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Properties

Label 16.0.14366690463...4336.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{8}\cdot 13^{8}$
Root discriminant $21.01$
Ramified primes $2, 3, 13$
Class number $4$
Class group $[2, 2]$
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![100, -320, 424, -1600, 2788, 880, -1600, 140, 290, -72, -148, 24, 54, -4, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 - 4*x^13 + 54*x^12 + 24*x^11 - 148*x^10 - 72*x^9 + 290*x^8 + 140*x^7 - 1600*x^6 + 880*x^5 + 2788*x^4 - 1600*x^3 + 424*x^2 - 320*x + 100)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 - 4*x^13 + 54*x^12 + 24*x^11 - 148*x^10 - 72*x^9 + 290*x^8 + 140*x^7 - 1600*x^6 + 880*x^5 + 2788*x^4 - 1600*x^3 + 424*x^2 - 320*x + 100, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} - 4 x^{13} + 54 x^{12} + 24 x^{11} - 148 x^{10} - 72 x^{9} + 290 x^{8} + 140 x^{7} - 1600 x^{6} + 880 x^{5} + 2788 x^{4} - 1600 x^{3} + 424 x^{2} - 320 x + 100 \)

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:  \(1436669046353440014336=2^{28}\cdot 3^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.01$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{78} a^{13} - \frac{17}{78} a^{11} + \frac{17}{78} a^{10} - \frac{11}{78} a^{9} + \frac{11}{78} a^{8} + \frac{10}{39} a^{7} - \frac{2}{39} a^{6} + \frac{5}{13} a^{5} - \frac{4}{39} a^{4} - \frac{4}{39} a^{3} + \frac{1}{39} a^{2} + \frac{11}{39} a + \frac{6}{13}$, $\frac{1}{4134} a^{14} + \frac{1}{2067} a^{13} - \frac{173}{4134} a^{12} - \frac{251}{4134} a^{11} - \frac{991}{4134} a^{10} - \frac{401}{4134} a^{9} - \frac{51}{1378} a^{8} + \frac{19}{689} a^{7} + \frac{908}{2067} a^{6} + \frac{23}{159} a^{5} - \frac{82}{689} a^{4} - \frac{982}{2067} a^{3} + \frac{10}{159} a^{2} + \frac{274}{2067} a - \frac{222}{689}$, $\frac{1}{2261399255736270} a^{15} + \frac{57061599323}{2261399255736270} a^{14} - \frac{303630295483}{1130699627868135} a^{13} + \frac{1915243264739}{57984596300930} a^{12} - \frac{366483958616761}{2261399255736270} a^{11} - \frac{161920189449567}{753799751912090} a^{10} + \frac{287768519084287}{2261399255736270} a^{9} + \frac{227711731566859}{1130699627868135} a^{8} + \frac{42914902931513}{226139925573627} a^{7} - \frac{41537971098487}{226139925573627} a^{6} - \frac{50733711824636}{226139925573627} a^{5} + \frac{107036816995429}{226139925573627} a^{4} + \frac{410783159366354}{1130699627868135} a^{3} - \frac{8006154321224}{226139925573627} a^{2} + \frac{3221499748652}{1130699627868135} a - \frac{105961016364085}{226139925573627}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{79685535337}{13381060684830} a^{15} - \frac{1800349780187}{173953788902790} a^{14} - \frac{3480312352888}{86976894451395} a^{13} - \frac{5666925651269}{173953788902790} a^{12} + \frac{9336762776179}{28992298150465} a^{11} + \frac{21425702348987}{86976894451395} a^{10} - \frac{25511183626328}{28992298150465} a^{9} - \frac{126296795421007}{173953788902790} a^{8} + \frac{28294968337222}{17395378890279} a^{7} + \frac{8233350159064}{5798459630093} a^{6} - \frac{164889651957428}{17395378890279} a^{5} + \frac{13832282255268}{5798459630093} a^{4} + \frac{1673745126757334}{86976894451395} a^{3} - \frac{79957723111114}{17395378890279} a^{2} - \frac{19179540606248}{86976894451395} a - \frac{20445056037875}{17395378890279} \) (order $12$)
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:  \( 18089.3875494 \)
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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.0.832.1, 4.0.7488.4, \(\Q(\zeta_{12})\), 8.0.728911872.3 x2, 8.0.56070144.5, 8.0.4211490816.1 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
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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$