Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9)
gp: K = bnfinit(y^16 - 8*y^14 - 9*y^13 + 18*y^12 + 51*y^11 - 29*y^10 - 108*y^9 - 23*y^8 + 171*y^7 + 99*y^6 - 111*y^5 - 84*y^4 - 45*y^3 + 9*y^2 + 18*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9)
\( x^{16} - 8 x^{14} - 9 x^{13} + 18 x^{12} + 51 x^{11} - 29 x^{10} - 108 x^{9} - 23 x^{8} + 171 x^{7} + \cdots + 9 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(44499647686531640625\)
\(\medspace = 3^{12}\cdot 5^{8}\cdot 11^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.91\) | 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: | $3^{3/4}5^{1/2}11^{1/2}\approx 16.90527678711191$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}$, $\frac{1}{345}a^{14}-\frac{22}{345}a^{13}-\frac{8}{69}a^{12}-\frac{82}{345}a^{11}+\frac{37}{345}a^{10}-\frac{27}{115}a^{9}-\frac{89}{345}a^{8}-\frac{169}{345}a^{7}-\frac{61}{345}a^{6}+\frac{167}{345}a^{5}-\frac{44}{345}a^{4}+\frac{1}{23}a^{3}-\frac{9}{23}a^{2}+\frac{1}{23}a-\frac{2}{115}$, $\frac{1}{7645978659825}a^{15}-\frac{296115592}{2548659553275}a^{14}+\frac{740656696963}{7645978659825}a^{13}+\frac{56583131432}{849553184425}a^{12}-\frac{22764806036}{169910636885}a^{11}-\frac{41389091341}{849553184425}a^{10}+\frac{442781414486}{1529195731965}a^{9}-\frac{1046248247536}{2548659553275}a^{8}+\frac{75153932006}{1529195731965}a^{7}-\frac{3449746738}{12679898275}a^{6}+\frac{263772745427}{849553184425}a^{5}-\frac{4412954488}{12679898275}a^{4}-\frac{22743614153}{101946382131}a^{3}-\frac{4641697066}{169910636885}a^{2}-\frac{206891870519}{849553184425}a+\frac{290719203846}{849553184425}$
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: | $9$ | 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: |
$\frac{291105609442}{7645978659825}a^{15}-\frac{67089345749}{2548659553275}a^{14}-\frac{2407694789969}{7645978659825}a^{13}-\frac{323658728468}{2548659553275}a^{12}+\frac{170387882461}{169910636885}a^{11}+\frac{3884842244339}{2548659553275}a^{10}-\frac{4075149402592}{1529195731965}a^{9}-\frac{9635959688897}{2548659553275}a^{8}+\frac{3785321274461}{1529195731965}a^{7}+\frac{303472114192}{38039694825}a^{6}-\frac{852744050206}{849553184425}a^{5}-\frac{331841384818}{38039694825}a^{4}-\frac{31868821175}{101946382131}a^{3}+\frac{294738309333}{169910636885}a^{2}+\frac{1789456736852}{849553184425}a-\frac{75244256598}{849553184425}$, $\frac{57917776726}{7645978659825}a^{15}-\frac{58983065492}{2548659553275}a^{14}-\frac{231764879462}{7645978659825}a^{13}+\frac{272692798771}{2548659553275}a^{12}+\frac{19140991219}{169910636885}a^{11}-\frac{563154197998}{2548659553275}a^{10}-\frac{1261149143149}{1529195731965}a^{9}+\frac{2849955376564}{2548659553275}a^{8}+\frac{63319743172}{66486770955}a^{7}-\frac{31972043039}{38039694825}a^{6}-\frac{2450472266698}{849553184425}a^{5}+\frac{75794966486}{38039694825}a^{4}+\frac{258810673435}{101946382131}a^{3}-\frac{434206670346}{169910636885}a^{2}+\frac{221394464681}{849553184425}a-\frac{295727211404}{849553184425}$, $\frac{34863575218}{1529195731965}a^{15}+\frac{728324954}{101946382131}a^{14}-\frac{286412859872}{1529195731965}a^{13}-\frac{5959853744}{22162256985}a^{12}+\frac{65869719611}{169910636885}a^{11}+\frac{233074330636}{169910636885}a^{10}-\frac{23045559766}{66486770955}a^{9}-\frac{1505337926612}{509731910655}a^{8}-\frac{1968350131292