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Properties

Label 16.0.37319027463...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $39.65$
Ramified primes $3, 5, 13$
Class number $208$ (GRH)
Class group $[2, 104]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 152, 1088, 3230, 7555, 8383, 9134, 2315, 6474, 855, 1996, -219, 370, -15, 22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 22*x^14 - 15*x^13 + 370*x^12 - 219*x^11 + 1996*x^10 + 855*x^9 + 6474*x^8 + 2315*x^7 + 9134*x^6 + 8383*x^5 + 7555*x^4 + 3230*x^3 + 1088*x^2 + 152*x + 16)
 
gp: K = bnfinit(x^16 - x^15 + 22*x^14 - 15*x^13 + 370*x^12 - 219*x^11 + 1996*x^10 + 855*x^9 + 6474*x^8 + 2315*x^7 + 9134*x^6 + 8383*x^5 + 7555*x^4 + 3230*x^3 + 1088*x^2 + 152*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 22 x^{14} - 15 x^{13} + 370 x^{12} - 219 x^{11} + 1996 x^{10} + 855 x^{9} + 6474 x^{8} + 2315 x^{7} + 9134 x^{6} + 8383 x^{5} + 7555 x^{4} + 3230 x^{3} + 1088 x^{2} + 152 x + 16 \)

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:  \(37319027463036582275390625=3^{8}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.65$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(64,·)$, $\chi_{195}(1,·)$, $\chi_{195}(194,·)$, $\chi_{195}(131,·)$, $\chi_{195}(8,·)$, $\chi_{195}(73,·)$, $\chi_{195}(14,·)$, $\chi_{195}(79,·)$, $\chi_{195}(83,·)$, $\chi_{195}(148,·)$, $\chi_{195}(47,·)$, $\chi_{195}(112,·)$, $\chi_{195}(116,·)$, $\chi_{195}(181,·)$, $\chi_{195}(122,·)$, $\chi_{195}(187,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} + \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{3}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6832} a^{12} + \frac{9}{3416} a^{11} + \frac{365}{3416} a^{10} - \frac{317}{6832} a^{9} - \frac{4}{61} a^{8} + \frac{127}{854} a^{7} - \frac{3}{976} a^{6} - \frac{561}{3416} a^{5} + \frac{83}{3416} a^{4} - \frac{871}{6832} a^{3} + \frac{263}{854} a^{2} + \frac{190}{427} a + \frac{331}{854}$, $\frac{1}{6832} a^{13} - \frac{3}{976} a^{11} + \frac{317}{3416} a^{10} - \frac{293}{6832} a^{9} + \frac{135}{1708} a^{8} - \frac{1229}{6832} a^{7} - \frac{93}{854} a^{6} - \frac{1415}{6832} a^{5} + \frac{419}{3416} a^{4} + \frac{1129}{6832} a^{3} - \frac{85}{244} a^{2} - \frac{52}{427} a - \frac{407}{854}$, $\frac{1}{13664} a^{14} - \frac{1}{13664} a^{12} + \frac{5}{488} a^{11} - \frac{211}{13664} a^{10} + \frac{89}{6832} a^{9} + \frac{59}{13664} a^{8} + \frac{78}{427} a^{7} + \frac{1581}{13664} a^{6} + \frac{113}{488} a^{5} - \frac{1529}{13664} a^{4} - \frac{2641}{6832} a^{3} + \frac{459}{1708} a^{2} + \frac{361}{1708} a + \frac{321}{854}$, $\frac{1}{252487646272736} a^{15} - \frac{81575825}{7890238946023} a^{14} + \frac{769274079}{36069663753248} a^{13} - \frac{1222171313}{63121911568184} a^{12} - \frac{35266109839}{4139141742176} a^{11} + \frac{805569754137}{9017415938312} a^{10} - \frac{1944511138533}{252487646272736} a^{9} - \frac{894585491985}{7890238946023} a^{8} + \frac{50716372151355}{252487646272736} a^{7} + \frac{13534451851359}{63121911568184} a^{6} - \frac{53044949618041}{252487646272736} a^{5} + \frac{13537723235963}{63121911568184} a^{4} - \frac{3652024289927}{31560955784092} a^{3} + \frac{13244661183801}{31560955784092} a^{2} + \frac{2708746940937}{7890238946023} a + \frac{162394140775}{1127176992289}$

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

Class group and class number

$C_{2}\times C_{104}$, which has order $208$ (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:  \( \frac{338956626409}{36069663753248} a^{15} - \frac{47681578125}{4508707969156} a^{14} + \frac{7515382212485}{36069663753248} a^{13} - \frac{754578187495}{4508707969156} a^{12} + \frac{126404786385895}{36069663753248} a^{11} - \frac{45117193664137}{18034831876624} a^{10} + \frac{691823617577905}{36069663753248} a^{9} + \frac{6275048236910}{1127176992289} a^{8} + \frac{2191277838906415}{36069663753248} a^{7} + \frac{265168896865}{18478311349} a^{6} + \frac{3105601000723861}{36069663753248} a^{5} + \frac{1237294688841905}{18034831876624} a^{4} + \frac{1179778690335745}{18034831876624} a^{3} + \frac{26945925799755}{1127176992289} a^{2} + \frac{10993603275760}{1127176992289} a + \frac{3084568479081}{2254353984578} \) (order $6$)
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:  \( 136143.590528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{13})\), 4.4.274625.2, 4.0.2471625.1, 4.4.274625.1, 4.0.2471625.2, 8.0.1445900625.1, 8.0.6108930140625.3, 8.0.6108930140625.1, 8.8.75418890625.1, 8.0.6108930140625.4, 8.0.6108930140625.5, 8.0.6108930140625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$