Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 16*x^13 - 4*x^12 + 32*x^11 - 96*x^10 + 168*x^9 - 128*x^8 - 144*x^7 + 300*x^6 - 112*x^5 - 88*x^4 + 56*x^3 + 8*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 8*y^15 + 20*y^14 - 16*y^13 - 4*y^12 + 32*y^11 - 96*y^10 + 168*y^9 - 128*y^8 - 144*y^7 + 300*y^6 - 112*y^5 - 88*y^4 + 56*y^3 + 8*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 20*x^14 - 16*x^13 - 4*x^12 + 32*x^11 - 96*x^10 + 168*x^9 - 128*x^8 - 144*x^7 + 300*x^6 - 112*x^5 - 88*x^4 + 56*x^3 + 8*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 20*x^14 - 16*x^13 - 4*x^12 + 32*x^11 - 96*x^10 + 168*x^9 - 128*x^8 - 144*x^7 + 300*x^6 - 112*x^5 - 88*x^4 + 56*x^3 + 8*x^2 - 8*x + 1)
\( x^{16} - 8 x^{15} + 20 x^{14} - 16 x^{13} - 4 x^{12} + 32 x^{11} - 96 x^{10} + 168 x^{9} - 128 x^{8} + \cdots + 1 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2393397489569403764736\)
\(\medspace = 2^{52}\cdot 3^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.69\) | 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: | $2^{13/4}3^{3/4}\approx 21.686448086636275$ | ||
| Ramified primes: |
\(2\), \(3\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{1655262856227}a^{15}-\frac{67458952034}{1655262856227}a^{14}-\frac{96593215316}{1655262856227}a^{13}+\frac{168227253872}{1655262856227}a^{12}-\frac{51895102103}{1655262856227}a^{11}-\frac{343192761905}{1655262856227}a^{10}-\frac{13523421936}{551754285409}a^{9}-\frac{260618529550}{1655262856227}a^{8}-\frac{121813614307}{1655262856227}a^{7}-\frac{146680315578}{551754285409}a^{6}-\frac{211426679072}{1655262856227}a^{5}+\frac{719816825056}{1655262856227}a^{4}-\frac{654409508755}{1655262856227}a^{3}-\frac{770888388068}{1655262856227}a^{2}+\frac{323550606364}{1655262856227}a-\frac{665039146241}{1655262856227}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{2497351368391}{1655262856227}a^{15}-\frac{6478764800062}{551754285409}a^{14}+\frac{45715061794114}{1655262856227}a^{13}-\frac{10000395571075}{551754285409}a^{12}-\frac{16353181884920}{1655262856227}a^{11}+\frac{25320302634569}{551754285409}a^{10}-\frac{223273616615717}{1655262856227}a^{9}+\frac{371173942221224}{1655262856227}a^{8}-\frac{79793927508173}{551754285409}a^{7}-\frac{409729422731677}{1655262856227}a^{6}+\frac{657301038938179}{1655262856227}a^{5}-\frac{45142698825193}{551754285409}a^{4}-\frac{80779236777828}{551754285409}a^{3}+\frac{84327215840501}{1655262856227}a^{2}+\frac{38703869613530}{1655262856227}a-\frac{10750579865530}{1655262856227}$, $\frac{684717069826}{1655262856227}a^{15}-\frac{5077882047542}{1655262856227}a^{14}+\frac{10623514116088}{1655262856227}a^{13}-\frac{1323716539800}{551754285409}a^{12}-\frac{2229634116914}{551754285409}a^{11}+\frac{18665300951375}{1655262856227}a^{10}-\frac{54041022500462}{1655262856227}a^{9}+\frac{26972386698411}{551754285409}a^{8}-\frac{32269462837028}{1655262856227}a^{7}-\frac{129791823289162}{1655262856227}a^{6}+\frac{45043029599276}{551754285409}a^{5}+\frac{20620307242667}{1655262856227}a^{4}-\frac{22075686352836}{551754285409}a^{3}-\frac{3838360927057}{1655262856227}a^{2}+\frac{3541810229082}{551754285409}a+\frac{1670822136863}{1655262856227}$, $\frac{1065017185835}{1655262856227}a^{15}-\frac{8741853784168}{1655262856227}a^{14}+\frac{7582729077733}{551754285409}a^{13}-\frac{19168470822806}{1655262856227}a^{12}-\frac{5043983424937}{1655262856227}a^{11}+\frac{35779903708366}{1655262856227}a^{10}-\frac{106516525170517}{1655262856227}a^{9}+\frac{192164730969238}{1655262856227}a^{8}-\frac{150035377959436}{1655262856227}a^{7}-\frac{156588930614108}{1655262856227}a^{6}+\frac{359586218723