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Properties

Label 16.0.12566407444...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 5^{8}\cdot 17^{8}$
Root discriminant $135.27$
Ramified primes $2, 5, 17$
Class number $1958400$ (GRH)
Class group $[4, 4, 60, 2040]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![82125187631, -28954052376, 37498043644, -18047994784, 11477990842, -3097554280, 1292149168, -232686904, 71059739, -8913832, 2108384, -182456, 34474, -1904, 292, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 292*x^14 - 1904*x^13 + 34474*x^12 - 182456*x^11 + 2108384*x^10 - 8913832*x^9 + 71059739*x^8 - 232686904*x^7 + 1292149168*x^6 - 3097554280*x^5 + 11477990842*x^4 - 18047994784*x^3 + 37498043644*x^2 - 28954052376*x + 82125187631)
 
gp: K = bnfinit(x^16 - 8*x^15 + 292*x^14 - 1904*x^13 + 34474*x^12 - 182456*x^11 + 2108384*x^10 - 8913832*x^9 + 71059739*x^8 - 232686904*x^7 + 1292149168*x^6 - 3097554280*x^5 + 11477990842*x^4 - 18047994784*x^3 + 37498043644*x^2 - 28954052376*x + 82125187631, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 292 x^{14} - 1904 x^{13} + 34474 x^{12} - 182456 x^{11} + 2108384 x^{10} - 8913832 x^{9} + 71059739 x^{8} - 232686904 x^{7} + 1292149168 x^{6} - 3097554280 x^{5} + 11477990842 x^{4} - 18047994784 x^{3} + 37498043644 x^{2} - 28954052376 x + 82125187631 \)

