Properties

Label 16.0.20822535093...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 31^{8}$
Root discriminant $18.62$
Ramified primes $5, 31$
Class number $3$
Class group $[3]$
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - x^13 - 8*x^12 - 46*x^11 + 93*x^10 - 78*x^9 + 80*x^8 + 271*x^7 + 434*x^6 - 866*x^5 - 47*x^4 - 776*x^3 - 960*x^2 - 512*x + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -512, -960, -776, -47, -866, 434, 271, 80, -78, 93, -46, -8, -1, 2, -2, 1]);
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - x^{13} - 8 x^{12} - 46 x^{11} + 93 x^{10} - 78 x^{9} + 80 x^{8} + 271 x^{7} + 434 x^{6} - 866 x^{5} - 47 x^{4} - 776 x^{3} - 960 x^{2} - 512 x + 4096 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(208225350937744140625=5^{12}\cdot 31^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $18.62$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $16$
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41} a^{10} + \frac{2}{41} a^{5} - \frac{3}{41}$, $\frac{1}{41} a^{11} + \frac{2}{41} a^{6} - \frac{3}{41} a$, $\frac{1}{1025} a^{12} + \frac{11}{1025} a^{11} + \frac{1}{1025} a^{10} - \frac{6}{25} a^{8} + \frac{207}{1025} a^{7} + \frac{22}{1025} a^{6} - \frac{162}{1025} a^{5} + \frac{1}{25} a^{4} + \frac{4}{25} a^{3} - \frac{44}{1025} a^{2} - \frac{64}{205} a + \frac{161}{1025}$, $\frac{1}{155800} a^{13} - \frac{29}{77900} a^{12} + \frac{421}{77900} a^{11} - \frac{1569}{155800} a^{10} - \frac{7}{475} a^{9} - \frac{21647}{77900} a^{8} - \frac{53211}{155800} a^{7} - \frac{4563}{15580} a^{6} - \frac{6532}{19475} a^{5} - \frac{173}{760} a^{4} + \frac{4809}{15580} a^{3} - \frac{38617}{77900} a^{2} - \frac{7159}{155800} a - \frac{7873}{19475}$, $\frac{1}{1246400} a^{14} - \frac{1}{623200} a^{13} - \frac{63}{623200} a^{12} - \frac{1537}{1246400} a^{11} + \frac{83}{31160} a^{10} + \frac{13973}{124640} a^{9} + \frac{46649}{249280} a^{8} - \frac{1597}{32800} a^{7} + \frac{25889}{77900} a^{6} + \frac{434639}{1246400} a^{5} + \frac{49253}{124640} a^{4} + \frac{13077}{32800} a^{3} - \frac{537583}{1246400} a^{2} + \frac{72639}{155800} a + \frac{7209}{19475}$, $\frac{1}{49856000} a^{15} - \frac{1}{4985600} a^{14} + \frac{1}{608000} a^{13} + \frac{22383}{49856000} a^{12} - \frac{2339}{389500} a^{11} - \frac{9323}{4985600} a^{10} - \frac{2571827}{49856000} a^{9} + \frac{213789}{608000} a^{8} + \frac{17478}{97375} a^{7} + \frac{1116943}{49856000} a^{6} - \frac{719431}{4985600} a^{5} + \frac{8035687}{24928000} a^{4} + \frac{38521}{1216000} a^{3} - \frac{92129}{3116000} a^{2} + \frac{367003}{779000} a + \frac{23341}{97375}$

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

Class group and class number

$C_{3}$, which has order $3$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{7}{124640} a^{15} + \frac{21}{249280} a^{14} - \frac{7}{24928} a^{13} + \frac{43}{15580} a^{12} + \frac{7}{249280} a^{11} + \frac{189}{62320} a^{10} - \frac{7}{3280} a^{9} + \frac{3017}{249280} a^{8} - \frac{15179}{124640} a^{7} + \frac{7}{124640} a^{6} - \frac{12453}{249280} a^{5} - \frac{609}{24928} a^{4} - \frac{1099}{15580} a^{3} + \frac{192809}{249280} a^{2} + \frac{336}{3895} a + \frac{896}{3895} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5085.12397787 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2^2:C_4$ (as 16T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-155}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{5}, \sqrt{-31})\), 4.2.775.1 x2, 4.0.4805.1 x2, 4.0.120125.1 x2, 4.2.3875.1 x2, \(\Q(\zeta_{5})\), 4.4.120125.1, 8.0.577200625.1, 8.0.14430015625.2, 8.0.14430015625.1, 8.0.15015625.1 x2, 8.4.14430015625.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: 8.4.14430015625.1, 8.0.15015625.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$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}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$