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Properties

Label 16.0.14111142256...0561.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 37^{12}$
Root discriminant $49.76$
Ramified primes $11, 37$
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![343, -3535, 15142, -29340, 10082, 35383, -1048, -142, 5343, -4138, 1296, -307, -8, 8, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + 8*x^13 - 8*x^12 - 307*x^11 + 1296*x^10 - 4138*x^9 + 5343*x^8 - 142*x^7 - 1048*x^6 + 35383*x^5 + 10082*x^4 - 29340*x^3 + 15142*x^2 - 3535*x + 343)
 
gp: K = bnfinit(x^16 - 3*x^15 + 2*x^14 + 8*x^13 - 8*x^12 - 307*x^11 + 1296*x^10 - 4138*x^9 + 5343*x^8 - 142*x^7 - 1048*x^6 + 35383*x^5 + 10082*x^4 - 29340*x^3 + 15142*x^2 - 3535*x + 343, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 2 x^{14} + 8 x^{13} - 8 x^{12} - 307 x^{11} + 1296 x^{10} - 4138 x^{9} + 5343 x^{8} - 142 x^{7} - 1048 x^{6} + 35383 x^{5} + 10082 x^{4} - 29340 x^{3} + 15142 x^{2} - 3535 x + 343 \)

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:  \(1411114225648575427835680561=11^{8}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.76$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{141} a^{13} - \frac{1}{47} a^{12} - \frac{5}{141} a^{11} - \frac{21}{47} a^{10} + \frac{7}{47} a^{9} - \frac{17}{141} a^{8} + \frac{12}{47} a^{7} + \frac{65}{141} a^{6} + \frac{28}{141} a^{5} + \frac{13}{47} a^{4} + \frac{61}{141} a^{3} - \frac{37}{141} a^{2} + \frac{17}{47} a - \frac{18}{47}$, $\frac{1}{382815} a^{14} + \frac{197}{382815} a^{13} - \frac{3599}{76563} a^{12} + \frac{39874}{382815} a^{11} + \frac{166162}{382815} a^{10} + \frac{250}{543} a^{9} - \frac{103333}{382815} a^{8} + \frac{45364}{127605} a^{7} - \frac{181646}{382815} a^{6} + \frac{181654}{382815} a^{5} - \frac{3414}{42535} a^{4} + \frac{11389}{25521} a^{3} + \frac{55349}{382815} a^{2} - \frac{16282}{76563} a - \frac{38201}{382815}$, $\frac{1}{332499552807003353625865095} a^{15} + \frac{7395901926330889676}{66499910561400670725173019} a^{14} + \frac{77170308832800574468472}{110833184269001117875288365} a^{13} - \frac{15797308282087683188984341}{332499552807003353625865095} a^{12} - \frac{10684874416254637613786042}{110833184269001117875288365} a^{11} + \frac{41435325032962961783521721}{332499552807003353625865095} a^{10} + \frac{45771038596439160380932697}{332499552807003353625865095} a^{9} + \frac{66180127686410733014739103}{332499552807003353625865095} a^{8} + \frac{30896446250850536821765649}{66499910561400670725173019} a^{7} - \frac{43358847166422547553139713}{110833184269001117875288365} a^{6} + \frac{158781520833888707675111816}{332499552807003353625865095} a^{5} + \frac{15212555039754813399319793}{36944394756333705958429455} a^{4} + \frac{2394772236288025878317347}{7074458570361773481401385} a^{3} - \frac{68102927287306780284884548}{332499552807003353625865095} a^{2} - \frac{338545197832436599912379}{36944394756333705958429455} a + \frac{13336602791297414560492121}{47499936115286193375123585}$

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:  \( 3048431.55304 \) (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

\(\Q(\sqrt{37}) \), 4.0.50653.1, 4.2.15059.1, 4.2.557183.1, 8.4.1015264874437.1, 8.0.1015264874437.1, 8.0.310452895489.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$37$37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$