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:  \(12566407444765825894344294400000000=2^{62}\cdot 5^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.27$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2720=2^{5}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2720}(1,·)$, $\chi_{2720}(2379,·)$, $\chi_{2720}(1089,·)$, $\chi_{2720}(1291,·)$, $\chi_{2720}(1361,·)$, $\chi_{2720}(1699,·)$, $\chi_{2720}(409,·)$, $\chi_{2720}(2651,·)$, $\chi_{2720}(611,·)$, $\chi_{2720}(2449,·)$, $\chi_{2720}(681,·)$, $\chi_{2720}(339,·)$, $\chi_{2720}(1769,·)$, $\chi_{2720}(2041,·)$, $\chi_{2720}(1019,·)$, $\chi_{2720}(1971,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{17} a^{4} - \frac{2}{17} a^{3} - \frac{1}{17} a^{2} + \frac{2}{17} a + \frac{1}{17}$, $\frac{1}{17} a^{5} - \frac{5}{17} a^{3} + \frac{5}{17} a + \frac{2}{17}$, $\frac{1}{17} a^{6} + \frac{7}{17} a^{3} - \frac{5}{17} a + \frac{5}{17}$, $\frac{1}{17} a^{7} - \frac{3}{17} a^{3} + \frac{2}{17} a^{2} + \frac{8}{17} a - \frac{7}{17}$, $\frac{1}{289} a^{8} - \frac{4}{289} a^{7} + \frac{2}{289} a^{6} + \frac{8}{289} a^{5} - \frac{5}{289} a^{4} - \frac{8}{289} a^{3} + \frac{2}{289} a^{2} + \frac{4}{289} a + \frac{1}{289}$, $\frac{1}{289} a^{9} + \frac{3}{289} a^{7} - \frac{1}{289} a^{6} - \frac{7}{289} a^{5} + \frac{6}{289} a^{4} - \frac{98}{289} a^{3} + \frac{12}{289} a^{2} + \frac{8}{17} a + \frac{55}{289}$, $\frac{1}{289} a^{10} - \frac{6}{289} a^{7} + \frac{4}{289} a^{6} - \frac{1}{289} a^{5} + \frac{2}{289} a^{4} - \frac{49}{289} a^{3} + \frac{11}{289} a^{2} + \frac{77}{289} a + \frac{31}{289}$, $\frac{1}{289} a^{11} - \frac{3}{289} a^{7} - \frac{6}{289} a^{6} - \frac{1}{289} a^{5} + \frac{6}{289} a^{4} - \frac{122}{289} a^{3} + \frac{38}{289} a^{2} - \frac{98}{289} a + \frac{74}{289}$, $\frac{1}{4913} a^{12} - \frac{6}{4913} a^{11} - \frac{8}{4913} a^{10} - \frac{7}{4913} a^{9} + \frac{4}{4913} a^{8} - \frac{91}{4913} a^{7} + \frac{58}{4913} a^{6} + \frac{125}{4913} a^{5} - \frac{47}{4913} a^{4} - \frac{1829}{4913} a^{3} - \frac{603}{4913} a^{2} + \frac{567}{4913} a - \frac{1716}{4913}$, $\frac{1}{4913} a^{13} + \frac{7}{4913} a^{11} - \frac{4}{4913} a^{10} - \frac{4}{4913} a^{9} + \frac{1}{4913} a^{8} + \frac{39}{4913} a^{7} - \frac{105}{4913} a^{6} + \frac{40}{4913} a^{5} - \frac{105}{4913} a^{4} - \frac{79}{289} a^{3} + \frac{570}{4913} a^{2} + \frac{2043}{4913} a + \frac{754}{4913}$, $\frac{1}{201124593788475101400289} a^{14} - \frac{7}{201124593788475101400289} a^{13} + \frac{1142297365547990704}{11830858458145594200017} a^{12} - \frac{116514331285895051717}{201124593788475101400289} a^{11} - \frac{204014225988407678216}{201124593788475101400289} a^{10} + \frac{320615291951898316}{201124593788475101400289} a^{9} + \frac{163635889663114268736}{201124593788475101400289} a^{8} + \frac{2933405058805054166566}{201124593788475101400289} a^{7} - \frac{3792451557348165994814}{201124593788475101400289} a^{6} - \frac{1712562623229148397031}{201124593788475101400289} a^{5} - \frac{3766930288852355518381}{201124593788475101400289} a^{4} + \frac{34858874638969784345248}{201124593788475101400289} a^{3} + \frac{32325470611321057901839}{201124593788475101400289} a^{2} - \frac{74627309852144357782528}{201124593788475101400289} a - \frac{1814098529450546310863}{6487890122208874238719}$, $\frac{1}{14138644023292814004306127903793} a^{15} + \frac{35148961}{14138644023292814004306127903793} a^{14} + \frac{1198456279106523697168803631}{14138644023292814004306127903793} a^{13} + \frac{87100369249151493460722856}{14138644023292814004306127903793} a^{12} - \frac{12698760204803821078029689262}{14138644023292814004306127903793} a^{11} - \frac{22369388923521085035958480627}{14138644023292814004306127903793} a^{10} + \frac{1336606601096137614012183702}{831684942546636117900360464929} a^{9} + \frac{5557252613371188963356198760}{14138644023292814004306127903793} a^{8} - \frac{372498580281186834425299432294}{14138644023292814004306127903793} a^{7} + \frac{170542073812094891414132690112}{14138644023292814004306127903793} a^{6} + \frac{16046852859153822279642658279}{831684942546636117900360464929} a^{5} - \frac{198290403171442567247103191883}{14138644023292814004306127903793} a^{4} + \frac{4713817704198784783879224582657}{14138644023292814004306127903793} a^{3} + \frac{4675171167589462963222009790842}{14138644023292814004306127903793} a^{2} - \frac{5094065758653782550779123193278}{14138644023292814004306127903793} a + \frac{141570685476382513422153456038}{456085291073961742074391222703}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{60}\times C_{2040}$, which has order $1958400$ (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:  \( 12198.951274811623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{16})^+\), 4.4.51200.1, 8.8.2621440000.1, 8.0.112099988602880000.1, 8.0.179359981764608.32

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
5Data not computed
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$