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Properties

Label 16.0.39087700648...2896.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 13^{12}$
Root discriminant $19.36$
Ramified primes $2, 13$
Class number $1$
Class group Trivial
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![9, -78, 332, -898, 1679, -2216, 2026, -1136, 156, 342, -306, 88, 46, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 46*x^12 + 88*x^11 - 306*x^10 + 342*x^9 + 156*x^8 - 1136*x^7 + 2026*x^6 - 2216*x^5 + 1679*x^4 - 898*x^3 + 332*x^2 - 78*x + 9)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 46*x^12 + 88*x^11 - 306*x^10 + 342*x^9 + 156*x^8 - 1136*x^7 + 2026*x^6 - 2216*x^5 + 1679*x^4 - 898*x^3 + 332*x^2 - 78*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 46 x^{12} + 88 x^{11} - 306 x^{10} + 342 x^{9} + 156 x^{8} - 1136 x^{7} + 2026 x^{6} - 2216 x^{5} + 1679 x^{4} - 898 x^{3} + 332 x^{2} - 78 x + 9 \)

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:  \(390877006486250192896=2^{24}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.36$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{477} a^{14} - \frac{7}{477} a^{13} - \frac{4}{159} a^{12} + \frac{4}{477} a^{11} + \frac{25}{159} a^{10} + \frac{31}{477} a^{9} + \frac{59}{477} a^{8} + \frac{67}{477} a^{7} + \frac{25}{477} a^{6} - \frac{142}{477} a^{5} + \frac{158}{477} a^{4} + \frac{28}{159} a^{3} - \frac{112}{477} a^{2} - \frac{77}{159} a - \frac{8}{53}$, $\frac{1}{60579} a^{15} + \frac{56}{60579} a^{14} + \frac{61}{20193} a^{13} + \frac{668}{20193} a^{12} - \frac{1852}{20193} a^{11} + \frac{331}{20193} a^{10} + \frac{7}{53} a^{9} - \frac{4484}{60579} a^{8} - \frac{4499}{60579} a^{7} - \frac{176}{20193} a^{6} + \frac{17129}{60579} a^{5} + \frac{656}{6731} a^{4} + \frac{4253}{20193} a^{3} + \frac{2624}{60579} a^{2} - \frac{2950}{6731} a + \frac{2411}{6731}$

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:  \( \frac{27208}{60579} a^{15} - \frac{68020}{20193} a^{14} + \frac{662842}{60579} a^{13} - \frac{404521}{20193} a^{12} + \frac{235270}{20193} a^{11} + \frac{2658436}{60579} a^{10} - \frac{55145}{477} a^{9} + \frac{682622}{6731} a^{8} + \frac{749882}{6731} a^{7} - \frac{27410375}{60579} a^{6} + \frac{42547822}{60579} a^{5} - \frac{13783325}{20193} a^{4} + \frac{27639383}{60579} a^{3} - \frac{12554732}{60579} a^{2} + \frac{1264723}{20193} a - \frac{66227}{6731} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15199.5247179 \)
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

\(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{13})\), 4.2.2704.1 x2, 4.0.832.1 x2, 4.0.140608.1 x2, 4.2.35152.1 x2, 4.4.35152.1, 4.0.2197.1, 8.0.116985856.1, 8.0.19770609664.3, 8.0.1235663104.1, 8.4.19770609664.1 x2, 8.0.1235663104.2 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$