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Properties

Label 16.0.12251765391...000.64
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $135.06$
Ramified primes $2, 3, 5, 19$
Class number $5939200$ (GRH)
Class group $[8, 16, 46400]$ (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![204097777726, -54121458632, 60672375788, -13574268328, 7990422842, -1506813672, 609155508, -95871272, 29390151, -3775144, 917820, -92040, 18070, -1288, 204, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 204*x^14 - 1288*x^13 + 18070*x^12 - 92040*x^11 + 917820*x^10 - 3775144*x^9 + 29390151*x^8 - 95871272*x^7 + 609155508*x^6 - 1506813672*x^5 + 7990422842*x^4 - 13574268328*x^3 + 60672375788*x^2 - 54121458632*x + 204097777726)
 
gp: K = bnfinit(x^16 - 8*x^15 + 204*x^14 - 1288*x^13 + 18070*x^12 - 92040*x^11 + 917820*x^10 - 3775144*x^9 + 29390151*x^8 - 95871272*x^7 + 609155508*x^6 - 1506813672*x^5 + 7990422842*x^4 - 13574268328*x^3 + 60672375788*x^2 - 54121458632*x + 204097777726, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 204 x^{14} - 1288 x^{13} + 18070 x^{12} - 92040 x^{11} + 917820 x^{10} - 3775144 x^{9} + 29390151 x^{8} - 95871272 x^{7} + 609155508 x^{6} - 1506813672 x^{5} + 7990422842 x^{4} - 13574268328 x^{3} + 60672375788 x^{2} - 54121458632 x + 204097777726 \)

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:  \(12251765391792386665453977600000000=2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.06$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4560=2^{4}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4560}(1,·)$, $\chi_{4560}(1331,·)$, $\chi_{4560}(4559,·)$, $\chi_{4560}(4369,·)$, $\chi_{4560}(3419,·)$, $\chi_{4560}(3421,·)$, $\chi_{4560}(3611,·)$, $\chi_{4560}(2471,·)$, $\chi_{4560}(2089,·)$, $\chi_{4560}(2279,·)$, $\chi_{4560}(3229,·)$, $\chi_{4560}(1139,·)$, $\chi_{4560}(1141,·)$, $\chi_{4560}(2281,·)$, $\chi_{4560}(191,·)$, $\chi_{4560}(949,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23} a^{11} + \frac{6}{23} a^{10} + \frac{6}{23} a^{9} - \frac{3}{23} a^{8} - \frac{4}{23} a^{7} + \frac{1}{23} a^{6} - \frac{3}{23} a^{5} - \frac{9}{23} a^{4} - \frac{7}{23} a^{3} + \frac{8}{23} a^{2} - \frac{8}{23} a + \frac{6}{23}$, $\frac{1}{23} a^{12} - \frac{7}{23} a^{10} + \frac{7}{23} a^{9} - \frac{9}{23} a^{8} + \frac{2}{23} a^{7} - \frac{9}{23} a^{6} + \frac{9}{23} a^{5} + \frac{1}{23} a^{4} + \frac{4}{23} a^{3} - \frac{10}{23} a^{2} + \frac{8}{23} a + \frac{10}{23}$, $\frac{1}{23} a^{13} + \frac{3}{23} a^{10} + \frac{10}{23} a^{9} + \frac{4}{23} a^{8} + \frac{9}{23} a^{7} - \frac{7}{23} a^{6} + \frac{3}{23} a^{5} + \frac{10}{23} a^{4} + \frac{10}{23} a^{3} - \frac{5}{23} a^{2} - \frac{4}{23}$, $\frac{1}{38172124832866311923743} a^{14} - \frac{7}{38172124832866311923743} a^{13} - \frac{3772257284175616109}{537635561026286083433} a^{12} - \frac{52675998370157621116}{38172124832866311923743} a^{11} + \frac{925016731174271059896}{2245419107815665407279} a^{10} - \frac{416261164496496313910}{38172124832866311923743} a^{9} - \frac{11469461482946390074863}{38172124832866311923743} a^{8} - \frac{5313066309034238325590}{38172124832866311923743} a^{7} + \frac{212680523937275429124}{2245419107815665407279} a^{6} + \frac{3758746529575987425605}{38172124832866311923743} a^{5} - \frac{14107996509730451624447}{38172124832866311923743} a^{4} + \frac{3018381636153011202403}{38172124832866311923743} a^{3} + \frac{14749382411043366855277}{38172124832866311923743} a^{2} + \frac{10675819035233187910738}{38172124832866311923743} a - \frac{10246277994358326113958}{38172124832866311923743}$, $\frac{1}{185669482391935571261269418201} a^{15} + \frac{2431996}{185669482391935571261269418201} a^{14} + \frac{2457995790983674670608727906}{185669482391935571261269418201} a^{13} - \frac{483164406665551482240419547}{185669482391935571261269418201} a^{12} + \frac{143282238311372282403661915}{10921734258349151250662906953} a^{11} - \frac{52379157090131964151752458663}{185669482391935571261269418201} a^{10} + \frac{67298649107235524480330331107}{185669482391935571261269418201} a^{9} - \frac{32256236481690874985308578473}{185669482391935571261269418201} a^{8} + \frac{544076421763937679452848930}{10921734258349151250662906953} a^{7} + \frac{43388031794563122680212915682}{185669482391935571261269418201} a^{6} + \frac{24278567561198739301369492144}{185669482391935571261269418201} a^{5} + \frac{11929788760348834292711182717}{185669482391935571261269418201} a^{4} - \frac{58870510450735828547662737911}{185669482391935571261269418201} a^{3} - \frac{66640464237262551204928394396}{185669482391935571261269418201} a^{2} - \frac{48188935313516938768099435810}{185669482391935571261269418201} a - \frac{14504332053930508855903169}{10921734258349151250662906953}$

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

Class group and class number

$C_{8}\times C_{16}\times C_{46400}$, which has order $5939200$ (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:  \( 11964.310642723332 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-570}) \), \(\Q(\sqrt{-190}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}, \sqrt{-95})\), \(\Q(\sqrt{2}, \sqrt{-285})\), \(\Q(\sqrt{6}, \sqrt{-190})\), \(\Q(\sqrt{2}, \sqrt{-95})\), \(\Q(\sqrt{6}, \sqrt{-95})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-190})\), 4.0.166348800.2, \(\Q(\zeta_{16})^+\), 4.4.18432.1, 4.0.18483200.2, 8.0.432373800960000.273, 8.0.110687693045760000.20, 8.0.110687693045760000.9, 8.0.27671923261440000.184, 8.0.341628682240000.90, 8.0.110687693045760000.18, \(\Q(\zeta_{48})^+\)

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 R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$