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Properties

Label 16.0.490184257318560000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{12}\cdot 5^{4}\cdot 7^{8}$
Root discriminant $12.75$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $C_2^4:C_2^2$ (as 16T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 20, -80, 239, -314, 246, 9, -200, 216, -87, -38, 80, -59, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 59*x^13 + 80*x^12 - 38*x^11 - 87*x^10 + 216*x^9 - 200*x^8 + 9*x^7 + 246*x^6 - 314*x^5 + 239*x^4 - 80*x^3 + 20*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 7*x^15 + 26*x^14 - 59*x^13 + 80*x^12 - 38*x^11 - 87*x^10 + 216*x^9 - 200*x^8 + 9*x^7 + 246*x^6 - 314*x^5 + 239*x^4 - 80*x^3 + 20*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 26 x^{14} - 59 x^{13} + 80 x^{12} - 38 x^{11} - 87 x^{10} + 216 x^{9} - 200 x^{8} + 9 x^{7} + 246 x^{6} - 314 x^{5} + 239 x^{4} - 80 x^{3} + 20 x^{2} - 4 x + 1 \)

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:  \(490184257318560000=2^{8}\cdot 3^{12}\cdot 5^{4}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.75$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{21} a^{14} + \frac{2}{21} a^{13} + \frac{1}{21} a^{12} - \frac{10}{21} a^{11} - \frac{4}{21} a^{10} + \frac{2}{7} a^{9} - \frac{1}{21} a^{8} + \frac{5}{21} a^{7} - \frac{10}{21} a^{5} + \frac{2}{21} a^{4} - \frac{2}{7} a^{3} + \frac{1}{21} a^{2} - \frac{3}{7} a + \frac{8}{21}$, $\frac{1}{3410802801} a^{15} - \frac{21715640}{3410802801} a^{14} + \frac{6281380}{487257543} a^{13} - \frac{19894547}{162419181} a^{12} - \frac{1383001402}{3410802801} a^{11} - \frac{1213293472}{3410802801} a^{10} + \frac{703176568}{3410802801} a^{9} + \frac{1423831964}{3410802801} a^{8} + \frac{802941959}{3410802801} a^{7} + \frac{33146194}{3410802801} a^{6} + \frac{32331230}{487257543} a^{5} - \frac{181390693}{487257543} a^{4} + \frac{227732161}{1136934267} a^{3} + \frac{701548408}{3410802801} a^{2} - \frac{219589109}{3410802801} a - \frac{1181791013}{3410802801}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{461464}{2055939} a^{15} + \frac{2922277}{2055939} a^{14} - \frac{9849427}{2055939} a^{13} + \frac{19388090}{2055939} a^{12} - \frac{19685239}{2055939} a^{11} - \frac{4232279}{2055939} a^{10} + \frac{15567249}{685313} a^{9} - \frac{68609461}{2055939} a^{8} + \frac{28757191}{2055939} a^{7} + \frac{14749843}{685313} a^{6} - \frac{100926331}{2055939} a^{5} + \frac{66967003}{2055939} a^{4} - \frac{28134224}{2055939} a^{3} - \frac{15175229}{2055939} a^{2} + \frac{3008117}{2055939} a + \frac{494659}{2055939} \) (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:  \( 430.060191481 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2$ (as 16T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^4:C_2^2$
Character table for $C_2^4:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), 4.0.189.1 x2, 4.2.1323.1 x2, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.77792400.1, 8.0.1750329.1, 8.0.700131600.1

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 R R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$