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Properties

Label 16.8.33634369411...3609.3
Degree $16$
Signature $[8, 4]$
Discriminant $61^{6}\cdot 97^{14}$
Root discriminant $255.81$
Ramified primes $61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-49669687, 247945353, -399339032, 195632765, 58963430, -52474780, -1736631, -275150, 1231086, 412377, -171530, -8920, 1901, 512, 114, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687)
 
gp: K = bnfinit(x^16 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 114 x^{14} + 512 x^{13} + 1901 x^{12} - 8920 x^{11} - 171530 x^{10} + 412377 x^{9} + 1231086 x^{8} - 275150 x^{7} - 1736631 x^{6} - 52474780 x^{5} + 58963430 x^{4} + 195632765 x^{3} - 399339032 x^{2} + 247945353 x - 49669687 \)

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:  \(336343694112121707284133254922278323609=61^{6}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $255.81$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{4} a^{9} - \frac{1}{6} a^{8} - \frac{1}{4} a^{6} + \frac{1}{3} a^{4} + \frac{1}{12} a^{3} + \frac{1}{3} a^{2} + \frac{1}{12}$, $\frac{1}{732} a^{13} + \frac{5}{366} a^{12} - \frac{1}{6} a^{11} - \frac{83}{244} a^{10} - \frac{121}{366} a^{9} + \frac{17}{61} a^{8} + \frac{103}{244} a^{7} - \frac{24}{61} a^{6} - \frac{56}{183} a^{5} + \frac{37}{732} a^{4} + \frac{19}{183} a^{3} - \frac{8}{61} a^{2} + \frac{301}{732} a - \frac{19}{61}$, $\frac{1}{732} a^{14} + \frac{11}{366} a^{12} - \frac{83}{244} a^{11} + \frac{74}{183} a^{10} - \frac{76}{183} a^{9} - \frac{23}{732} a^{8} + \frac{47}{122} a^{7} - \frac{68}{183} a^{6} + \frac{27}{244} a^{5} - \frac{25}{366} a^{4} + \frac{10}{61} a^{3} + \frac{41}{732} a^{2} - \frac{155}{366} a + \frac{82}{183}$, $\frac{1}{498574475690546600855273799950463789879388754022246437535276} a^{15} + \frac{87458594817287912690638910183752911152350910381701226795}{166191491896848866951757933316821263293129584674082145845092} a^{14} + \frac{50593823673965907296578396919762167742578615947861890075}{124643618922636650213818449987615947469847188505561609383819} a^{13} + \frac{2137656729372000519890034176608168860649917306962640547535}{124643618922636650213818449987615947469847188505561609383819} a^{12} - \frac{187459716951935239499680042720765320602021776759745554546955}{498574475690546600855273799950463789879388754022246437535276} a^{11} - \frac{7350015266246890042670516441812020126509047263193471450710}{41547872974212216737939483329205315823282396168520536461273} a^{10} + \frac{88748590865226405448706766993316331204149523227707952001089}{249287237845273300427636899975231894939694377011123218767638} a^{9} + \frac{101450872377074843570764949083007162417216536976361385332625}{498574475690546600855273799950463789879388754022246437535276} a^{8} - \frac{102646844239695565924070892769007682618206255730759293639}{124643618922636650213818449987615947469847188505561609383819} a^{7} - \frac{13216409113111458314475656238010645073741742732178388200836}{41547872974212216737939483329205315823282396168520536461273} a^{6} - \frac{225921665798595003409400438673133108735034100778859600694457}{498574475690546600855273799950463789879388754022246437535276} a^{5} + \frac{58610112584964940411416983000904596130140852742201023590553}{124643618922636650213818449987615947469847188505561609383819} a^{4} + \frac{52517360699395179681821182067026897191075893165243545312312}{124643618922636650213818449987615947469847188505561609383819} a^{3} + \frac{61212338419738804162619600062995951197445497011004559838309}{166191491896848866951757933316821263293129584674082145845092} a^{2} - \frac{24864542828534511740434224669423056877640443752731337775328}{124643618922636650213818449987615947469847188505561609383819} a + \frac{1411312758274789827258592277778321893719502824293573517789}{11594755248617362810587762789545669532078808233075498547332}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 9467964682180 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
61Data not computed
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$