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Properties

Label 16.0.12543905625...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 23^{8}$
Root discriminant $27.77$
Ramified primes $3, 5, 23$
Class number $15$ (GRH)
Class group $[15]$ (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![1679616, -279936, 279936, -132192, 22032, 10584, -2808, 4998, -2129, 833, -78, 49, 17, -17, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616)
 
gp: K = bnfinit(x^16 - x^15 + 6*x^14 - 17*x^13 + 17*x^12 + 49*x^11 - 78*x^10 + 833*x^9 - 2129*x^8 + 4998*x^7 - 2808*x^6 + 10584*x^5 + 22032*x^4 - 132192*x^3 + 279936*x^2 - 279936*x + 1679616, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 6 x^{14} - 17 x^{13} + 17 x^{12} + 49 x^{11} - 78 x^{10} + 833 x^{9} - 2129 x^{8} + 4998 x^{7} - 2808 x^{6} + 10584 x^{5} + 22032 x^{4} - 132192 x^{3} + 279936 x^{2} - 279936 x + 1679616 \)

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:  \(125439056256992431640625=3^{8}\cdot 5^{12}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.77$
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:  $3, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(345=3\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{345}(1,·)$, $\chi_{345}(323,·)$, $\chi_{345}(68,·)$, $\chi_{345}(137,·)$, $\chi_{345}(139,·)$, $\chi_{345}(206,·)$, $\chi_{345}(208,·)$, $\chi_{345}(277,·)$, $\chi_{345}(22,·)$, $\chi_{345}(344,·)$, $\chi_{345}(91,·)$, $\chi_{345}(229,·)$, $\chi_{345}(298,·)$, $\chi_{345}(47,·)$, $\chi_{345}(116,·)$, $\chi_{345}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{684} a^{10} - \frac{1}{36} a^{9} - \frac{1}{3} a^{8} + \frac{1}{36} a^{7} + \frac{17}{36} a^{6} - \frac{257}{684} a^{5} + \frac{1}{3} a^{4} - \frac{13}{36} a^{3} - \frac{5}{36} a^{2} + \frac{1}{3} a + \frac{7}{19}$, $\frac{1}{4104} a^{11} - \frac{1}{4104} a^{10} + \frac{1}{36} a^{9} + \frac{73}{216} a^{8} + \frac{35}{216} a^{7} + \frac{1453}{4104} a^{6} + \frac{293}{684} a^{5} + \frac{95}{216} a^{4} + \frac{13}{216} a^{3} + \frac{5}{36} a^{2} + \frac{15}{38} a + \frac{2}{19}$, $\frac{1}{123120} a^{12} + \frac{1}{24624} a^{11} + \frac{1}{6480} a^{9} + \frac{613}{1296} a^{8} - \frac{6709}{24624} a^{7} + \frac{1}{570} a^{6} - \frac{269}{1296} a^{5} - \frac{229}{1296} a^{4} + \frac{1}{5} a^{3} - \frac{29}{684} a^{2} + \frac{17}{114} a - \frac{1}{5}$, $\frac{1}{2219853600} a^{13} + \frac{7421}{2219853600} a^{12} + \frac{1261}{12332520} a^{11} - \frac{189881}{2219853600} a^{10} - \frac{6081799}{116834400} a^{9} + \frac{209839703}{443970720} a^{8} - \frac{24907349}{61662600} a^{7} + \frac{274669241}{2219853600} a^{6} - \frac{46734367}{443970720} a^{5} - \frac{1107749}{3245400} a^{4} + \frac{901616}{2569275} a^{3} - \frac{4643}{114190} a^{2} + \frac{77621}{285475} a - \frac{83614}{285475}$, $\frac{1}{13319121600} a^{14} - \frac{1}{13319121600} a^{13} - \frac{737}{2219853600} a^{12} + \frac{1149859}{13319121600} a^{11} - \frac{1171999}{13319121600} a^{10} - \frac{458842163}{13319121600} a^{9} - \frac{159295099}{2219853600} a^{8} - \frac{4019426851}{13319121600} a^{7} - \frac{2348716577}{13319121600} a^{6} + \frac{750753959}{2219853600} a^{5} - \frac{59664203}{123325200} a^{4} + \frac{1024343}{6851400} a^{3} + \frac{122706}{285475} a^{2} - \frac{413651}{1712850} a - \frac{25622}{285475}$, $\frac{1}{79914729600} a^{15} - \frac{1}{79914729600} a^{14} + \frac{1}{13319121600} a^{13} + \frac{13573}{4206038400} a^{12} - \frac{114661}{4206038400} a^{11} + \frac{5015569}{79914729600} a^{10} - \frac{820750117}{13319121600} a^{9} - \frac{3458710831}{79914729600} a^{8} + \frac{414681349}{4206038400} a^{7} + \frac{11167403}{701006400} a^{6} + \frac{126726281}{369975600} a^{5} + \frac{179200459}{369975600} a^{4} - \frac{22444417}{61662600} a^{3} + \frac{63389}{135225} a^{2} - \frac{20998}{45075} a + \frac{91743}{285475}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  \( -\frac{1}{2055420} a^{15} + \frac{11}{10277100} a^{14} - \frac{11}{1712850} a^{12} + \frac{29}{10277100} a^{11} - \frac{151}{2055420} a^{10} + \frac{36}{285475} a^{9} - \frac{151}{540900} a^{8} - \frac{73}{342570} a^{7} - \frac{216}{285475} a^{6} - \frac{906}{285475} a^{5} + \frac{612}{57095} a^{4} - \frac{216}{15025} a^{3} + \frac{1609901}{10277100} a^{2} - \frac{7776}{57095} a - \frac{46656}{285475} \) (order $30$)
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:  \( 93215.7577208 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{345}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-3}, \sqrt{-115})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-23})\), \(\Q(\sqrt{5}, \sqrt{69})\), \(\Q(\sqrt{-15}, \sqrt{-23})\), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\sqrt{-15}, \sqrt{69})\), 4.4.66125.1, 4.0.595125.1, \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), 8.0.14166950625.1, 8.0.354173765625.2, \(\Q(\zeta_{15})\), 8.8.354173765625.1, 8.0.354173765625.3, 8.0.4372515625.1, 8.0.354173765625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$