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Properties

Label 16.0.48825707677...5649.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{15}\cdot 157^{5}$
Root discriminant $53.77$
Ramified primes $13, 157$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, -36450, 131013, -256167, 275834, -265473, 153595, -41054, 7423, 4056, -2587, 741, 13, -34, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 25*x^14 - 34*x^13 + 13*x^12 + 741*x^11 - 2587*x^10 + 4056*x^9 + 7423*x^8 - 41054*x^7 + 153595*x^6 - 265473*x^5 + 275834*x^4 - 256167*x^3 + 131013*x^2 - 36450*x + 59049)
 
gp: K = bnfinit(x^16 - 6*x^15 + 25*x^14 - 34*x^13 + 13*x^12 + 741*x^11 - 2587*x^10 + 4056*x^9 + 7423*x^8 - 41054*x^7 + 153595*x^6 - 265473*x^5 + 275834*x^4 - 256167*x^3 + 131013*x^2 - 36450*x + 59049, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 25 x^{14} - 34 x^{13} + 13 x^{12} + 741 x^{11} - 2587 x^{10} + 4056 x^{9} + 7423 x^{8} - 41054 x^{7} + 153595 x^{6} - 265473 x^{5} + 275834 x^{4} - 256167 x^{3} + 131013 x^{2} - 36450 x + 59049 \)

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:  \(4882570767744501515595495649=13^{15}\cdot 157^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.77$
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:  $13, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{4}{9} a^{10} - \frac{1}{3} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{2}{9} a^{5} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{1536607190577289558108626512866230344781093} a^{15} + \frac{24209645683961063914230575437529143339861}{512202396859096519369542170955410114927031} a^{14} - \frac{90796444011872011718272235968208867882513}{1536607190577289558108626512866230344781093} a^{13} + \frac{100446644446928084787327040863477147259385}{1536607190577289558108626512866230344781093} a^{12} + \frac{119579814000815846167257635763228865780165}{1536607190577289558108626512866230344781093} a^{11} + \frac{15960767521914265168363290096535751068609}{512202396859096519369542170955410114927031} a^{10} - \frac{300351854592265385305173961898484216213559}{1536607190577289558108626512866230344781093} a^{9} - \frac{56010059777638177044580178698501761789427}{512202396859096519369542170955410114927031} a^{8} + \frac{160319824042148221198731613545387655883374}{1536607190577289558108626512866230344781093} a^{7} + \frac{186389379314048938527822379767661123914748}{1536607190577289558108626512866230344781093} a^{6} - \frac{487548058868316761015225058848400549891680}{1536607190577289558108626512866230344781093} a^{5} + \frac{83980081552078825657150261753255289947430}{170734132286365506456514056985136704975677} a^{4} + \frac{410448239738765460788959876074460185323}{1536607190577289558108626512866230344781093} a^{3} - \frac{57681162960928327191370695670839192673532}{170734132286365506456514056985136704975677} a^{2} + \frac{20866804081672499276895885263469041141761}{170734132286365506456514056985136704975677} a + \frac{308421002430112969408841051689003129481}{2107828793658833413043383419569588950317}$

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:  $7$
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:  \( 127001679.49 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.9851517169.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
13Data not computed
$157$157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.4.3.2$x^{4} - 3925$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.2.2$x^{4} - 157 x^{2} + 147894$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$