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Properties

Label 16.8.62141154520...5001.1
Degree $16$
Signature $[8, 4]$
Discriminant $41^{15}\cdot 43^{12}$
Root discriminant $545.87$
Ramified primes $41, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8241104367689, 2007029914720, -2031620275545, 44395141190, -10127121036, -14781801288, 5419873627, -416484476, 96731439, 5241530, -1363335, 171756, -28535, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 28535*x^12 + 171756*x^11 - 1363335*x^10 + 5241530*x^9 + 96731439*x^8 - 416484476*x^7 + 5419873627*x^6 - 14781801288*x^5 - 10127121036*x^4 + 44395141190*x^3 - 2031620275545*x^2 + 2007029914720*x + 8241104367689)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 28535*x^12 + 171756*x^11 - 1363335*x^10 + 5241530*x^9 + 96731439*x^8 - 416484476*x^7 + 5419873627*x^6 - 14781801288*x^5 - 10127121036*x^4 + 44395141190*x^3 - 2031620275545*x^2 + 2007029914720*x + 8241104367689, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 28535 x^{12} + 171756 x^{11} - 1363335 x^{10} + 5241530 x^{9} + 96731439 x^{8} - 416484476 x^{7} + 5419873627 x^{6} - 14781801288 x^{5} - 10127121036 x^{4} + 44395141190 x^{3} - 2031620275545 x^{2} + 2007029914720 x + 8241104367689 \)

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:  \(62141154520506997217088824525941184562505001=41^{15}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $545.87$
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:  $41, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{43} a^{4} - \frac{2}{43} a^{3} - \frac{20}{43} a^{2} + \frac{21}{43} a - \frac{8}{43}$, $\frac{1}{43} a^{5} + \frac{19}{43} a^{3} - \frac{19}{43} a^{2} - \frac{9}{43} a - \frac{16}{43}$, $\frac{1}{43} a^{6} + \frac{19}{43} a^{3} - \frac{16}{43} a^{2} + \frac{15}{43} a - \frac{20}{43}$, $\frac{1}{43} a^{7} - \frac{21}{43} a^{3} + \frac{8}{43} a^{2} + \frac{11}{43} a - \frac{20}{43}$, $\frac{1}{3698} a^{8} - \frac{2}{1849} a^{7} + \frac{7}{3698} a^{6} - \frac{7}{3698} a^{5} - \frac{1}{3698} a^{4} + \frac{9}{3698} a^{3} + \frac{574}{1849} a^{2} - \frac{1153}{3698} a + \frac{1827}{3698}$, $\frac{1}{3698} a^{9} - \frac{9}{3698} a^{7} + \frac{21}{3698} a^{6} - \frac{29}{3698} a^{5} + \frac{5}{3698} a^{4} + \frac{592}{1849} a^{3} - \frac{259}{3698} a^{2} + \frac{913}{3698} a - \frac{44}{1849}$, $\frac{1}{3698} a^{10} - \frac{15}{3698} a^{7} + \frac{17}{1849} a^{6} + \frac{14}{1849} a^{5} - \frac{29}{3698} a^{4} + \frac{83}{1849} a^{3} + \frac{409}{3698} a^{2} + \frac{457}{3698} a - \frac{1187}{3698}$, $\frac{1}{3698} a^{11} - \frac{13}{1849} a^{7} - \frac{39}{3698} a^{6} + \frac{19}{1849} a^{5} - \frac{21}{3698} a^{4} + \frac{444}{1849} a^{3} - \frac{1587}{3698} a^{2} - \frac{168}{1849} a - \frac{115}{3698}$, $\frac{1}{159014} a^{12} - \frac{3}{79507} a^{11} - \frac{5}{159014} a^{10} - \frac{3}{79507} a^{9} - \frac{2}{79507} a^{8} + \frac{122}{79507} a^{7} + \frac{323}{79507} a^{6} + \frac{1093}{159014} a^{5} - \frac{785}{159014} a^{4} - \frac{23165}{159014} a^{3} + \frac{77123}{159014} a^{2} + \frac{38093}{79507} a + \frac{19655}{159014}$, $\frac{1}{159014} a^{13} + \frac{1}{79507} a^{11} + \frac{7}{159014} a^{10} + \frac{3}{159014} a^{9} + \frac{5}{159014} a^{8} + \frac{410}{79507} a^{7} + \frac{227}{79507} a^{6} + \frac{1473}{159014} a^{5} - \frac{11}{159014} a^{4} - \frac{39254}{79507} a^{3} - \frac{2661}{159014} a^{2} - \frac{14820}{79507} a - \frac{60993}{159014}$, $\frac{1}{8118869055645235773138574} a^{14} - \frac{7}{8118869055645235773138574} a^{13} - \frac{18408103118835894891}{8118869055645235773138574} a^{12} + \frac{110448618713015369437}{8118869055645235773138574} a^{11} - \frac{12659142598350642796}{4059434527822617886569287} a^{10} - \frac{442927122776233896023}{4059434527822617886569287} a^{9} - \frac{417472694190381277690}{4059434527822617886569287} a^{8} + \frac{9869841832681026037889}{8118869055645235773138574} a^{7} + \frac{15682779365012915829657}{4059434527822617886569287} a^{6} + \frac{27617131514597408341682}{4059434527822617886569287} a^{5} - \frac{29298004109489842706983}{4059434527822617886569287} a^{4} + \frac{3941121007018390749598115}{8118869055645235773138574} a^{3} + \frac{2123327008297280811495701}{8118869055645235773138574} a^{2} + \frac{507714196193622482559691}{8118869055645235773138574} a - \frac{1345682171501305328303192}{4059434527822617886569287}$, $\frac{1}{519031179858344277740975897246} a^{15} + \frac{31957}{519031179858344277740975897246} a^{14} - \frac{698281788717431062925355}{519031179858344277740975897246} a^{13} + \frac{2468943067859398909351}{1867018632583972222089841357} a^{12} + \frac{34585802587252952533990794}{259515589929172138870487948623} a^{11} - \frac{3509737657086623157079954}{259515589929172138870487948623} a^{10} + \frac{4488443365059647942976125}{259515589929172138870487948623} a^{9} - \frac{29991360243583746656422931}{259515589929172138870487948623} a^{8} - \frac{1601036701952068788694196821}{519031179858344277740975897246} a^{7} - \frac{5016625066974797036202976963}{519031179858344277740975897246} a^{6} - \frac{2765330434591790378804707188}{259515589929172138870487948623} a^{5} + \frac{494506839515085078853614394}{259515589929172138870487948623} a^{4} + \frac{125553087674105671553260862097}{519031179858344277740975897246} a^{3} - \frac{164585041214565505166317551295}{519031179858344277740975897246} a^{2} - \frac{117907380182291504231283681800}{259515589929172138870487948623} a + \frac{69237296863076320429389885849}{519031179858344277740975897246}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 5540181992490000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.665826106298636681.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R R $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
41Data not computed
$43$43.8.6.3$x^{8} - 43 x^{4} + 5547$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
43.8.6.3$x^{8} - 43 x^{4} + 5547$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$