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Properties

Label 16.8.70699102466...7153.2
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 89^{12}$
Root discriminant $412.66$
Ramified primes $17, 89$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-863507151949, -2616854225292, -1915541871643, -253210003245, -65890867646, -10149468346, 3402810507, -277864511, 168854201, 1891798, -365263, 60091, -26298, 21, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 26298*x^12 + 60091*x^11 - 365263*x^10 + 1891798*x^9 + 168854201*x^8 - 277864511*x^7 + 3402810507*x^6 - 10149468346*x^5 - 65890867646*x^4 - 253210003245*x^3 - 1915541871643*x^2 - 2616854225292*x - 863507151949)
 
gp: K = bnfinit(x^16 - 4*x^15 - x^14 + 21*x^13 - 26298*x^12 + 60091*x^11 - 365263*x^10 + 1891798*x^9 + 168854201*x^8 - 277864511*x^7 + 3402810507*x^6 - 10149468346*x^5 - 65890867646*x^4 - 253210003245*x^3 - 1915541871643*x^2 - 2616854225292*x - 863507151949, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - x^{14} + 21 x^{13} - 26298 x^{12} + 60091 x^{11} - 365263 x^{10} + 1891798 x^{9} + 168854201 x^{8} - 277864511 x^{7} + 3402810507 x^{6} - 10149468346 x^{5} - 65890867646 x^{4} - 253210003245 x^{3} - 1915541871643 x^{2} - 2616854225292 x - 863507151949 \)

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:  \(706991024666918541719408792174287819307153=17^{15}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $412.66$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{89} a^{8} - \frac{2}{89} a^{7} + \frac{42}{89} a^{6} - \frac{39}{89} a^{5} + \frac{42}{89} a^{4} + \frac{39}{89} a^{3} + \frac{42}{89} a^{2} + \frac{2}{89} a + \frac{1}{89}$, $\frac{1}{89} a^{9} + \frac{38}{89} a^{7} - \frac{44}{89} a^{6} - \frac{36}{89} a^{5} + \frac{34}{89} a^{4} + \frac{31}{89} a^{3} - \frac{3}{89} a^{2} + \frac{5}{89} a + \frac{2}{89}$, $\frac{1}{1157} a^{10} - \frac{6}{1157} a^{9} - \frac{4}{1157} a^{8} - \frac{544}{1157} a^{7} - \frac{112}{1157} a^{6} - \frac{515}{1157} a^{5} - \frac{513}{1157} a^{4} + \frac{398}{1157} a^{3} - \frac{406}{1157} a^{2} - \frac{379}{1157} a - \frac{11}{89}$, $\frac{1}{1157} a^{11} - \frac{1}{1157} a^{9} + \frac{4}{1157} a^{8} + \frac{433}{1157} a^{7} + \frac{295}{1157} a^{6} + \frac{453}{1157} a^{5} - \frac{470}{1157} a^{4} + \frac{45}{1157} a^{3} + \frac{266}{1157} a^{2} + \frac{79}{1157} a - \frac{16}{89}$, $\frac{1}{2314} a^{12} - \frac{1}{2314} a^{10} + \frac{2}{1157} a^{9} - \frac{9}{2314} a^{8} + \frac{11}{1157} a^{7} - \frac{378}{1157} a^{6} - \frac{587}{2314} a^{5} + \frac{575}{1157} a^{4} + \frac{383}{2314} a^{3} - \frac{565}{1157} a^{2} + \frac{5}{178} a + \frac{55}{178}$, $\frac{1}{2314} a^{13} - \frac{1}{2314} a^{11} - \frac{11}{2314} a^{9} + \frac{6}{1157} a^{8} + \frac{242}{1157} a^{7} - \frac{87}{2314} a^{6} + \frac{266}{1157} a^{5} + \frac{459}{2314} a^{4} + \frac{43}{1157} a^{3} + \frac{675}{2314} a^{2} - \frac{265}{2314} a + \frac{19}{89}$, $\frac{1}{4891671439694} a^{14} - \frac{173237253}{2445835719847} a^{13} + \frac{151945010}{2445835719847} a^{12} + \frac{43367210}{188141209219} a^{11} - \frac{653754311}{2445835719847} a^{10} - \frac{2885667592}{2445835719847} a^{9} - \frac{20785119617}{4891671439694} a^{8} - \frac{385586102643}{4891671439694} a^{7} - \frac{5931394646}{188141209219} a^{6} - \frac{8470094954}{2445835719847} a^{5} - \frac{1100921711193}{2445835719847} a^{4} - \frac{1133346925629}{2445835719847} a^{3} - \frac{1777583812359}{4891671439694} a^{2} - \frac{2066727168281}{4891671439694} a - \frac{187118638157}{376282418438}$, $\frac{1}{341726038360900966300494291148095907260640622385406711677275731934} a^{15} + \frac{3033231755155855589227824749987049975983888265432082}{170863019180450483150247145574047953630320311192703355838637865967} a^{14} + \frac{8304229849341684084744466506373176762773177126745833551551794}{170863019180450483150247145574047953630320311192703355838637865967} a^{13} - \frac{69189116769418299856945619475016501952890167429166927693954425}{341726038360900966300494291148095907260640622385406711677275731934} a^{12} - \frac{15326823270595268664244252714195414567761112919309385406498223}{170863019180450483150247145574047953630320311192703355838637865967} a^{11} + \frac{113501379212621465683171873927942859625778842001348881821013743}{341726038360900966300494291148095907260640622385406711677275731934} a^{10} - \frac{124618205634227564234806143058571083578081426087965044123033473}{341726038360900966300494291148095907260640622385406711677275731934} a^{9} + \frac{496890274736385801901556423617196799372977988437180751627809146}{170863019180450483150247145574047953630320311192703355838637865967} a^{8} + \frac{17259923457992106649978744063325999147943659405438031847104383589}{170863019180450483150247145574047953630320311192703355838637865967} a^{7} + \frac{34040272117769887633064725977286160352184370134105287207787039874}{170863019180450483150247145574047953630320311192703355838637865967} a^{6} - \frac{100579956993892556290676749615621679868452687250364855102018562775}{341726038360900966300494291148095907260640622385406711677275731934} a^{5} + \frac{29886233288735051836741065662172525797985550100888118867617301873}{170863019180450483150247145574047953630320311192703355838637865967} a^{4} - \frac{65517803475988226068733151662373859557438255695268018245981326330}{170863019180450483150247145574047953630320311192703355838637865967} a^{3} - \frac{80600455206224262463603577253515948480395890362965387322038905941}{341726038360900966300494291148095907260640622385406711677275731934} a^{2} + \frac{27643757910315700676879387017527363843977564523656859209855443521}{170863019180450483150247145574047953630320311192703355838637865967} a - \frac{1137434898700779974595496244730269734597428078170071221657764799}{26286618335453920484653407011391992866203124798877439359790440918}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 214439499042000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.25745567912986193.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$89$89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$