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Properties

Label 16.0.84693947642...6001.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 173^{14}$
Root discriminant $857.02$
Ramified primes $13, 173$
Class number $688781200$ (GRH)
Class group $[2, 10, 34439060]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5843269449320273, -776582504211170, 386408516466130, -50798315471152, 10084531101885, -1299529050323, 115359230062, -11274491499, 568783688, 25016421, -1062900, 676587, 7659, 2682, 206, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 206*x^14 + 2682*x^13 + 7659*x^12 + 676587*x^11 - 1062900*x^10 + 25016421*x^9 + 568783688*x^8 - 11274491499*x^7 + 115359230062*x^6 - 1299529050323*x^5 + 10084531101885*x^4 - 50798315471152*x^3 + 386408516466130*x^2 - 776582504211170*x + 5843269449320273)
 
gp: K = bnfinit(x^16 - 3*x^15 + 206*x^14 + 2682*x^13 + 7659*x^12 + 676587*x^11 - 1062900*x^10 + 25016421*x^9 + 568783688*x^8 - 11274491499*x^7 + 115359230062*x^6 - 1299529050323*x^5 + 10084531101885*x^4 - 50798315471152*x^3 + 386408516466130*x^2 - 776582504211170*x + 5843269449320273, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 206 x^{14} + 2682 x^{13} + 7659 x^{12} + 676587 x^{11} - 1062900 x^{10} + 25016421 x^{9} + 568783688 x^{8} - 11274491499 x^{7} + 115359230062 x^{6} - 1299529050323 x^{5} + 10084531101885 x^{4} - 50798315471152 x^{3} + 386408516466130 x^{2} - 776582504211170 x + 5843269449320273 \)

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:  \(84693947642368239934810426203628826170679406001=13^{14}\cdot 173^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $857.02$
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:  $13, 173$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} + \frac{3}{13} a^{10} - \frac{1}{13} a^{9} - \frac{4}{13} a^{8} + \frac{6}{13} a^{7} + \frac{3}{13} a^{6} + \frac{1}{13} a^{5} - \frac{2}{13} a^{4} - \frac{3}{13} a^{3} + \frac{2}{13} a^{2} + \frac{5}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{12} + \frac{3}{13} a^{10} - \frac{1}{13} a^{9} + \frac{5}{13} a^{8} - \frac{2}{13} a^{7} + \frac{5}{13} a^{6} - \frac{5}{13} a^{5} + \frac{3}{13} a^{4} - \frac{2}{13} a^{3} - \frac{1}{13} a^{2} - \frac{5}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{13} + \frac{3}{13} a^{10} - \frac{5}{13} a^{9} - \frac{3}{13} a^{8} - \frac{1}{13} a^{6} + \frac{4}{13} a^{4} - \frac{5}{13} a^{3} + \frac{2}{13} a^{2} - \frac{6}{13} a - \frac{4}{13}$, $\frac{1}{169} a^{14} + \frac{2}{169} a^{13} + \frac{5}{169} a^{12} - \frac{3}{169} a^{11} + \frac{11}{169} a^{10} - \frac{77}{169} a^{9} + \frac{43}{169} a^{8} + \frac{31}{169} a^{7} + \frac{31}{169} a^{6} - \frac{66}{169} a^{5} - \frac{48}{169} a^{4} - \frac{3}{13} a^{3} - \frac{32}{169} a^{2} + \frac{46}{169} a + \frac{42}{169}$, $\frac{1}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{15} - \frac{8230980177778735784087790573360686529085488582931727378427576312065571550346964982550222413654500055}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{14} - \frac{12588194468304927905034931308848908725971732781294765671419500363011802183517404892133829996775806155}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{13} - \frac{161397058752933213441824515031431748235121122772804854885959647473865015670967955563165781717154595660}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{12} - \frac{46636287327990257943574721959697836742350246425472303259644977462422749203018476651146858719657026924}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{11} - \frac{124038704038272713731817052403242609198550961436382384871430601574065545417987418575482347925818941133}{380214290081574584995623613460166474232630800813684520030764186125781378498891646726978271184861865097} a^{10} - \frac{2146105112677440287950104718440895775521953398954207794417965089574096360438430249253201758853490842616}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{9} + \frac{737345765530198886741254350148958991062061219765152053724800213814979560747840469494946336781967370275}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{8} - \frac{430468512431655249490370441660336591810108604176138680705173625336874743930727932441008188450418921176}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{7} - \frac{1992859616478215928679023037486848364572501254468712220680965117485503155561827051262466239579662158135}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{6} + \frac{738952034573954818411431117235709986157352213617031731442910534312507589963161092229743248201324422396}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{5} + \frac{2150207543291587384492001381165915691083882085602823229616911431133285595558169007501100742206900515427}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{4} + \frac{1460174194261387527613447027375724682272178128818265848592613592522348556211858979207044706744816782249}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{3} - \frac{2400612784727229848571226358881376595170164593183720964908425283962541589172248862408523846492890893926}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261} a^{2} + \frac{113788553556413276378595842498129965718974400061130157503051046782304109596403083008857791002332557723}{380214290081574584995623613460166474232630800813684520030764186125781378498891646726978271184861865097} a - \frac{306432940040928367436173576097903931538293827579691915498066160247071712088740349160982300693482226772}{4942785771060469604943106974982164165024200410577898760399934419635157920485591407450717525403204246261}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{34439060}$, which has order $688781200$ (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:  \( 67707997.6241 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{2249}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{173}) \), \(\Q(\sqrt{13}, \sqrt{173})\), 8.8.129400731862107174001.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$173$173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$