138}{1655262856227}a^{5}-\frac{125290019886044}{1655262856227}a^{4}-\frac{37800189839377}{551754285409}a^{3}+\frac{65304683614655}{1655262856227}a^{2}+\frac{19267904261072}{1655262856227}a-\frac{2411991903231}{551754285409}$, $\frac{3514604155837}{1655262856227}a^{15}-\frac{8894833021380}{551754285409}a^{14}+\frac{19865880833574}{551754285409}a^{13}-\frac{11088930644159}{551754285409}a^{12}-\frac{24891132513767}{1655262856227}a^{11}+\frac{101158134526801}{1655262856227}a^{10}-\frac{297220994765011}{1655262856227}a^{9}+\frac{473696406735416}{1655262856227}a^{8}-\frac{270860120695150}{1655262856227}a^{7}-\frac{595449403005611}{1655262856227}a^{6}+\frac{801240754633531}{1655262856227}a^{5}-\frac{95920845238703}{1655262856227}a^{4}-\frac{319423242303191}{1655262856227}a^{3}+\frac{67880691911519}{1655262856227}a^{2}+\frac{47139365472542}{1655262856227}a-\frac{10336130750465}{1655262856227}$, $\frac{742226841473}{551754285409}a^{15}-\frac{17262389383408}{1655262856227}a^{14}+\frac{13431770668427}{551754285409}a^{13}-\frac{25941217134514}{1655262856227}a^{12}-\frac{14540963131894}{1655262856227}a^{11}+\frac{22303294643307}{551754285409}a^{10}-\frac{197206600824400}{1655262856227}a^{9}+\frac{326280259510165}{1655262856227}a^{8}-\frac{69252179839632}{551754285409}a^{7}-\frac{365463349045907}{1655262856227}a^{6}+\frac{571522652891573}{1655262856227}a^{5}-\frac{37277981203606}{551754285409}a^{4}-\frac{211751607890344}{1655262856227}a^{3}+\frac{66800128695794}{1655262856227}a^{2}+\frac{10818271784158}{551754285409}a-\frac{8538091055899}{1655262856227}$, $\frac{784313605751}{551754285409}a^{15}-\frac{6054654488300}{551754285409}a^{14}+\frac{14016839242880}{551754285409}a^{13}-\frac{26494045620769}{1655262856227}a^{12}-\frac{15394152806810}{1655262856227}a^{11}+\frac{70028154305246}{1655262856227}a^{10}-\frac{206530919111128}{1655262856227}a^{9}+\frac{339580878194489}{1655262856227}a^{8}-\frac{213138959135690}{1655262856227}a^{7}-\frac{386574984573812}{1655262856227}a^{6}+\frac{589304320465096}{1655262856227}a^{5}-\frac{109376921513929}{1655262856227}a^{4}-\frac{221072203930978}{1655262856227}a^{3}+\frac{66385246830380}{1655262856227}a^{2}+\frac{33952454381140}{1655262856227}a-\frac{9034015404259}{1655262856227}$, $\frac{179788821539}{551754285409}a^{15}-\frac{4069490976931}{1655262856227}a^{14}+\frac{8911270912201}{1655262856227}a^{13}-\frac{4440667307038}{1655262856227}a^{12}-\frac{4164989442907}{1655262856227}a^{11}+\frac{14389621866428}{1655262856227}a^{10}-\frac{43680900593114}{1655262856227}a^{9}+\frac{69446683401190}{1655262856227}a^{8}-\frac{36336966788984}{1655262856227}a^{7}-\frac{92236354892647}{1655262856227}a^{6}+\frac{110640321334190}{1655262856227}a^{5}+\frac{5544671977898}{1655262856227}a^{4}-\frac{50223479698586}{1655262856227}a^{3}-\frac{4340809262728}{1655262856227}a^{2}+\frac{5429603235460}{551754285409}a-\frac{436865803336}{551754285409}$, $\frac{4127542826831}{1655262856227}a^{15}-\frac{31987770842105}{1655262856227}a^{14}+\frac{74576569061912}{1655262856227}a^{13}-\frac{47570098696739}{1655262856227}a^{12}-\frac{9374679795706}{551754285409}a^{11}+\frac{41713207972295}{551754285409}a^{10}-\frac{121746362274900}{551754285409}a^{9}+\frac{200915628512187}{551754285409}a^{8}-\frac{126470911531715}{551754285409}a^{7}-\frac{229151686263329}{551754285409}a^{6}+\frac{355762274209966}{551754285409}a^{5}-\frac{67410405603033}{551754285409}a^{4}-\frac{411165118220165}{1655262856227}a^{3}+\frac{132058220572376}{1655262856227}a^{2}+\frac{63360355447540}{1655262856227}a-\frac{18231723644290}{1655262856227}$, $\frac{3270807172024}{551754285409}a^{15}-\frac{76542788791865}{1655262856227}a^{14}+\frac{180932908852948}{1655262856227}a^{13}-\frac{120547685261810}{1655262856227}a^{12}-\frac{63980495118695}{1655262856227}a^{11}+\frac{301348304037827}{1655262856227}a^{10}-\frac{293783739112516}{551754285409}a^{9}+\frac{14\!