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Properties

Label 16.0.43153395314...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 17^{14}$
Root discriminant $46.20$
Ramified primes $3, 5, 17$
Class number $816$ (GRH)
Class group $[816]$ (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![838099, -435519, 779262, -388515, 371982, -170636, 115811, -47784, 25370, -9195, 4000, -1250, 439, -110, 32, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 32*x^14 - 110*x^13 + 439*x^12 - 1250*x^11 + 4000*x^10 - 9195*x^9 + 25370*x^8 - 47784*x^7 + 115811*x^6 - 170636*x^5 + 371982*x^4 - 388515*x^3 + 779262*x^2 - 435519*x + 838099)
 
gp: K = bnfinit(x^16 - 6*x^15 + 32*x^14 - 110*x^13 + 439*x^12 - 1250*x^11 + 4000*x^10 - 9195*x^9 + 25370*x^8 - 47784*x^7 + 115811*x^6 - 170636*x^5 + 371982*x^4 - 388515*x^3 + 779262*x^2 - 435519*x + 838099, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 32 x^{14} - 110 x^{13} + 439 x^{12} - 1250 x^{11} + 4000 x^{10} - 9195 x^{9} + 25370 x^{8} - 47784 x^{7} + 115811 x^{6} - 170636 x^{5} + 371982 x^{4} - 388515 x^{3} + 779262 x^{2} - 435519 x + 838099 \)

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:  \(431533953146964646550390625=3^{8}\cdot 5^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.20$
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$$\chi_{255}(1,·)$, $\chi_{255}(196,·)$, $\chi_{255}(134,·)$, $\chi_{255}(76,·)$, $\chi_{255}(16,·)$, $\chi_{255}(149,·)$, $\chi_{255}(151,·)$, $\chi_{255}(89,·)$, $\chi_{255}(166,·)$, $\chi_{255}(104,·)$, $\chi_{255}(106,·)$, $\chi_{255}(239,·)$, $\chi_{255}(179,·)$, $\chi_{255}(121,·)$, $\chi_{255}(59,·)$, $\chi_{255}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{62485422378861066211994309922614986} a^{15} - \frac{11073328978532777113928198053345223}{62485422378861066211994309922614986} a^{14} + \frac{4980209845197566291944824051405881}{31242711189430533105997154961307493} a^{13} + \frac{1047192353030113998380314424807098}{31242711189430533105997154961307493} a^{12} + \frac{25260357289969062764394391319042915}{62485422378861066211994309922614986} a^{11} - \frac{13404368108086888847221598926799184}{31242711189430533105997154961307493} a^{10} - \frac{622755134151681018294989969141253}{2403285476110041008153627304715961} a^{9} - \frac{19587533753081436362868083869740649}{62485422378861066211994309922614986} a^{8} + \frac{13426590312789899810469879007484612}{31242711189430533105997154961307493} a^{7} - \frac{8263540162632786334606102687471095}{31242711189430533105997154961307493} a^{6} - \frac{17067630231946755769270533548762725}{62485422378861066211994309922614986} a^{5} - \frac{13568868542319050186792039612187610}{31242711189430533105997154961307493} a^{4} + \frac{4114500197560367625668534004188380}{31242711189430533105997154961307493} a^{3} + \frac{9585204617408514450909706062547057}{62485422378861066211994309922614986} a^{2} - \frac{1155527490396789597225687861781891}{2403285476110041008153627304715961} a - \frac{18974543803297872046579357065575031}{62485422378861066211994309922614986}$

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

Class group and class number

$C_{816}$, which has order $816$ (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:  \( 3640.012213375973 \) (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

\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{17})\), 4.4.4913.1, 4.0.1105425.1, 8.0.1221964430625.2, \(\Q(\zeta_{17})^+\), 8.0.20773395320625.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
5Data not computed
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$