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Properties

Label 16.0.20392810905...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{4}\cdot 5^{12}\cdot 7^{4}$
Root discriminant $28.63$
Ramified primes $2, 3, 5, 7$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3:(C_2\times C_4)$ (as 16T68)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26896, 40672, 6704, -23872, -19576, 1072, 8896, 2256, -1896, -816, 156, 120, 32, -4, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 4*x^13 + 32*x^12 + 120*x^11 + 156*x^10 - 816*x^9 - 1896*x^8 + 2256*x^7 + 8896*x^6 + 1072*x^5 - 19576*x^4 - 23872*x^3 + 6704*x^2 + 40672*x + 26896)
 
gp: K = bnfinit(x^16 - 10*x^14 - 4*x^13 + 32*x^12 + 120*x^11 + 156*x^10 - 816*x^9 - 1896*x^8 + 2256*x^7 + 8896*x^6 + 1072*x^5 - 19576*x^4 - 23872*x^3 + 6704*x^2 + 40672*x + 26896, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} - 4 x^{13} + 32 x^{12} + 120 x^{11} + 156 x^{10} - 816 x^{9} - 1896 x^{8} + 2256 x^{7} + 8896 x^{6} + 1072 x^{5} - 19576 x^{4} - 23872 x^{3} + 6704 x^{2} + 40672 x + 26896 \)

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:  \(203928109056000000000000=2^{32}\cdot 3^{4}\cdot 5^{12}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.63$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{40} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{1}{20} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{200} a^{13} + \frac{1}{100} a^{12} + \frac{2}{25} a^{11} - \frac{3}{25} a^{10} + \frac{7}{100} a^{9} + \frac{1}{50} a^{8} - \frac{3}{25} a^{7} - \frac{6}{25} a^{5} - \frac{2}{25} a^{4} + \frac{1}{25} a^{3} - \frac{7}{25} a^{2} + \frac{8}{25} a + \frac{12}{25}$, $\frac{1}{200} a^{14} + \frac{1}{100} a^{12} - \frac{2}{25} a^{11} + \frac{3}{50} a^{10} + \frac{2}{25} a^{9} - \frac{3}{50} a^{8} - \frac{3}{50} a^{7} - \frac{6}{25} a^{6} - \frac{1}{10} a^{5} + \frac{1}{10} a^{4} + \frac{1}{25} a^{3} + \frac{2}{25} a^{2} + \frac{11}{25} a + \frac{6}{25}$, $\frac{1}{1703064234367203707800} a^{15} + \frac{2429683071041003}{1661526082309467032} a^{14} - \frac{202193209797047707}{1703064234367203707800} a^{13} - \frac{8051963539580138417}{851532117183601853900} a^{12} + \frac{6686567890153343646}{212883029295900463475} a^{11} + \frac{12570190285148392183}{425766058591800926950} a^{10} + \frac{35048078316775695181}{851532117183601853900} a^{9} - \frac{29952647174401391449}{851532117183601853900} a^{8} + \frac{73869882534394096167}{425766058591800926950} a^{7} - \frac{9844662987215279791}{85153211718360185390} a^{6} - \frac{46475492292230733356}{212883029295900463475} a^{5} - \frac{30206075829563784837}{425766058591800926950} a^{4} + \frac{5517826613166788243}{212883029295900463475} a^{3} - \frac{7086627891603827976}{212883029295900463475} a^{2} - \frac{18540174816196632416}{212883029295900463475} a - \frac{752904299254380463}{5192269007217084475}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -\frac{5652554362912655}{34061284687344074156} a^{15} + \frac{580689817879877}{1661526082309467032} a^{14} + \frac{110808001518052297}{85153211718360185390} a^{13} - \frac{632749150086898429}{340612846873440741560} a^{12} - \frac{177334673175233189}{42576605859180092695} a^{11} - \frac{2369366101480687611}{170306423436720370780} a^{10} + \frac{370889132287630639}{42576605859180092695} a^{9} + \frac{6866360472536101039}{42576605859180092695} a^{8} + \frac{2679612637354399991}{42576605859180092695} a^{7} - \frac{5789564544642229510}{8515321171836018539} a^{6} - \frac{29098523012519561358}{42576605859180092695} a^{5} + \frac{122971326663897851823}{85153211718360185390} a^{4} + \frac{113352238211501398348}{42576605859180092695} a^{3} + \frac{20840886199975033884}{42576605859180092695} a^{2} - \frac{199232465589699312146}{42576605859180092695} a - \frac{4424707689453691934}{1038453801443416895} \) (order $10$)
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:  \( 115957.767519 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:(C_2\times C_4)$ (as 16T68):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 34 conjugacy class representatives for $C_2^3:(C_2\times C_4)$
Character table for $C_2^3:(C_2\times C_4)$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.18063360000.1, 8.8.451584000000.1, 8.0.64000000.2

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

Sibling fields

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 R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$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$$[\ ]_{4}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$