\cdots\!83}{1655262856227}a^{8}-\frac{958407010967615}{1655262856227}a^{7}-\frac{536528072698396}{551754285409}a^{6}+\frac{26\!\cdots\!28}{1655262856227}a^{5}-\frac{564935346894049}{1655262856227}a^{4}-\frac{328600913978411}{551754285409}a^{3}+\frac{354758792905519}{1655262856227}a^{2}+\frac{50040465543753}{551754285409}a-\frac{48007253606608}{1655262856227}$, $\frac{9665946}{166542193}a^{15}-\frac{232510478}{499626579}a^{14}+\frac{204506428}{166542193}a^{13}-\frac{672970051}{499626579}a^{12}+\frac{71347158}{166542193}a^{11}+\frac{1003493290}{499626579}a^{10}-\frac{3131650900}{499626579}a^{9}+\frac{1873090868}{166542193}a^{8}-\frac{5730495826}{499626579}a^{7}-\frac{1964833574}{499626579}a^{6}+\frac{3054336444}{166542193}a^{5}-\frac{9595503329}{499626579}a^{4}+\frac{301296794}{499626579}a^{3}+\frac{4015536776}{499626579}a^{2}-\frac{1585061464}{499626579}a-\frac{39171697}{499626579}$, $\frac{1526540447224}{551754285409}a^{15}-\frac{35426352848308}{1655262856227}a^{14}+\frac{27448609911307}{551754285409}a^{13}-\frac{52435262755283}{1655262856227}a^{12}-\frac{9978371979568}{551754285409}a^{11}+\frac{136938038235167}{1655262856227}a^{10}-\frac{403737519935528}{1655262856227}a^{9}+\frac{221953712568218}{551754285409}a^{8}-\frac{420895498654586}{1655262856227}a^{7}-\frac{752038333619719}{1655262856227}a^{6}+\frac{386942324452223}{551754285409}a^{5}-\frac{221210865124747}{1655262856227}a^{4}-\frac{432823811821322}{1655262856227}a^{3}+\frac{133185375526174}{1655262856227}a^{2}+\frac{66407269733614}{1655262856227}a-\frac{19227369316385}{1655262856227}$
| 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: | \( 107901.929008 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 107901.929008 \cdot 1}{2\cdot\sqrt{2393397489569403764736}}\cr\approx \mathstrut & 0.439998540912 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 16*x^13 - 4*x^12 + 32*x^11 - 96*x^10 + 168*x^9 - 128*x^8 - 144*x^7 + 300*x^6 - 112*x^5 - 88*x^4 + 56*x^3 + 8*x^2 - 8*x + 1)
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 - 8*x^15 + 20*x^14 - 16*x^13 - 4*x^12 + 32*x^11 - 96*x^10 + 168*x^9 - 128*x^8 - 144*x^7 + 300*x^6 - 112*x^5 - 88*x^4 + 56*x^3 + 8*x^2 - 8*x + 1, 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 - 8*x^15 + 20*x^14 - 16*x^13 - 4*x^12 + 32*x^11 - 96*x^10 + 168*x^9 - 128*x^8 - 144*x^7 + 300*x^6 - 112*x^5 - 88*x^4 + 56*x^3 + 8*x^2 - 8*x + 1);
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 - 8*x^15 + 20*x^14 - 16*x^13 - 4*x^12 + 32*x^11 - 96*x^10 + 168*x^9 - 128*x^8 - 144*x^7 + 300*x^6 - 112*x^5 - 88*x^4 + 56*x^3 + 8*x^2 - 8*x + 1);
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
$C_2^2:Q_8$ (as 16T31):
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 14 conjugacy class representatives for $C_2^2:Q_8$ |
| Character table for $C_2^2:Q_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.2.2048.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.2.18432.3, 8.4.1358954496.2, 8.4.764411904.3, 8.8.12230590464.1 |
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 16 sibling: | 16.0.2393397489569403764736.4 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.52.16 | $x^{16} + 8 x^{15} + 12 x^{12} + 8 x^{11} + 4 x^{10} + 2 x^{8} + 8 x^{6} + 8 x^{5} + 14$ | $16$ | $1$ | $52$ | 16T31 | $[2, 3, 3, 4]^{2}$ |
|
\(3\)
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |