Properties

Label 16.0.345...625.8
Degree $16$
Signature $[0, 8]$
Discriminant $3.455\times 10^{24}$
Root discriminant \(34.17\)
Ramified primes $5,29$
Class number $36$ (GRH)
Class group [3, 12] (GRH)
Galois group $C_2^3:C_4$ (as 16T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295)
 
gp: K = bnfinit(y^16 - 3*y^15 - 3*y^14 + 12*y^13 - 19*y^12 + 100*y^11 - 70*y^10 + 95*y^9 - 90*y^8 - 1065*y^7 + 3906*y^6 - 6078*y^5 + 6817*y^4 - 7738*y^3 + 11971*y^2 - 13835*y + 7295, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295)
 

\( x^{16} - 3 x^{15} - 3 x^{14} + 12 x^{13} - 19 x^{12} + 100 x^{11} - 70 x^{10} + 95 x^{9} - 90 x^{8} - 1065 x^{7} + 3906 x^{6} - 6078 x^{5} + 6817 x^{4} - 7738 x^{3} + 11971 x^{2} + \cdots + 7295 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3455222492240908603515625\) \(\medspace = 5^{10}\cdot 29^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(34.17\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{3/4}29^{3/4}\approx 41.78553833475025$
Ramified primes:   \(5\), \(29\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{15}a^{10}+\frac{1}{15}a^{9}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{15}a^{2}+\frac{1}{3}a$, $\frac{1}{90}a^{11}-\frac{1}{30}a^{10}-\frac{7}{45}a^{9}+\frac{7}{45}a^{8}+\frac{1}{3}a^{7}-\frac{1}{10}a^{6}+\frac{7}{30}a^{5}-\frac{7}{15}a^{4}+\frac{13}{45}a^{3}-\frac{2}{5}a^{2}+\frac{7}{18}a-\frac{1}{18}$, $\frac{1}{270}a^{12}-\frac{1}{270}a^{11}-\frac{1}{135}a^{10}-\frac{13}{135}a^{9}-\frac{16}{135}a^{8}+\frac{11}{90}a^{7}-\frac{23}{90}a^{6}+\frac{1}{3}a^{5}+\frac{5}{27}a^{4}-\frac{19}{135}a^{3}+\frac{7}{54}a^{2}+\frac{19}{54}a+\frac{8}{27}$, $\frac{1}{810}a^{13}-\frac{1}{270}a^{11}-\frac{1}{81}a^{10}-\frac{4}{81}a^{9}+\frac{1}{810}a^{8}+\frac{7}{45}a^{7}+\frac{133}{270}a^{6}-\frac{13}{81}a^{5}-\frac{5}{27}a^{4}-\frac{109}{270}a^{3}+\frac{56}{405}a^{2}-\frac{55}{162}a-\frac{19}{81}$, $\frac{1}{36231300}a^{14}+\frac{421}{36231300}a^{13}+\frac{2917}{6038550}a^{12}+\frac{3683}{5175900}a^{11}-\frac{155333}{36231300}a^{10}-\frac{151253}{1341900}a^{9}+\frac{4341553}{36231300}a^{8}-\frac{104687}{3019275}a^{7}-\frac{12706993}{36231300}a^{6}+\frac{7116239}{36231300}a^{5}-\frac{1480939}{4025700}a^{4}-\frac{10042343}{36231300}a^{3}-\frac{5364983}{18115650}a^{2}-\frac{4339}{115020}a+\frac{2087761}{7246260}$, $\frac{1}{18559048649400}a^{15}+\frac{27613}{9279524324700}a^{14}-\frac{82056103}{261395051400}a^{13}-\frac{29330143909}{18559048649400}a^{12}+\frac{733165981}{257764564575}a^{11}-\frac{295738418}{39320018325}a^{10}-\frac{203432990143}{1325646332100}a^{9}-\frac{77483822099}{18559048649400}a^{8}+\frac{4678994671067}{18559048649400}a^{7}+\frac{2895889593797}{9279524324700}a^{6}+\frac{187156274968}{2319881081175}a^{5}-\frac{1247677378249}{9279524324700}a^{4}-\frac{216431730283}{687372172200}a^{3}+\frac{262324849369}{3711809729880}a^{2}-\frac{56060675653}{463976216235}a-\frac{268168794079}{742361945976}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$, $5$

