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Properties

Label 16.0.55002674780...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{8}\cdot 29^{8}$
Root discriminant $96.33$
Ramified primes $2, 5, 29$
Class number $204000$ (GRH)
Class group $[5, 10, 4080]$ (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![702991681, -407897504, 502723444, -203161896, 136884272, -46753560, 23236220, -7244720, 2702073, -646472, 183344, -31416, 6874, -784, 132, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 132*x^14 - 784*x^13 + 6874*x^12 - 31416*x^11 + 183344*x^10 - 646472*x^9 + 2702073*x^8 - 7244720*x^7 + 23236220*x^6 - 46753560*x^5 + 136884272*x^4 - 203161896*x^3 + 502723444*x^2 - 407897504*x + 702991681)
 
gp: K = bnfinit(x^16 - 8*x^15 + 132*x^14 - 784*x^13 + 6874*x^12 - 31416*x^11 + 183344*x^10 - 646472*x^9 + 2702073*x^8 - 7244720*x^7 + 23236220*x^6 - 46753560*x^5 + 136884272*x^4 - 203161896*x^3 + 502723444*x^2 - 407897504*x + 702991681, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 132 x^{14} - 784 x^{13} + 6874 x^{12} - 31416 x^{11} + 183344 x^{10} - 646472 x^{9} + 2702073 x^{8} - 7244720 x^{7} + 23236220 x^{6} - 46753560 x^{5} + 136884272 x^{4} - 203161896 x^{3} + 502723444 x^{2} - 407897504 x + 702991681 \)

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:  \(55002674780385421400473600000000=2^{48}\cdot 5^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.33$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2320=2^{4}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{2320}(1,·)$, $\chi_{2320}(579,·)$, $\chi_{2320}(581,·)$, $\chi_{2320}(1159,·)$, $\chi_{2320}(1161,·)$, $\chi_{2320}(1739,·)$, $\chi_{2320}(1741,·)$, $\chi_{2320}(2319,·)$, $\chi_{2320}(349,·)$, $\chi_{2320}(929,·)$, $\chi_{2320}(1509,·)$, $\chi_{2320}(231,·)$, $\chi_{2320}(2089,·)$, $\chi_{2320}(811,·)$, $\chi_{2320}(1391,·)$, $\chi_{2320}(1971,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{15} a^{8} - \frac{4}{15} a^{7} - \frac{2}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{5} a^{3} - \frac{2}{15} a - \frac{4}{15}$, $\frac{1}{15} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{3} a^{5} + \frac{2}{15} a^{4} + \frac{1}{5} a^{3} - \frac{2}{15} a^{2} + \frac{1}{5} a - \frac{1}{15}$, $\frac{1}{1905} a^{10} - \frac{1}{381} a^{9} - \frac{38}{1905} a^{8} + \frac{182}{1905} a^{7} - \frac{2}{381} a^{6} - \frac{628}{1905} a^{5} + \frac{358}{1905} a^{4} + \frac{553}{1905} a^{3} + \frac{328}{1905} a^{2} - \frac{247}{635} a - \frac{187}{381}$, $\frac{1}{1905} a^{11} - \frac{21}{635} a^{9} - \frac{8}{1905} a^{8} + \frac{60}{127} a^{7} - \frac{226}{635} a^{6} - \frac{877}{1905} a^{5} + \frac{146}{635} a^{4} - \frac{239}{635} a^{3} + \frac{899}{1905} a^{2} - \frac{166}{381} a - \frac{173}{381}$, $\frac{1}{295275} a^{12} - \frac{2}{98425} a^{11} - \frac{37}{295275} a^{10} + \frac{16}{19685} a^{9} + \frac{1172}{98425} a^{8} - \frac{1038}{19685} a^{7} - \frac{145151}{295275} a^{6} - \frac{33176}{98425} a^{5} + \frac{145663}{295275} a^{4} + \frac{52726}{295275} a^{3} + \frac{18548}{98425} a^{2} + \frac{834}{98425} a - \frac{4394}{9525}$, $\frac{1}{295275} a^{13} - \frac{73}{295275} a^{11} + \frac{6}{98425} a^{10} + \frac{1652}{98425} a^{9} + \frac{1842}{98425} a^{8} + \frac{56704}{295275} a^{7} - \frac{28203}{98425} a^{6} + \frac{27809}{59055} a^{5} + \frac{40879}{295275} a^{4} + \frac{33}{127} a^{3} + \frac{13697}{98425} a^{2} - \frac{121202}{295275} a + \frac{737}{3175}$, $\frac{1}{66868846319028075} a^{14} - \frac{1}{9552692331289725} a^{13} + \frac{106529078641}{66868846319028075} a^{12} - \frac{127834894351}{13373769263805615} a^{11} - \frac{1332772000183}{66868846319028075} a^{10} + \frac{12522959325169}{66868846319028075} a^{9} - \frac{494968365248519}{66868846319028075} a^{8} + \frac{1897704785854043}{66868846319028075} a^{7} + \frac{1517049633679294}{66868846319028075} a^{6} - \frac{3711162767724106}{22289615439676025} a^{5} - \frac{10407865336516727}{22289615439676025} a^{4} + \frac{1087972471818513}{4457923087935205} a^{3} + \frac{12499512519681167}{66868846319028075} a^{2} + \frac{10607541119546953}{66868846319028075} a + \frac{81992034969676}{308151365525475}$, $\frac{1}{421662172938144106587675} a^{15} + \frac{3152897}{421662172938144106587675} a^{14} - \frac{100324110794019652}{421662172938144106587675} a^{13} + \frac{39138098653103959}{60237453276877729512525} a^{12} + \frac{2010369607043491639}{60237453276877729512525} a^{11} + \frac{2895009247280165104}{20079151092292576504175} a^{10} - \frac{42658303660178053109}{60237453276877729512525} a^{9} - \frac{9465014809347768233144}{421662172938144106587675} a^{8} - \frac{66061895430069012980498}{140554057646048035529225} a^{7} - \frac{16830287027029331539718}{60237453276877729512525} a^{6} - \frac{1978811413377270580969}{4015830218458515300835} a^{5} - \frac{1449723999763428710761}{60237453276877729512525} a^{4} + \frac{10615600593738103519798}{60237453276877729512525} a^{3} - \frac{17445827301105788256104}{60237453276877729512525} a^{2} - \frac{210544018287472652682523}{421662172938144106587675} a - \frac{837019104455011049084}{1943143654092829984275}$

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

Class group and class number

$C_{5}\times C_{10}\times C_{4080}$, which has order $204000$ (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

$C_2^2\times C_4$ (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-58}) \), \(\Q(\sqrt{-145}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{-290}) \), \(\Q(\sqrt{10}, \sqrt{-58})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-29})\), \(\Q(\sqrt{2}, \sqrt{-29})\), \(\Q(\sqrt{5}, \sqrt{-58})\), \(\Q(\sqrt{2}, \sqrt{-145})\), \(\Q(\sqrt{5}, \sqrt{-29})\), 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.0.43059200.3, 4.0.1722368.6, 8.0.28970229760000.62, 8.8.2621440000.1, 8.0.1854094704640000.93, 8.0.7416378818560000.44, 8.0.11866206109696.2, 8.0.7416378818560000.41, 8.0.7416378818560000.32

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$