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Properties

Label 16.0.37038789304...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 79^{8}$
Root discriminant $29.72$
Ramified primes $5, 79$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![101, -1019, 4722, -11431, 17574, -19304, 15113, -8154, 3538, -1931, 1308, -508, 115, -42, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 42*x^13 + 115*x^12 - 508*x^11 + 1308*x^10 - 1931*x^9 + 3538*x^8 - 8154*x^7 + 15113*x^6 - 19304*x^5 + 17574*x^4 - 11431*x^3 + 4722*x^2 - 1019*x + 101)
 
gp: K = bnfinit(x^16 - 8*x^15 + 26*x^14 - 42*x^13 + 115*x^12 - 508*x^11 + 1308*x^10 - 1931*x^9 + 3538*x^8 - 8154*x^7 + 15113*x^6 - 19304*x^5 + 17574*x^4 - 11431*x^3 + 4722*x^2 - 1019*x + 101, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 26 x^{14} - 42 x^{13} + 115 x^{12} - 508 x^{11} + 1308 x^{10} - 1931 x^{9} + 3538 x^{8} - 8154 x^{7} + 15113 x^{6} - 19304 x^{5} + 17574 x^{4} - 11431 x^{3} + 4722 x^{2} - 1019 x + 101 \)

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:  \(370387893043593994140625=5^{12}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.72$
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:  $5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{11} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} - \frac{1}{10} a^{6} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{13} + \frac{2}{5} a^{11} - \frac{1}{2} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{10} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{165971628890} a^{14} - \frac{7}{165971628890} a^{13} - \frac{6554516703}{165971628890} a^{12} + \frac{39327100309}{165971628890} a^{11} - \frac{60409941119}{165971628890} a^{10} - \frac{58448714071}{165971628890} a^{9} + \frac{3777319673}{33194325778} a^{8} + \frac{43857479091}{165971628890} a^{7} + \frac{10565098837}{33194325778} a^{6} + \frac{5769411169}{165971628890} a^{5} - \frac{64012313149}{165971628890} a^{4} + \frac{4857593279}{165971628890} a^{3} - \frac{36892396493}{165971628890} a^{2} + \frac{60794205143}{165971628890} a + \frac{4091278281}{165971628890}$, $\frac{1}{39667219304710} a^{15} + \frac{56}{19833609652355} a^{14} + \frac{345263161901}{19833609652355} a^{13} - \frac{652481962787}{19833609652355} a^{12} - \frac{447573026752}{3966721930471} a^{11} + \frac{691646497524}{19833609652355} a^{10} - \frac{1791941736253}{19833609652355} a^{9} + \frac{548183478259}{19833609652355} a^{8} + \frac{3034264662343}{19833609652355} a^{7} + \frac{1917811555806}{19833609652355} a^{6} + \frac{1409944388183}{3966721930471} a^{5} - \frac{1129717783281}{3966721930471} a^{4} + \frac{9399020190804}{19833609652355} a^{3} + \frac{7694014267304}{19833609652355} a^{2} - \frac{5558930232468}{19833609652355} a - \frac{10334488088189}{39667219304710}$

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:  \( 78703.2144954 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1975.1, 8.0.121719003125.1, 8.2.7703734375.1, 8.2.1540746875.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.4.3.1$x^{4} + 158$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
79.4.3.1$x^{4} + 158$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$