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Properties

Label 16.0.16777216000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 5^{12}$
Root discriminant $15.91$
Ramified primes $2, 5$
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![1, -8, 10, 176, 318, 156, 134, 44, -45, 48, 78, -164, 152, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 152*x^12 - 164*x^11 + 78*x^10 + 48*x^9 - 45*x^8 + 44*x^7 + 134*x^6 + 156*x^5 + 318*x^4 + 176*x^3 + 10*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 152*x^12 - 164*x^11 + 78*x^10 + 48*x^9 - 45*x^8 + 44*x^7 + 134*x^6 + 156*x^5 + 318*x^4 + 176*x^3 + 10*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 152 x^{12} - 164 x^{11} + 78 x^{10} + 48 x^{9} - 45 x^{8} + 44 x^{7} + 134 x^{6} + 156 x^{5} + 318 x^{4} + 176 x^{3} + 10 x^{2} - 8 x + 1 \)

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:  \(16777216000000000000=2^{36}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.91$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{29} a^{11} + \frac{8}{29} a^{10} - \frac{10}{29} a^{9} - \frac{8}{29} a^{8} + \frac{14}{29} a^{7} + \frac{7}{29} a^{6} + \frac{1}{29} a^{5} + \frac{2}{29} a^{4} + \frac{4}{29} a^{3} + \frac{2}{29} a^{2} - \frac{1}{29} a + \frac{9}{29}$, $\frac{1}{145} a^{12} - \frac{1}{145} a^{11} - \frac{24}{145} a^{10} - \frac{1}{29} a^{9} - \frac{1}{145} a^{8} - \frac{32}{145} a^{7} - \frac{33}{145} a^{6} + \frac{22}{145} a^{5} - \frac{14}{145} a^{4} - \frac{34}{145} a^{3} - \frac{19}{145} a^{2} - \frac{8}{29} a + \frac{6}{145}$, $\frac{1}{145} a^{13} + \frac{26}{145} a^{10} + \frac{34}{145} a^{9} + \frac{57}{145} a^{8} - \frac{1}{29} a^{7} + \frac{19}{145} a^{6} + \frac{33}{145} a^{5} + \frac{2}{145} a^{4} + \frac{47}{145} a^{3} - \frac{9}{145} a^{2} - \frac{59}{145} a - \frac{59}{145}$, $\frac{1}{8845} a^{14} + \frac{8}{8845} a^{13} + \frac{9}{8845} a^{12} + \frac{72}{8845} a^{11} - \frac{2434}{8845} a^{10} + \frac{894}{8845} a^{9} + \frac{727}{8845} a^{8} + \frac{2346}{8845} a^{7} + \frac{853}{8845} a^{6} + \frac{1534}{8845} a^{5} + \frac{917}{8845} a^{4} - \frac{3489}{8845} a^{3} + \frac{678}{8845} a^{2} + \frac{3259}{8845} a + \frac{1382}{8845}$, $\frac{1}{4069947145} a^{15} - \frac{187648}{4069947145} a^{14} + \frac{332371}{813989429} a^{13} + \frac{2244937}{813989429} a^{12} + \frac{2619437}{369995195} a^{11} - \frac{137168096}{4069947145} a^{10} - \frac{356122526}{4069947145} a^{9} - \frac{147100108}{813989429} a^{8} + \frac{76934713}{4069947145} a^{7} - \frac{335629169}{4069947145} a^{6} - \frac{1680188021}{4069947145} a^{5} - \frac{29748149}{68982155} a^{4} - \frac{1736855193}{4069947145} a^{3} + \frac{66071128}{140343005} a^{2} - \frac{830720623}{4069947145} a + \frac{394771009}{4069947145}$

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:  \( -\frac{172220600}{813989429} a^{15} + \frac{1392593530}{813989429} a^{14} - \frac{5642966290}{813989429} a^{13} + \frac{15051342165}{813989429} a^{12} - \frac{2534754170}{73999039} a^{11} + \frac{31760476427}{813989429} a^{10} - \frac{18307145800}{813989429} a^{9} - \frac{4057456195}{813989429} a^{8} + \frac{6199199890}{813989429} a^{7} - \frac{7731706815}{813989429} a^{6} - \frac{22394469822}{813989429} a^{5} - \frac{430014405}{13796431} a^{4} - \frac{54138498180}{813989429} a^{3} - \frac{26898163155}{813989429} a^{2} - \frac{3223932500}{813989429} a + \frac{457802201}{813989429} \) (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:  \( 1039.87631431 \) (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, 4.0.32000.1 x2, 4.2.8000.1 x2, 4.0.8000.2, 4.4.8000.1, 8.0.2560000.1, 8.0.1024000000.6, 8.0.1024000000.2, 8.0.1024000000.5 x2, 8.4.1024000000.6 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ 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
$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}$