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Properties

Label 16.0.67184640000...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{14}$
Root discriminant $20.03$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_8:C_2^2$ (as 16T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 270, 387, -180, -627, -300, 469, 380, -60, -240, -31, 60, 33, -10, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 10*x^13 + 33*x^12 + 60*x^11 - 31*x^10 - 240*x^9 - 60*x^8 + 380*x^7 + 469*x^6 - 300*x^5 - 627*x^4 - 180*x^3 + 387*x^2 + 270*x + 81)
 
gp: K = bnfinit(x^16 - 8*x^14 - 10*x^13 + 33*x^12 + 60*x^11 - 31*x^10 - 240*x^9 - 60*x^8 + 380*x^7 + 469*x^6 - 300*x^5 - 627*x^4 - 180*x^3 + 387*x^2 + 270*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 10 x^{13} + 33 x^{12} + 60 x^{11} - 31 x^{10} - 240 x^{9} - 60 x^{8} + 380 x^{7} + 469 x^{6} - 300 x^{5} - 627 x^{4} - 180 x^{3} + 387 x^{2} + 270 x + 81 \)

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:  \(671846400000000000000=2^{24}\cdot 3^{8}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.03$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{30} a^{12} + \frac{2}{15} a^{11} - \frac{1}{10} a^{10} - \frac{1}{15} a^{9} + \frac{13}{30} a^{8} + \frac{4}{15} a^{7} + \frac{1}{10} a^{6} + \frac{4}{15} a^{5} - \frac{1}{15} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{3}{10}$, $\frac{1}{90} a^{13} + \frac{1}{90} a^{11} - \frac{1}{9} a^{10} - \frac{1}{10} a^{9} + \frac{2}{5} a^{8} + \frac{41}{90} a^{7} + \frac{1}{15} a^{6} + \frac{1}{15} a^{5} + \frac{13}{45} a^{4} + \frac{1}{9} a^{3} - \frac{4}{15} a^{2} + \frac{1}{6} a + \frac{2}{5}$, $\frac{1}{90} a^{14} + \frac{1}{90} a^{12} - \frac{1}{9} a^{11} - \frac{1}{10} a^{10} + \frac{1}{15} a^{9} + \frac{41}{90} a^{8} + \frac{1}{15} a^{7} - \frac{4}{15} a^{6} + \frac{13}{45} a^{5} + \frac{4}{9} a^{4} + \frac{2}{5} a^{3} - \frac{1}{6} a^{2} + \frac{2}{5} a$, $\frac{1}{127311676830} a^{15} + \frac{136702309}{42437225610} a^{14} - \frac{252409036}{63655838415} a^{13} + \frac{1596628901}{127311676830} a^{12} - \frac{3473563651}{21218612805} a^{11} + \frac{1811728697}{42437225610} a^{10} - \frac{9118737839}{63655838415} a^{9} - \frac{5134198847}{14145741870} a^{8} - \frac{4060795211}{42437225610} a^{7} + \frac{18957907396}{63655838415} a^{6} + \frac{27629543888}{63655838415} a^{5} - \frac{631618379}{1414574187} a^{4} - \frac{17603992817}{42437225610} a^{3} + \frac{836400079}{14145741870} a^{2} - \frac{4013465003}{14145741870} a - \frac{2841161}{2357623645}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8277.91651278 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-6}) \), 4.2.24000.1 x2, 4.0.9000.1 x2, \(\Q(\sqrt{-6}, \sqrt{10})\), 8.2.8640000000.1 x2, 8.2.8640000000.2 x2, 8.0.5184000000.12

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$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$5.8.7.4$x^{8} + 40$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.4$x^{8} + 40$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$