Class group and class number

$C_{3}\times C_{12}$, which has order $36$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{43002619}{314560146600}a^{15}-\frac{4703803}{11234290950}a^{14}-\frac{40346629}{62912029320}a^{13}+\frac{697970633}{314560146600}a^{12}-\frac{2331163}{2097067644}a^{11}+\frac{50308423}{4493716380}a^{10}-\frac{72469196}{7864003665}a^{9}-\frac{494493907}{62912029320}a^{8}-\frac{255157603}{12582405864}a^{7}-\frac{213729949}{1572800733}a^{6}+\frac{12864434131}{22468581900}a^{5}-\frac{3076459669}{5617145475}a^{4}+\frac{5394615697}{20970676440}a^{3}-\frac{272064137257}{314560146600}a^{2}+\frac{8760643561}{6291202932}a-\frac{80883951443}{62912029320}$, $\frac{3278631151}{3711809729880}a^{15}-\frac{1193242691}{662823166050}a^{14}-\frac{86751197473}{18559048649400}a^{13}+\frac{115487433109}{18559048649400}a^{12}-\frac{27855683893}{3093174774900}a^{11}+\frac{1805234669}{22468581900}a^{10}+\frac{49281095746}{2319881081175}a^{9}+\frac{1556654274881}{18559048649400}a^{8}-\frac{888759001943}{18559048649400}a^{7}-\frac{5035169650369}{4639762162350}a^{6}+\frac{3000190293527}{1325646332100}a^{5}-\frac{946304012657}{331411583025}a^{4}+\frac{16999226875133}{6186349549800}a^{3}-\frac{64155151625837}{18559048649400}a^{2}+\frac{12051838181503}{1855904864940}a-\frac{18398004265363}{3711809729880}$, $\frac{29705009}{157280073300}a^{15}-\frac{137908217}{157280073300}a^{14}-\frac{18929332}{39320018325}a^{13}+\frac{644617567}{157280073300}a^{12}-\frac{123824321}{52426691100}a^{11}+\frac{3885354029}{157280073300}a^{10}-\frac{783216391}{22468581900}a^{9}-\frac{2364131597}{78640036650}a^{8}-\frac{20663563283}{157280073300}a^{7}-\frac{57873903271}{157280073300}a^{6}+\frac{153124611217}{157280073300}a^{5}-\frac{248389508461}{157280073300}a^{4}+\frac{22611399362}{13106672775}a^{3}-\frac{270814325459}{157280073300}a^{2}+\frac{76150566911}{31456014660}a-\frac{23271911884}{7864003665}$, $\frac{102412637}{314560146600}a^{15}-\frac{203761459}{157280073300}a^{14}-\frac{50452543}{44937163800}a^{13}+\frac{1987205767}{314560146600}a^{12}-\frac{15175283}{4368890925}a^{11}+\frac{2823074417}{78640036650}a^{10}-\frac{6931898657}{157280073300}a^{9}-\frac{11928995923}{314560146600}a^{8}-\frac{47706066641}{314560146600}a^{7}-\frac{11320985453}{22468581900}a^{6}+\frac{121587825067}{78640036650}a^{5}-\frac{334530379193}{157280073300}a^{4}+\frac{69288091127}{34951127400}a^{3}-\frac{32547711527}{12582405864}a^{2}+\frac{29988446179}{7864003665}a-\frac{65994255167}{12582405864}$, $\frac{22087141961}{18559048649400}a^{15}-\frac{542779756}{331411583025}a^{14}-\frac{24756334979}{3711809729880}a^{13}+\frac{72419889907}{18559048649400}a^{12}-\frac{9156529537}{618634954980}a^{11}+\frac{433321759}{4493716380}a^{10}+\frac{79081862413}{927952432470}a^{9}+\frac{777849970411}{3711809729880}a^{8}+\frac{760146206123}{3711809729880}a^{7}-\frac{519307171687}{463976216235}a^{6}+\frac{3493790813009}{1325646332100}a^{5}-\frac{1773731756227}{662823166050}a^{4}+\frac{700077629323}{247453981992}a^{3}-\frac{46422988848083}{18559048649400}a^{2}+\frac{2348884354079}{371180972988}a-\frac{12317624273617}{3711809729880}$, $\frac{4205992087}{6186349549800}a^{15}+\frac{1732753151}{3093174774900}a^{14}-\frac{9539028937}{2062116516600}a^{13}-\frac{47960312863}{6186349549800}a^{12}-\frac{9313619002}{773293693725}a^{11}+\frac{59624401}{1456296975}a^{10}+\frac{514987056053}{3093174774900}a^{9}+\frac{667375564789}{2062116516600}a^{8}+\frac{2973212177849}{6186349549800}a^{7}-\frac{827460480761}{3093174774900}a^{6}+\frac{1118840561}{9546835725}a^{5}+\frac{1730565481937}{3093174774900}a^{4}+\frac{6705146463413}{6186349549800}a^{3}+\frac{10415581909}{9164962296}a^{2}+\frac{259450602584}{154658738745}a+\frac{80774564455}{27494886888}$, $\frac{52192844}{331411583025}a^{15}-\frac{228570382}{463976216235}a^{14}-\frac{2977924139}{4639762162350}a^{13}+\frac{6272142862}{2319881081175}a^{12}-\frac{232409273}{73647018450}a^{11}+\frac{558164884}{39320018325}a^{10}-\frac{14672145343}{2319881081175}a^{9}-\frac{36827104817}{4639762162350}a^{8}+\frac{7404692188}{2319881081175}a^{7}-\frac{1054397714813}{4639762162350}a^{6}+\frac{317454292247}{463976216235}a^{5}-\frac{2246066604511}{2319881081175}a^{4}+\frac{236123974063}{515529129150}a^{3}+\frac{1039605804238}{2319881081175}a^{2}-\frac{106588020491}{132564633210}a+\frac{102791134157}{463976216235}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 73371.5821514 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 73371.5821514 \cdot 36}{2\cdot\sqrt{3455222492240908603515625}}\cr\approx \mathstrut & 1.72584316534 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^3:C_4$ (as 16T33):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3:C_4$
Character table for $C_2^3:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.121945.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.609725.1 x2, 8.0.371764575625.3 x2, 8.4.64097340625.2 x2, 8.0.371764575625.5

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.4.64097340625.2, 8.0.11051265625.2, 8.0.371764575625.3, 8.4.9294114390625.2
Degree 16 siblings: 16.0.86380562306022715087890625.4, 16.0.102711726879931884765625.4, 16.8.86380562306022715087890625.3
Minimal sibling: 8.0.11051265625.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.2.0.1}{2} }^{8}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
\(29\) Copy content Toggle raw display 29.8.6.1$x^{8} + 96 x^{7} + 3464 x^{6} + 55872 x^{5} + 345682 x^{4} + 114528 x^{3} + 113384 x^{2} + 1587648 x + 9488961$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} + 96 x^{7} + 3464 x^{6} + 55872 x^{5} + 345682 x^{4} + 114528 x^{3} + 113384 x^{2} + 1587648 x + 9488961$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$