Properties

Label 16.0.11173814592...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $23.88$
Ramified primes $3, 5, 17$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 16384, -16384, -13312, -3328, -3136, 2112, 2548, 381, -637, 132, 49, -13, 13, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536)
 
gp: K = bnfinit(x^16 - x^15 - 4*x^14 + 13*x^13 - 13*x^12 + 49*x^11 + 132*x^10 - 637*x^9 + 381*x^8 + 2548*x^7 + 2112*x^6 - 3136*x^5 - 3328*x^4 - 13312*x^3 - 16384*x^2 + 16384*x + 65536, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 4 x^{14} + 13 x^{13} - 13 x^{12} + 49 x^{11} + 132 x^{10} - 637 x^{9} + 381 x^{8} + 2548 x^{7} + 2112 x^{6} - 3136 x^{5} - 3328 x^{4} - 13312 x^{3} - 16384 x^{2} + 16384 x + 65536 \)

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:  \(11173814592383056640625=3^{8}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.88$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(67,·)$, $\chi_{255}(137,·)$, $\chi_{255}(203,·)$, $\chi_{255}(16,·)$, $\chi_{255}(86,·)$, $\chi_{255}(152,·)$, $\chi_{255}(154,·)$, $\chi_{255}(101,·)$, $\chi_{255}(103,·)$, $\chi_{255}(169,·)$, $\chi_{255}(239,·)$, $\chi_{255}(52,·)$, $\chi_{255}(118,·)$, $\chi_{255}(188,·)$, $\chi_{255}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{464} a^{10} - \frac{1}{16} a^{9} - \frac{1}{2} a^{8} + \frac{1}{16} a^{7} + \frac{3}{16} a^{6} - \frac{51}{464} a^{5} - \frac{1}{2} a^{4} - \frac{1}{16} a^{3} - \frac{3}{16} a^{2} - \frac{1}{2} a - \frac{6}{29}$, $\frac{1}{1856} a^{11} - \frac{1}{1856} a^{10} - \frac{1}{16} a^{9} - \frac{31}{64} a^{8} + \frac{15}{64} a^{7} + \frac{65}{1856} a^{6} - \frac{67}{464} a^{5} - \frac{17}{64} a^{4} - \frac{31}{64} a^{3} - \frac{3}{16} a^{2} - \frac{35}{116} a - \frac{13}{29}$, $\frac{1}{37120} a^{12} - \frac{1}{7424} a^{11} + \frac{1}{1280} a^{9} + \frac{79}{256} a^{8} - \frac{799}{7424} a^{7} - \frac{1}{580} a^{6} - \frac{61}{256} a^{5} - \frac{31}{256} a^{4} + \frac{1}{5} a^{3} - \frac{181}{464} a^{2} - \frac{13}{116} a - \frac{1}{5}$, $\frac{1}{333337600} a^{13} + \frac{691}{333337600} a^{12} + \frac{2003}{8333440} a^{11} - \frac{83571}{333337600} a^{10} + \frac{484611}{11494400} a^{9} + \frac{7349657}{66667520} a^{8} + \frac{300577}{41667200} a^{7} - \frac{31071229}{333337600} a^{6} - \frac{11763643}{66667520} a^{5} - \frac{417823}{1436800} a^{4} - \frac{2359949}{5208400} a^{3} - \frac{40157}{520840} a^{2} + \frac{269351}{1302100} a + \frac{6199}{325525}$, $\frac{1}{1333350400} a^{14} - \frac{1}{1333350400} a^{13} - \frac{733}{333337600} a^{12} - \frac{30211}{1333350400} a^{11} + \frac{44851}{1333350400} a^{10} + \frac{73158017}{1333350400} a^{9} + \frac{108074349}{333337600} a^{8} + \frac{309809299}{1333350400} a^{7} + \frac{569216573}{1333350400} a^{6} - \frac{56562839}{333337600} a^{5} + \frac{9440161}{83334400} a^{4} + \frac{3961379}{10416800} a^{3} + \frac{190861}{5208400} a^{2} - \frac{198599}{1302100} a - \frac{104832}{325525}$, $\frac{1}{5333401600} a^{15} - \frac{1}{5333401600} a^{14} - \frac{1}{1333350400} a^{13} - \frac{18483}{5333401600} a^{12} + \frac{437811}{5333401600} a^{11} + \frac{3751729}{5333401600} a^{10} - \frac{35986863}{1333350400} a^{9} + \frac{1782429379}{5333401600} a^{8} + \frac{2062447421}{5333401600} a^{7} - \frac{570676547}{1333350400} a^{6} + \frac{10694327}{41667200} a^{5} + \frac{16746493}{41667200} a^{4} - \frac{536557}{1302100} a^{3} + \frac{979683}{2604200} a^{2} + \frac{312751}{1302100} a - \frac{120538}{325525}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( \frac{44541}{5333401600} a^{15} - \frac{44541}{1066680320} a^{14} - \frac{441}{20833600} a^{13} + \frac{44541}{183910400} a^{12} - \frac{579033}{1066680320} a^{11} + \frac{4498641}{5333401600} a^{10} - \frac{44541}{83334400} a^{9} - \frac{13251653}{1066680320} a^{8} + \frac{4498641}{183910400} a^{7} + \frac{44541}{5208400} a^{6} - \frac{4498641}{66667520} a^{5} - \frac{8061921}{83334400} a^{4} - \frac{3346749}{20833600} a^{3} + \frac{400869}{1302100} a + \frac{44541}{65105} \) (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:  \( 38248.3956886 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-3}, \sqrt{85})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-51})\), \(\Q(\sqrt{-15}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-15}, \sqrt{-51})\), 4.4.325125.1, 4.0.36125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 8.0.4228250625.1, 8.0.105706265625.2, \(\Q(\zeta_{15})\), 8.0.105706265625.3, 8.0.105706265625.1, 8.8.105706265625.1, 8.0.1305015625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$