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Properties

Label 16.0.66787881998...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 149^{6}$
Root discriminant $26.70$
Ramified primes $5, 149$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2501, -14964, 35166, -41026, 23133, -154, -10546, 7987, -1842, -884, 1171, -388, -92, 67, -6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 6*x^14 + 67*x^13 - 92*x^12 - 388*x^11 + 1171*x^10 - 884*x^9 - 1842*x^8 + 7987*x^7 - 10546*x^6 - 154*x^5 + 23133*x^4 - 41026*x^3 + 35166*x^2 - 14964*x + 2501)
 
gp: K = bnfinit(x^16 - 3*x^15 - 6*x^14 + 67*x^13 - 92*x^12 - 388*x^11 + 1171*x^10 - 884*x^9 - 1842*x^8 + 7987*x^7 - 10546*x^6 - 154*x^5 + 23133*x^4 - 41026*x^3 + 35166*x^2 - 14964*x + 2501, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 6 x^{14} + 67 x^{13} - 92 x^{12} - 388 x^{11} + 1171 x^{10} - 884 x^{9} - 1842 x^{8} + 7987 x^{7} - 10546 x^{6} - 154 x^{5} + 23133 x^{4} - 41026 x^{3} + 35166 x^{2} - 14964 x + 2501 \)

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:  \(66787881998297119140625=5^{14}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.70$
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:  $5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{662} a^{13} + \frac{82}{331} a^{12} - \frac{148}{331} a^{11} + \frac{251}{662} a^{10} - \frac{86}{331} a^{9} + \frac{163}{331} a^{8} - \frac{305}{662} a^{7} + \frac{164}{331} a^{5} - \frac{69}{662} a^{4} - \frac{135}{331} a^{3} + \frac{106}{331} a^{2} + \frac{251}{662} a - \frac{7}{331}$, $\frac{1}{662} a^{14} - \frac{25}{331} a^{12} - \frac{193}{662} a^{11} - \frac{146}{331} a^{10} + \frac{34}{331} a^{9} - \frac{147}{662} a^{8} - \frac{146}{331} a^{7} + \frac{164}{331} a^{6} - \frac{239}{662} a^{5} - \frac{104}{331} a^{4} + \frac{69}{331} a^{3} - \frac{93}{662} a^{2} - \frac{67}{331} a + \frac{155}{331}$, $\frac{1}{60276464031934961515000802} a^{15} + \frac{14384460951051441562091}{30138232015967480757500401} a^{14} + \frac{11832759103330604175359}{30138232015967480757500401} a^{13} + \frac{2944194244503640925827208}{30138232015967480757500401} a^{12} + \frac{478574877380309606961363}{2739839274178861887045491} a^{11} - \frac{14076115277434862692194612}{30138232015967480757500401} a^{10} - \frac{16608593210829020648807}{30138232015967480757500401} a^{9} + \frac{12046789942952034392157859}{30138232015967480757500401} a^{8} + \frac{4039042758407509482508540}{30138232015967480757500401} a^{7} - \frac{4186637559602451745427672}{30138232015967480757500401} a^{6} - \frac{7552869404126682570848008}{30138232015967480757500401} a^{5} + \frac{4595846044765253700148434}{30138232015967480757500401} a^{4} + \frac{13616394533177477164590086}{30138232015967480757500401} a^{3} + \frac{8376596529465301574808784}{30138232015967480757500401} a^{2} - \frac{380403822793688134732707}{2739839274178861887045491} a - \frac{27816628055499288553641537}{60276464031934961515000802}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{31408882359628814141947}{5479678548357723774090982} a^{15} + \frac{77106728506800341032003}{5479678548357723774090982} a^{14} + \frac{228396337659458042625165}{5479678548357723774090982} a^{13} - \frac{1977668046113522540266641}{5479678548357723774090982} a^{12} + \frac{1830453175033177485047935}{5479678548357723774090982} a^{11} + \frac{13080229401241108546994329}{5479678548357723774090982} a^{10} - \frac{29681592701491452973296791}{5479678548357723774090982} a^{9} + \frac{12425438694469638498754345}{5479678548357723774090982} a^{8} + \frac{63921875803621012074836221}{5479678548357723774090982} a^{7} - \frac{216447522530377546415681253}{5479678548357723774090982} a^{6} + \frac{216398702177129694059384821}{5479678548357723774090982} a^{5} + \frac{113721352652663577933418163}{5479678548357723774090982} a^{4} - \frac{663161923407707258883137951}{5479678548357723774090982} a^{3} + \frac{938051238299683552053472989}{5479678548357723774090982} a^{2} - \frac{619460392324464953825859435}{5479678548357723774090982} a + \frac{78615522814862078562897165}{2739839274178861887045491} \) (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:  \( 59134.6269298 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.18625.1, 4.0.3725.1, 8.0.258433515625.1, 8.8.258433515625.1, 8.0.346890625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$149$149.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
149.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
149.8.6.2$x^{8} + 745 x^{4} + 199809$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$