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Properties

Label 16.0.14367274730...1281.5
Degree $16$
Signature $[0, 8]$
Discriminant $29^{12}\cdot 67^{8}$
Root discriminant $102.29$
Ramified primes $29, 67$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![165230275, 65807565, -36133339, -17591757, 7124578, 4806966, -678604, -680824, 104169, 63465, -16438, -4365, 1379, 197, -57, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 57*x^14 + 197*x^13 + 1379*x^12 - 4365*x^11 - 16438*x^10 + 63465*x^9 + 104169*x^8 - 680824*x^7 - 678604*x^6 + 4806966*x^5 + 7124578*x^4 - 17591757*x^3 - 36133339*x^2 + 65807565*x + 165230275)
 
gp: K = bnfinit(x^16 - 4*x^15 - 57*x^14 + 197*x^13 + 1379*x^12 - 4365*x^11 - 16438*x^10 + 63465*x^9 + 104169*x^8 - 680824*x^7 - 678604*x^6 + 4806966*x^5 + 7124578*x^4 - 17591757*x^3 - 36133339*x^2 + 65807565*x + 165230275, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 57 x^{14} + 197 x^{13} + 1379 x^{12} - 4365 x^{11} - 16438 x^{10} + 63465 x^{9} + 104169 x^{8} - 680824 x^{7} - 678604 x^{6} + 4806966 x^{5} + 7124578 x^{4} - 17591757 x^{3} - 36133339 x^{2} + 65807565 x + 165230275 \)

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:  \(143672747301451241912563249451281=29^{12}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.29$
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:  $29, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{145} a^{12} + \frac{3}{145} a^{11} + \frac{2}{145} a^{9} + \frac{42}{145} a^{8} - \frac{18}{145} a^{7} + \frac{31}{145} a^{6} + \frac{37}{145} a^{5} + \frac{11}{145} a^{4} + \frac{39}{145} a^{3} - \frac{4}{145} a^{2} + \frac{6}{29} a + \frac{13}{29}$, $\frac{1}{725} a^{13} + \frac{2}{725} a^{12} - \frac{32}{725} a^{11} + \frac{31}{725} a^{10} + \frac{69}{725} a^{9} - \frac{263}{725} a^{8} - \frac{9}{725} a^{7} - \frac{23}{725} a^{6} + \frac{119}{725} a^{5} - \frac{59}{725} a^{4} + \frac{189}{725} a^{3} - \frac{82}{725} a^{2} + \frac{238}{725} a - \frac{13}{145}$, $\frac{1}{725} a^{14} - \frac{1}{725} a^{12} + \frac{11}{145} a^{11} + \frac{7}{725} a^{10} - \frac{41}{725} a^{9} + \frac{247}{725} a^{8} - \frac{8}{29} a^{7} - \frac{8}{29} a^{6} - \frac{307}{725} a^{5} + \frac{112}{725} a^{4} - \frac{51}{145} a^{3} - \frac{318}{725} a^{2} - \frac{216}{725} a + \frac{46}{145}$, $\frac{1}{146525702752687986093732934322272992670195025} a^{15} + \frac{44506953779517756647866060310929050155029}{146525702752687986093732934322272992670195025} a^{14} + \frac{187898308025625220643939041755450143584}{146525702752687986093732934322272992670195025} a^{13} + \frac{387410776475493428411727591290496426466666}{146525702752687986093732934322272992670195025} a^{12} - \frac{7041043084275661388563797421578106562149568}{146525702752687986093732934322272992670195025} a^{11} + \frac{285815785219648132322392103413217033990242}{146525702752687986093732934322272992670195025} a^{10} + \frac{26637626528516246346362533725578049512991}{1422579638375611515473135284682262064759175} a^{9} - \frac{51509825676224035428296725215428725532641792}{146525702752687986093732934322272992670195025} a^{8} - \frac{2659783543350095619803861979879758821042815}{5861028110107519443749317372890919706807801} a^{7} + \frac{2540929833128018413941569725071698906252843}{146525702752687986093732934322272992670195025} a^{6} - \frac{14453802852364550187600851338115947941221286}{146525702752687986093732934322272992670195025} a^{5} - \frac{23160303465918007798164494102023529165848742}{146525702752687986093732934322272992670195025} a^{4} - \frac{14904352359682781335935565639839359554833123}{146525702752687986093732934322272992670195025} a^{3} - \frac{68453982561186455806823243554257416149984473}{146525702752687986093732934322272992670195025} a^{2} - \frac{47771940192189420512069048514836438786971794}{146525702752687986093732934322272992670195025} a - \frac{30236905022567771922356801654916829962137}{1274136545675547705162895081063243414523435}$

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

Class group and class number

$C_{4}$, 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:  \( -1 \) (order $2$)
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:  \( 1528503003.68 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 4.2.56347.1, 4.2.1634063.1, 8.0.413322645348029.1, 8.4.413322645348029.1, 8.0.2670161887969.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.8.6.1$x^{8} - 16147 x^{4} + 93083904$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$