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Properties

Label 16.0.10141545170...1216.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 7^{8}$
Root discriminant $17.80$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_2^2$ (as 16T149)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -16, 60, -144, 254, -384, 588, -900, 1207, -1300, 1096, -716, 362, -140, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 362*x^12 - 716*x^11 + 1096*x^10 - 1300*x^9 + 1207*x^8 - 900*x^7 + 588*x^6 - 384*x^5 + 254*x^4 - 144*x^3 + 60*x^2 - 16*x + 2)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 362*x^12 - 716*x^11 + 1096*x^10 - 1300*x^9 + 1207*x^8 - 900*x^7 + 588*x^6 - 384*x^5 + 254*x^4 - 144*x^3 + 60*x^2 - 16*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 140 x^{13} + 362 x^{12} - 716 x^{11} + 1096 x^{10} - 1300 x^{9} + 1207 x^{8} - 900 x^{7} + 588 x^{6} - 384 x^{5} + 254 x^{4} - 144 x^{3} + 60 x^{2} - 16 x + 2 \)

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:  \(101415451701035401216=2^{44}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.80$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2173} a^{14} - \frac{7}{2173} a^{13} + \frac{180}{2173} a^{12} - \frac{989}{2173} a^{11} - \frac{243}{2173} a^{10} - \frac{751}{2173} a^{9} - \frac{608}{2173} a^{8} + \frac{691}{2173} a^{7} - \frac{558}{2173} a^{6} + \frac{161}{2173} a^{5} - \frac{876}{2173} a^{4} + \frac{677}{2173} a^{3} + \frac{366}{2173} a^{2} - \frac{217}{2173} a - \frac{680}{2173}$, $\frac{1}{19557} a^{15} - \frac{1}{6519} a^{14} + \frac{4498}{19557} a^{13} - \frac{2987}{6519} a^{12} - \frac{2026}{19557} a^{11} + \frac{4796}{19557} a^{10} - \frac{5785}{19557} a^{9} - \frac{2029}{6519} a^{8} + \frac{8725}{19557} a^{7} + \frac{4448}{19557} a^{6} - \frac{8924}{19557} a^{5} - \frac{9346}{19557} a^{4} + \frac{11}{41} a^{3} + \frac{380}{2173} a^{2} - \frac{2689}{6519} a + \frac{5972}{19557}$

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:  \( -\frac{364724}{19557} a^{15} + \frac{911810}{6519} a^{14} - \frac{13218986}{19557} a^{13} + \frac{14812018}{6519} a^{12} - \frac{109737910}{19557} a^{11} + \frac{206034092}{19557} a^{10} - \frac{296154646}{19557} a^{9} + \frac{108350621}{6519} a^{8} - \frac{276321650}{19557} a^{7} + \frac{188679080}{19557} a^{6} - \frac{118992740}{19557} a^{5} + \frac{79826153}{19557} a^{4} - \frac{5805700}{2173} a^{3} + \frac{2896226}{2173} a^{2} - \frac{2876632}{6519} a + \frac{1421573}{19557} \) (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:  \( 18646.3078224 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T149):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), 4.0.25088.1, 4.4.25088.1, \(\Q(i, \sqrt{7})\), 8.0.314703872.5 x2, 8.4.10070523904.3 x2, 8.0.629407744.1

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
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.22.113$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 2$$8$$1$$22$$C_2^2 \wr C_2$$[2, 2, 3, 7/2]^{2}$
2.8.22.113$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 2$$8$$1$$22$$C_2^2 \wr C_2$$[2, 2, 3, 7/2]^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$