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Properties

Label 16.0.11819246862...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $56.82$
Ramified primes $2, 3, 5$
Class number $3920$ (GRH)
Class group $[7, 560]$ (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![1110334, -804304, 1108840, -686544, 551472, -292528, 174976, -79760, 38313, -14984, 5980, -1960, 630, -168, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 630*x^12 - 1960*x^11 + 5980*x^10 - 14984*x^9 + 38313*x^8 - 79760*x^7 + 174976*x^6 - 292528*x^5 + 551472*x^4 - 686544*x^3 + 1108840*x^2 - 804304*x + 1110334)
 
gp: K = bnfinit(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 630*x^12 - 1960*x^11 + 5980*x^10 - 14984*x^9 + 38313*x^8 - 79760*x^7 + 174976*x^6 - 292528*x^5 + 551472*x^4 - 686544*x^3 + 1108840*x^2 - 804304*x + 1110334, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 630 x^{12} - 1960 x^{11} + 5980 x^{10} - 14984 x^{9} + 38313 x^{8} - 79760 x^{7} + 174976 x^{6} - 292528 x^{5} + 551472 x^{4} - 686544 x^{3} + 1108840 x^{2} - 804304 x + 1110334 \)

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:  \(11819246862071129702400000000=2^{62}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.82$
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:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(389,·)$, $\chi_{480}(449,·)$, $\chi_{480}(329,·)$, $\chi_{480}(269,·)$, $\chi_{480}(209,·)$, $\chi_{480}(149,·)$, $\chi_{480}(89,·)$, $\chi_{480}(29,·)$, $\chi_{480}(421,·)$, $\chi_{480}(361,·)$, $\chi_{480}(301,·)$, $\chi_{480}(241,·)$, $\chi_{480}(181,·)$, $\chi_{480}(121,·)$, $\chi_{480}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1738215262991873} a^{14} - \frac{7}{1738215262991873} a^{13} - \frac{408104127315725}{1738215262991873} a^{12} + \frac{710409500902568}{1738215262991873} a^{11} - \frac{86752640465438}{1738215262991873} a^{10} + \frac{584834618855663}{1738215262991873} a^{9} + \frac{50960884146017}{1738215262991873} a^{8} + \frac{625222734226742}{1738215262991873} a^{7} - \frac{197490428479329}{1738215262991873} a^{6} - \frac{174967869422069}{1738215262991873} a^{5} + \frac{693447156218578}{1738215262991873} a^{4} - \frac{525297094536014}{1738215262991873} a^{3} - \frac{472580850245020}{1738215262991873} a^{2} - \frac{799681883885967}{1738215262991873} a - \frac{274740500306140}{1738215262991873}$, $\frac{1}{428464847681707718881} a^{15} + \frac{123241}{428464847681707718881} a^{14} + \frac{1014709609459075371}{428464847681707718881} a^{13} - \frac{56893833996961730267}{428464847681707718881} a^{12} - \frac{66927418083803477449}{428464847681707718881} a^{11} + \frac{161709899831857492798}{428464847681707718881} a^{10} + \frac{184345857256463000892}{428464847681707718881} a^{9} - \frac{79782800044554077891}{428464847681707718881} a^{8} - \frac{97800248538381437621}{428464847681707718881} a^{7} + \frac{20359468406009243407}{428464847681707718881} a^{6} + \frac{582854142404512359}{428464847681707718881} a^{5} - \frac{39321023917653797340}{428464847681707718881} a^{4} + \frac{116303260043956625696}{428464847681707718881} a^{3} + \frac{94870461773545997897}{428464847681707718881} a^{2} - \frac{46769343804835110921}{428464847681707718881} a + \frac{84525147950119343564}{428464847681707718881}$

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

Class group and class number

$C_{7}\times C_{560}$, which has order $3920$ (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:  \( 15753.94986242651 \) (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

\(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\zeta_{16})^+\), 4.0.460800.2, 8.0.212336640000.4, 8.0.108716359680000.13, \(\Q(\zeta_{32})^+\)

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
3Data not computed
5Data not computed