483}{539429} a^{3} - \frac{23909}{539429} a^{2} + \frac{73493}{539429} a + \frac{6211}{18601}$, $\frac{1}{204443591} a^{15} + \frac{182}{204443591} a^{14} - \frac{11497}{18585781} a^{13} - \frac{1009842}{204443591} a^{12} - \frac{387349}{204443591} a^{11} - \frac{1954068}{204443591} a^{10} + \frac{63048352}{204443591} a^{9} - \frac{83883975}{204443591} a^{8} - \frac{1390816}{18585781} a^{7} + \frac{88399356}{204443591} a^{6} + \frac{90666721}{204443591} a^{5} + \frac{33183793}{204443591} a^{4} + \frac{61107856}{204443591} a^{3} + \frac{41930901}{204443591} a^{2} - \frac{54556625}{204443591} a - \frac{9454479}{204443591}$

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:  \( -\frac{666308}{204443591} a^{15} + \frac{4997310}{204443591} a^{14} - \frac{18253460}{204443591} a^{13} + \frac{3895905}{18585781} a^{12} - \frac{78586880}{204443591} a^{11} + \frac{610368}{978199} a^{10} - \frac{157767755}{204443591} a^{9} + \frac{103002150}{204443591} a^{8} - \frac{37729460}{204443591} a^{7} + \frac{101766515}{204443591} a^{6} - \frac{190045228}{204443591} a^{5} + \frac{13758605}{18585781} a^{4} - \frac{9090945}{204443591} a^{3} - \frac{63312780}{204443591} a^{2} - \frac{77014375}{204443591} a + \frac{50467347}{204443591} \) (order $4$)
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:  \( 1146.87811803 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 4.2.400.1 x2, 4.0.320.1 x2, 4.0.72000.1 x2, 4.2.18000.1 x2, 4.0.18000.1, \(\Q(\zeta_{15})^+\), 8.0.2560000.1, 8.0.5184000000.15, 8.0.324000000.1, 8.0.5184000000.14 x2, 8.4.324000000.2 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$