↑

Properties

Label 16.4.39728624488...209.16
Degree $16$
Signature $[4, 6]$
Discriminant $71^{10}\cdot 73^{14}$
Root discriminant $612.98$
Ramified primes $71, 73$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T817

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![416833742984, -187686089996, -114722294026, 67298006331, -34228322558, 7724948460, -1868635159, -7899129, -22237642, -16109541, 167473, -304668, 22547, -1567, 284, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 284*x^14 - 1567*x^13 + 22547*x^12 - 304668*x^11 + 167473*x^10 - 16109541*x^9 - 22237642*x^8 - 7899129*x^7 - 1868635159*x^6 + 7724948460*x^5 - 34228322558*x^4 + 67298006331*x^3 - 114722294026*x^2 - 187686089996*x + 416833742984)
 
gp: K = bnfinit(x^16 - 2*x^15 + 284*x^14 - 1567*x^13 + 22547*x^12 - 304668*x^11 + 167473*x^10 - 16109541*x^9 - 22237642*x^8 - 7899129*x^7 - 1868635159*x^6 + 7724948460*x^5 - 34228322558*x^4 + 67298006331*x^3 - 114722294026*x^2 - 187686089996*x + 416833742984, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 284 x^{14} - 1567 x^{13} + 22547 x^{12} - 304668 x^{11} + 167473 x^{10} - 16109541 x^{9} - 22237642 x^{8} - 7899129 x^{7} - 1868635159 x^{6} + 7724948460 x^{5} - 34228322558 x^{4} + 67298006331 x^{3} - 114722294026 x^{2} - 187686089996 x + 416833742984 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(397286244885795491769071181517306466749106209=71^{10}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $612.98$
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:  $71, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{8} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{3552} a^{12} - \frac{15}{1184} a^{11} + \frac{25}{1776} a^{10} + \frac{1}{32} a^{9} - \frac{43}{1184} a^{8} + \frac{25}{296} a^{7} + \frac{77}{1184} a^{6} + \frac{275}{1184} a^{5} + \frac{79}{1776} a^{4} + \frac{1309}{3552} a^{3} - \frac{707}{3552} a^{2} + \frac{359}{888} a + \frac{43}{888}$, $\frac{1}{14208} a^{13} - \frac{1}{14208} a^{12} - \frac{47}{1776} a^{11} + \frac{91}{14208} a^{10} - \frac{191}{4736} a^{9} + \frac{103}{2368} a^{8} - \frac{15}{128} a^{7} + \frac{3}{128} a^{6} - \frac{77}{1776} a^{5} - \frac{1063}{14208} a^{4} + \frac{907}{4736} a^{3} + \frac{815}{7104} a^{2} + \frac{1631}{3552} a + \frac{391}{1776}$, $\frac{1}{4660224} a^{14} + \frac{7}{2330112} a^{13} + \frac{469}{4660224} a^{12} - \frac{87761}{4660224} a^{11} + \frac{15169}{291264} a^{10} - \frac{9911}{1553408} a^{9} - \frac{64933}{1553408} a^{8} + \frac{83885}{776704} a^{7} + \frac{139103}{4660224} a^{6} - \frac{706507}{4660224} a^{5} + \frac{6079}{72816} a^{4} - \frac{680941}{1553408} a^{3} - \frac{626407}{2330112} a^{2} + \frac{76105}{388352} a + \frac{260119}{582528}$, $\frac{1}{88620123658417497207783738440921360158446087822156913708384776916992} a^{15} - \frac{7099314873801013515202772357055058737216645270881342339411879}{88620123658417497207783738440921360158446087822156913708384776916992} a^{14} - \frac{68823970295119094102561453697851869358004588762834379532983441}{88620123658417497207783738440921360158446087822156913708384776916992} a^{13} + \frac{1477427216914493281133628334203547882741677024208152763531253275}{44310061829208748603891869220460680079223043911078456854192388458496} a^{12} + \frac{897454890767786546434038534262962344763167370145626532284616357799}{29540041219472499069261246146973786719482029274052304569461592305664} a^{11} - \frac{584151591535880726735685749617290223417437706274885192596593070183}{29540041219472499069261246146973786719482029274052304569461592305664} a^{10} + \frac{474361370912776216164206384439095871259288327121344632576194933551}{14770020609736249534630623073486893359741014637026152284730796152832} a^{9} + \frac{1528926085898456078910380329592930238454009966073283761277175167843}{29540041219472499069261246146973786719482029274052304569461592305664} a^{8} - \frac{8183876296840041623961265274317427919996087516551607831636984720775}{88620123658417497207783738440921360158446087822156913708384776916992} a^{7} - \frac{1632616690023996279497431863358677199516283940286019779206168269977}{14770020609736249534630623073486893359741014637026152284730796152832} a^{6} - \frac{19055907175956280885568196127920019995595486346504366958823449347929}{88620123658417497207783738440921360158446087822156913708384776916992} a^{5} - \frac{3035366586979357771173130001414201527689165675268653140329975288301}{29540041219472499069261246146973786719482029274052304569461592305664} a^{4} - \frac{8982286281535824334378168569978360804756179296940914310047295054633}{29540041219472499069261246146973786719482029274052304569461592305664} a^{3} + \frac{13106536983335455827930217693801252879678109509135688822432727360953}{44310061829208748603891869220460680079223043911078456854192388458496} a^{2} + \frac{3441572144366875530369803444989400085152438587850789027810940829885}{7385010304868124767315311536743446679870507318513076142365398076416} a - \frac{1677645804528261064753576916058103394363323194788244071262324638801}{3692505152434062383657655768371723339935253659256538071182699038208}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $9$
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:  \( 483266263146000000 \) (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

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.280732967047165372057.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
71Data not computed
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$