↑

Properties

Label 16.8.19524365719...9097.1
Degree $16$
Signature $[8, 4]$
Discriminant $23^{12}\cdot 73^{15}$
Root discriminant $586.36$
Ramified primes $23, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-716371879936, -243288205952, 721476532032, 621629449624, 206054540288, 40146705456, 5448645279, 499725828, 83558589, -4534424, -2592178, 43368, -49358, -1560, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 - 1560*x^13 - 49358*x^12 + 43368*x^11 - 2592178*x^10 - 4534424*x^9 + 83558589*x^8 + 499725828*x^7 + 5448645279*x^6 + 40146705456*x^5 + 206054540288*x^4 + 621629449624*x^3 + 721476532032*x^2 - 243288205952*x - 716371879936)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 - 1560*x^13 - 49358*x^12 + 43368*x^11 - 2592178*x^10 - 4534424*x^9 + 83558589*x^8 + 499725828*x^7 + 5448645279*x^6 + 40146705456*x^5 + 206054540288*x^4 + 621629449624*x^3 + 721476532032*x^2 - 243288205952*x - 716371879936, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} - 1560 x^{13} - 49358 x^{12} + 43368 x^{11} - 2592178 x^{10} - 4534424 x^{9} + 83558589 x^{8} + 499725828 x^{7} + 5448645279 x^{6} + 40146705456 x^{5} + 206054540288 x^{4} + 621629449624 x^{3} + 721476532032 x^{2} - 243288205952 x - 716371879936 \)

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:  \(195243657193163639221629075064503172225659097=23^{12}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $586.36$
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:  $23, 73$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4}$, $\frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{32} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{736} a^{8} - \frac{1}{368} a^{7} - \frac{7}{184} a^{6} + \frac{19}{368} a^{5} + \frac{75}{736} a^{4} + \frac{9}{184} a^{3} - \frac{11}{46} a^{2} + \frac{35}{92} a + \frac{1}{23}$, $\frac{1}{1472} a^{9} + \frac{7}{736} a^{7} - \frac{1}{23} a^{6} - \frac{79}{1472} a^{5} - \frac{17}{184} a^{4} - \frac{9}{46} a^{3} + \frac{37}{184} a^{2} - \frac{8}{23} a + \frac{1}{23}$, $\frac{1}{5888} a^{10} + \frac{1}{5888} a^{9} + \frac{1}{2944} a^{8} - \frac{13}{2944} a^{7} - \frac{175}{5888} a^{6} + \frac{65}{5888} a^{5} + \frac{129}{1472} a^{4} + \frac{39}{736} a^{3} + \frac{53}{736} a^{2} - \frac{27}{184} a - \frac{7}{23}$, $\frac{1}{47104} a^{11} - \frac{1}{23552} a^{10} + \frac{15}{47104} a^{9} + \frac{63}{47104} a^{7} + \frac{71}{23552} a^{6} + \frac{1745}{47104} a^{5} - \frac{59}{512} a^{4} - \frac{85}{736} a^{3} + \frac{725}{5888} a^{2} - \frac{687}{1472} a - \frac{43}{92}$, $\frac{1}{47104} a^{12} + \frac{3}{47104} a^{10} - \frac{5}{23552} a^{9} - \frac{17}{47104} a^{8} + \frac{39}{11776} a^{7} + \frac{1381}{47104} a^{6} - \frac{1181}{23552} a^{5} - \frac{401}{5888} a^{4} - \frac{83}{5888} a^{3} - \frac{13}{2944} a^{2} + \frac{157}{736} a + \frac{13}{46}$, $\frac{1}{188416} a^{13} - \frac{1}{188416} a^{12} + \frac{1}{188416} a^{11} + \frac{7}{188416} a^{10} - \frac{21}{188416} a^{9} - \frac{5}{8192} a^{8} - \frac{1621}{188416} a^{7} - \frac{6699}{188416} a^{6} - \frac{1}{256} a^{5} + \frac{2283}{23552} a^{4} + \frac{5017}{23552} a^{3} + \frac{45}{2944} a^{2} + \frac{665}{1472} a - \frac{27}{92}$, $\frac{1}{24950800384} a^{14} + \frac{95}{194928128} a^{13} + \frac{2359}{1559425024} a^{12} + \frac{13945}{3118850048} a^{11} - \frac{178719}{12475400192} a^{10} - \frac{755029}{3118850048} a^{9} - \frac{2109351}{3118850048} a^{8} + \frac{491575}{48732032} a^{7} - \frac{943670363}{24950800384} a^{6} - \frac{21645205}{3118850048} a^{5} + \frac{85696797}{779712512} a^{4} + \frac{754287537}{3118850048} a^{3} - \frac{9204517}{389856256} a^{2} - \frac{90957839}{194928128} a - \frac{232225}{529696}$, $\frac{1}{29037869109510285771107710451142463082704062376312832} a^{15} - \frac{455017814093768421329141004361393254147209}{29037869109510285771107710451142463082704062376312832} a^{14} + \frac{230846563283949932938223410799119665152736113}{113429176209024553793389493949775246416812743657472} a^{13} - \frac{25868216566839392158642879949171325581705919987}{3629733638688785721388463806392807885338007797039104} a^{12} + \frac{13323601261904385060189641243519072920032584533}{14518934554755142885553855225571231541352031188156416} a^{11} - \frac{257039106393930443953824510355439002976127354101}{14518934554755142885553855225571231541352031188156416} a^{10} - \frac{64119438065078451009229917984335081242480830779}{453716704836098215173557975799100985667250974629888} a^{9} + \frac{2353479978713642860669221137754511596812438111429}{3629733638688785721388463806392807885338007797039104} a^{8} + \frac{402816046678496404694731053693603292354991218890485}{29037869109510285771107710451142463082704062376312832} a^{7} + \frac{319000422810451218857138416218600588024893489387323}{29037869109510285771107710451142463082704062376312832} a^{6} + \frac{183535388068357782842204233472162049614481378865349}{3629733638688785721388463806392807885338007797039104} a^{5} - \frac{148858244410179309891143748634302782410822920487171}{3629733638688785721388463806392807885338007797039104} a^{4} + \frac{333653747781765893474018247227931344963370551242831}{3629733638688785721388463806392807885338007797039104} a^{3} - \frac{82233021330898506862671329597827990206746771789117}{453716704836098215173557975799100985667250974629888} a^{2} + \frac{75204969384912862868700029630006723776719555007215}{226858352418049107586778987899550492833625487314944} a + \frac{5561639245747263395586029003184852607841982752999}{14178647026128069224173686743721905802101592957184}$

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:  \( 6590405917470000000 \) (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

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.3091515048982623577.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R $16$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$

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
$23$23.8.6.3$x^{8} - 23 x^{4} + 3703$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
23.8.6.3$x^{8} - 23 x^{4} + 3703$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
73Data not computed