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Properties

Label 16.8.15231185783...5001.3
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 67^{6}$
Root discriminant $57.73$
Ramified primes $17, 67$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2529311, -6103952, 3589244, -921468, 89828, 285090, -277525, 75531, 8739, -10862, 4906, -312, -274, 61, -27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 27*x^14 + 61*x^13 - 274*x^12 - 312*x^11 + 4906*x^10 - 10862*x^9 + 8739*x^8 + 75531*x^7 - 277525*x^6 + 285090*x^5 + 89828*x^4 - 921468*x^3 + 3589244*x^2 - 6103952*x + 2529311)
 
gp: K = bnfinit(x^16 - 4*x^15 - 27*x^14 + 61*x^13 - 274*x^12 - 312*x^11 + 4906*x^10 - 10862*x^9 + 8739*x^8 + 75531*x^7 - 277525*x^6 + 285090*x^5 + 89828*x^4 - 921468*x^3 + 3589244*x^2 - 6103952*x + 2529311, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 27 x^{14} + 61 x^{13} - 274 x^{12} - 312 x^{11} + 4906 x^{10} - 10862 x^{9} + 8739 x^{8} + 75531 x^{7} - 277525 x^{6} + 285090 x^{5} + 89828 x^{4} - 921468 x^{3} + 3589244 x^{2} - 6103952 x + 2529311 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15231185783695887615175635001=17^{14}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.73$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{871766122556578321344770392844215658657166125969} a^{15} - \frac{195153356285425318265128018698185788762440326976}{871766122556578321344770392844215658657166125969} a^{14} + \frac{51060896996361024785254777858592605405805880194}{871766122556578321344770392844215658657166125969} a^{13} + \frac{3228001301819912646189344160505340384926401740}{8631347748084933874700696958853620382744219069} a^{12} + \frac{52650538847210725076298413591966023098742377279}{871766122556578321344770392844215658657166125969} a^{11} + \frac{340966566112697379308046279253464105625100882333}{871766122556578321344770392844215658657166125969} a^{10} - \frac{164993360040682309893082928997595457105397214514}{871766122556578321344770392844215658657166125969} a^{9} + \frac{256812847650573603779430022072655305103217463400}{871766122556578321344770392844215658657166125969} a^{8} + \frac{228474977276595078757572246535246243976723277580}{871766122556578321344770392844215658657166125969} a^{7} - \frac{21331686399735916950842181647974807729539956677}{871766122556578321344770392844215658657166125969} a^{6} + \frac{200738745729416331009562151149929437677829838736}{871766122556578321344770392844215658657166125969} a^{5} + \frac{404088681083995925255139420088609071289881180458}{871766122556578321344770392844215658657166125969} a^{4} + \frac{128091918794876086938585110181327023365361344262}{871766122556578321344770392844215658657166125969} a^{3} - \frac{420089593527896385967036298145200167946390327455}{871766122556578321344770392844215658657166125969} a^{2} - \frac{141542859122172957298300752762513091478079347915}{871766122556578321344770392844215658657166125969} a + \frac{308133233211545127818760807365341196395205175571}{871766122556578321344770392844215658657166125969}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 49687194.265 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed