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Properties

Label 16.0.21585269445...5616.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 43^{8}$
Root discriminant $90.86$
Ramified primes $2, 3, 43$
Class number $102000$ (GRH)
Class group $[5, 20400]$ (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![745807489, -343516544, 418672980, -160500584, 106654352, -34361912, 16097276, -4351376, 1570905, -351880, 101264, -18200, 4186, -560, 100, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 100*x^14 - 560*x^13 + 4186*x^12 - 18200*x^11 + 101264*x^10 - 351880*x^9 + 1570905*x^8 - 4351376*x^7 + 16097276*x^6 - 34361912*x^5 + 106654352*x^4 - 160500584*x^3 + 418672980*x^2 - 343516544*x + 745807489)
 
gp: K = bnfinit(x^16 - 8*x^15 + 100*x^14 - 560*x^13 + 4186*x^12 - 18200*x^11 + 101264*x^10 - 351880*x^9 + 1570905*x^8 - 4351376*x^7 + 16097276*x^6 - 34361912*x^5 + 106654352*x^4 - 160500584*x^3 + 418672980*x^2 - 343516544*x + 745807489, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 100 x^{14} - 560 x^{13} + 4186 x^{12} - 18200 x^{11} + 101264 x^{10} - 351880 x^{9} + 1570905 x^{8} - 4351376 x^{7} + 16097276 x^{6} - 34361912 x^{5} + 106654352 x^{4} - 160500584 x^{3} + 418672980 x^{2} - 343516544 x + 745807489 \)

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:  \(21585269445983519741468649455616=2^{48}\cdot 3^{8}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.86$
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:  $2, 3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2064=2^{4}\cdot 3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2064}(1,·)$, $\chi_{2064}(515,·)$, $\chi_{2064}(517,·)$, $\chi_{2064}(1031,·)$, $\chi_{2064}(1033,·)$, $\chi_{2064}(1547,·)$, $\chi_{2064}(1549,·)$, $\chi_{2064}(2063,·)$, $\chi_{2064}(85,·)$, $\chi_{2064}(601,·)$, $\chi_{2064}(1117,·)$, $\chi_{2064}(1633,·)$, $\chi_{2064}(431,·)$, $\chi_{2064}(947,·)$, $\chi_{2064}(1463,·)$, $\chi_{2064}(1979,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{111482369236598504473} a^{14} - \frac{1}{15926052748085500639} a^{13} + \frac{35798223975911111759}{111482369236598504473} a^{12} + \frac{8175394617730338483}{111482369236598504473} a^{11} + \frac{47255923522030406773}{111482369236598504473} a^{10} - \frac{51095206720616959689}{111482369236598504473} a^{9} - \frac{45732506868097428361}{111482369236598504473} a^{8} + \frac{3145537486876096116}{15926052748085500639} a^{7} - \frac{4951066255675920077}{15926052748085500639} a^{6} + \frac{23379910857920974545}{111482369236598504473} a^{5} + \frac{5533839188804212795}{111482369236598504473} a^{4} - \frac{7253320601936072250}{111482369236598504473} a^{3} + \frac{32223845367194337942}{111482369236598504473} a^{2} - \frac{35647401957342154264}{111482369236598504473} a - \frac{670527225833997700}{15926052748085500639}$, $\frac{1}{735671374215728874827795689} a^{15} + \frac{3299489}{735671374215728874827795689} a^{14} + \frac{307124152596560035042736663}{735671374215728874827795689} a^{13} + \frac{328823245484281194115401124}{735671374215728874827795689} a^{12} + \frac{146202224452488251906468543}{735671374215728874827795689} a^{11} - \frac{242759336333177191930786164}{735671374215728874827795689} a^{10} + \frac{61993852836778140633194985}{735671374215728874827795689} a^{9} + \frac{128554082725745186269052463}{735671374215728874827795689} a^{8} + \frac{49865690810176421982272142}{105095910602246982118256527} a^{7} - \frac{187159602877336805505443004}{735671374215728874827795689} a^{6} - \frac{306370242756455051583592280}{735671374215728874827795689} a^{5} + \frac{24240306022878004319032179}{105095910602246982118256527} a^{4} + \frac{51945305990330325058183094}{735671374215728874827795689} a^{3} - \frac{132691215619854201131305639}{735671374215728874827795689} a^{2} + \frac{95253774723874556872624466}{735671374215728874827795689} a + \frac{2818113017183724940783255}{6182112388367469536368031}$

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

Class group and class number

$C_{5}\times C_{20400}$, which has order $102000$ (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-258}) \), \(\Q(\sqrt{-86}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-129}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{3}, \sqrt{-86})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-43})\), \(\Q(\sqrt{2}, \sqrt{-129})\), \(\Q(\sqrt{6}, \sqrt{-43})\), \(\Q(\sqrt{2}, \sqrt{-43})\), \(\Q(\sqrt{6}, \sqrt{-86})\), 4.4.18432.1, \(\Q(\zeta_{16})^+\), 4.0.3786752.2, 4.0.34080768.2, 8.0.18148417929216.38, \(\Q(\zeta_{48})^+\), 8.0.4645994989879296.53, 8.0.4645994989879296.40, 8.0.4645994989879296.57, 8.0.1161498747469824.66, 8.0.14339490709504.11

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$