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Properties

Label 16.0.28179280429...0000.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 5^{12}$
Root discriminant $38.96$
Ramified primes $2, 3, 5$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22801, -20536, 35356, -47628, 32372, -8760, 398, 932, -468, -148, 380, -180, 25, 4, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 + 4*x^13 + 25*x^12 - 180*x^11 + 380*x^10 - 148*x^9 - 468*x^8 + 932*x^7 + 398*x^6 - 8760*x^5 + 32372*x^4 - 47628*x^3 + 35356*x^2 - 20536*x + 22801)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 + 4*x^13 + 25*x^12 - 180*x^11 + 380*x^10 - 148*x^9 - 468*x^8 + 932*x^7 + 398*x^6 - 8760*x^5 + 32372*x^4 - 47628*x^3 + 35356*x^2 - 20536*x + 22801, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} + 4 x^{13} + 25 x^{12} - 180 x^{11} + 380 x^{10} - 148 x^{9} - 468 x^{8} + 932 x^{7} + 398 x^{6} - 8760 x^{5} + 32372 x^{4} - 47628 x^{3} + 35356 x^{2} - 20536 x + 22801 \)

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:  \(28179280429056000000000000=2^{44}\cdot 3^{8}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.96$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(240=2^{4}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{240}(19,·)$, $\chi_{240}(1,·)$, $\chi_{240}(139,·)$, $\chi_{240}(137,·)$, $\chi_{240}(203,·)$, $\chi_{240}(17,·)$, $\chi_{240}(83,·)$, $\chi_{240}(91,·)$, $\chi_{240}(227,·)$, $\chi_{240}(113,·)$, $\chi_{240}(233,·)$, $\chi_{240}(107,·)$, $\chi_{240}(49,·)$, $\chi_{240}(211,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55596303289} a^{14} + \frac{16382379243}{55596303289} a^{13} + \frac{20953930284}{55596303289} a^{12} + \frac{6477145158}{55596303289} a^{11} + \frac{17955751672}{55596303289} a^{10} - \frac{24412065323}{55596303289} a^{9} - \frac{19918246386}{55596303289} a^{8} - \frac{13560435651}{55596303289} a^{7} - \frac{9393549066}{55596303289} a^{6} + \frac{12087710323}{55596303289} a^{5} + \frac{4688673615}{55596303289} a^{4} + \frac{5911424417}{55596303289} a^{3} - \frac{26399649497}{55596303289} a^{2} + \frac{26761162343}{55596303289} a + \frac{122407760}{368187439}$, $\frac{1}{238121384763772503024721} a^{15} - \frac{325682038}{1576962813005115914071} a^{14} - \frac{58472315091159780161786}{238121384763772503024721} a^{13} - \frac{42744824106939050964894}{238121384763772503024721} a^{12} - \frac{17090011930379281566809}{238121384763772503024721} a^{11} + \frac{67161471700856513011824}{238121384763772503024721} a^{10} + \frac{41496318655772378520907}{238121384763772503024721} a^{9} - \frac{85350038742924799364142}{238121384763772503024721} a^{8} + \frac{102741143127180109340461}{238121384763772503024721} a^{7} + \frac{49401750535699846830356}{238121384763772503024721} a^{6} - \frac{109320842214432228236295}{238121384763772503024721} a^{5} - \frac{60415989759252443052684}{238121384763772503024721} a^{4} + \frac{63870021587060321745550}{238121384763772503024721} a^{3} + \frac{16732124151306198874329}{238121384763772503024721} a^{2} - \frac{43231624262340546872328}{238121384763772503024721} a + \frac{294013733053906168877}{1576962813005115914071}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( 16694.3932435 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.2048.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.51200.2, 4.4.72000.1, \(\Q(\zeta_{15})^+\), 4.0.2304000.2, 4.0.2304000.1, 8.0.2621440000.1, 8.8.5184000000.1, 8.0.5308416000000.7

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
5Data not computed