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Properties

Label 16.8.14470590387...3936.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 17^{8}\cdot 16673^{4}$
Root discriminant $157.59$
Ramified primes $2, 17, 16673$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1439

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![321823, -2950448, 3299088, -1226092, 762925, -654124, -314000, 116768, -54078, 32796, 17150, 3056, 1725, 96, -82, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 82*x^14 + 96*x^13 + 1725*x^12 + 3056*x^11 + 17150*x^10 + 32796*x^9 - 54078*x^8 + 116768*x^7 - 314000*x^6 - 654124*x^5 + 762925*x^4 - 1226092*x^3 + 3299088*x^2 - 2950448*x + 321823)
 
gp: K = bnfinit(x^16 - 4*x^15 - 82*x^14 + 96*x^13 + 1725*x^12 + 3056*x^11 + 17150*x^10 + 32796*x^9 - 54078*x^8 + 116768*x^7 - 314000*x^6 - 654124*x^5 + 762925*x^4 - 1226092*x^3 + 3299088*x^2 - 2950448*x + 321823, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 82 x^{14} + 96 x^{13} + 1725 x^{12} + 3056 x^{11} + 17150 x^{10} + 32796 x^{9} - 54078 x^{8} + 116768 x^{7} - 314000 x^{6} - 654124 x^{5} + 762925 x^{4} - 1226092 x^{3} + 3299088 x^{2} - 2950448 x + 321823 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(144705903870176820092932857775783936=2^{28}\cdot 17^{8}\cdot 16673^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $157.59$
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:  $2, 17, 16673$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{3}{16} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a - \frac{1}{16}$, $\frac{1}{591156842933913836346292518975749799304812919920} a^{15} - \frac{2023608035497854597360940111360536403214974973}{591156842933913836346292518975749799304812919920} a^{14} + \frac{15384921223336267142074763890294243345008830081}{118231368586782767269258503795149959860962583984} a^{13} - \frac{46702581596363492952500358250445985085483355989}{591156842933913836346292518975749799304812919920} a^{12} + \frac{3873693639298113321650076266318025131384690269}{147789210733478459086573129743937449826203229980} a^{11} + \frac{11665366132468819290510705019801125217784179123}{147789210733478459086573129743937449826203229980} a^{10} - \frac{50250685075420336476854012405696912465743935559}{295578421466956918173146259487874899652406459960} a^{9} - \frac{36585925410983846812458802162557984256876412051}{295578421466956918173146259487874899652406459960} a^{8} - \frac{10074630243533916446458217042056064265397196491}{29557842146695691817314625948787489965240645996} a^{7} + \frac{61974744380156213823910086291951883812525189067}{147789210733478459086573129743937449826203229980} a^{6} + \frac{9964624459710439162431395842776136031944559929}{49263070244492819695524376581312483275401076660} a^{5} + \frac{1458058853274544673267332141901924638100145944}{36947302683369614771643282435984362456550807495} a^{4} + \frac{31468460069289080242409373629708748553877115029}{591156842933913836346292518975749799304812919920} a^{3} - \frac{62584164445650055454526007544411677550344952691}{197052280977971278782097506325249933101604306640} a^{2} - \frac{14647647573979676139992264925850075166672424483}{39410456195594255756419501265049986620320861328} a - \frac{113820061819281437619230629081339040992359694763}{591156842933913836346292518975749799304812919920}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 707932887569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1439:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1439 are not computed
Character table for t16n1439 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5703866912768.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{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}$
16673Data not computed