}{1529195731965}a^{7}+\frac{6865748770}{1521587793}a^{6}+\frac{119229343670}{33982127377}a^{5}-\frac{7119374903}{2535979655}a^{4}-\frac{330317646481}{101946382131}a^{3}-\frac{12132425497}{33982127377}a^{2}+\frac{24313299608}{169910636885}a-\frac{37360951754}{169910636885}$, $\frac{336457452487}{7645978659825}a^{15}-\frac{1932576098}{849553184425}a^{14}-\frac{2572311186629}{7645978659825}a^{13}-\frac{939772164973}{2548659553275}a^{12}+\frac{69275630407}{101946382131}a^{11}+\frac{5098527023269}{2548659553275}a^{10}-\frac{1689932554654}{1529195731965}a^{9}-\frac{3203831357824}{849553184425}a^{8}-\frac{1740858506587}{1529195731965}a^{7}+\frac{207769494527}{38039694825}a^{6}+\frac{9446218908667}{2548659553275}a^{5}-\frac{67179488963}{38039694825}a^{4}-\frac{247780480478}{101946382131}a^{3}-\frac{639521745622}{169910636885}a^{2}-\frac{261199593828}{849553184425}a+\frac{467335939982}{849553184425}$, $\frac{76653546817}{2548659553275}a^{15}+\frac{60613263713}{2548659553275}a^{14}-\frac{607416247189}{2548659553275}a^{13}-\frac{1095755203879}{2548659553275}a^{12}+\frac{30748979692}{101946382131}a^{11}+\frac{1461605294654}{849553184425}a^{10}+\frac{77477958221}{509731910655}a^{9}-\frac{8630394806431}{2548659553275}a^{8}-\frac{1044651033977}{509731910655}a^{7}+\frac{129618599546}{38039694825}a^{6}+\frac{12465647724016}{2548659553275}a^{5}-\frac{13270695583}{12679898275}a^{4}-\frac{1337967132}{1477483799}a^{3}-\frac{440480944636}{169910636885}a^{2}-\frac{2030425592944}{849553184425}a-\frac{110201509139}{849553184425}$, $\frac{104186303884}{1529195731965}a^{15}-\frac{4139535497}{509731910655}a^{14}-\frac{844141267454}{1529195731965}a^{13}-\frac{91910372347}{169910636885}a^{12}+\frac{233169468391}{169910636885}a^{11}+\frac{338385602066}{101946382131}a^{10}-\frac{4111311474023}{1529195731965}a^{9}-\frac{3769837259663}{509731910655}a^{8}+\frac{141564792013}{1529195731965}a^{7}+\frac{32437324222}{2535979655}a^{6}+\frac{696125702922}{169910636885}a^{5}-\frac{76909200638}{7607938965}a^{4}-\frac{470494632526}{101946382131}a^{3}+\frac{5140647230}{33982127377}a^{2}+\frac{43775347054}{169910636885}a+\frac{354689267277}{169910636885}$, $\frac{516755262818}{7645978659825}a^{15}-\frac{25319695586}{2548659553275}a^{14}-\frac{4006161408811}{7645978659825}a^{13}-\frac{437476714424}{849553184425}a^{12}+\frac{595468221688}{509731910655}a^{11}+\frac{2574457393342}{849553184425}a^{10}-\frac{3453782824469}{1529195731965}a^{9}-\frac{15452643599153}{2548659553275}a^{8}-\frac{154999141390}{305839146393}a^{7}+\frac{123995051021}{12679898275}a^{6}+\frac{10478285851223}{2548659553275}a^{5}-\frac{79070278564}{12679898275}a^{4}-\frac{212636391241}{101946382131}a^{3}-\frac{445830238058}{169910636885}a^{2}+\frac{278403258308}{849553184425}a+\frac{16789113356}{36937094975}$, $\frac{138093551798}{7645978659825}a^{15}-\frac{114565895071}{2548659553275}a^{14}-\frac{1094377884946}{7645978659825}a^{13}+\frac{494261939383}{2548659553275}a^{12}+\frac{119472942426}{169910636885}a^{11}+\frac{83924686962}{849553184425}a^{10}-\frac{4045302281444}{1529195731965}a^{9}-\frac{900073774258}{2548659553275}a^{8}+\frac{1294414049552}{305839146393}a^{7}+\frac{126417234418}{38039694825}a^{6}-\frac{5052322409899}{849553184425}a^{5}-\frac{57840775354}{12679898275}a^{4}+\frac{425815136378}{101946382131}a^{3}+\frac{305398137427}{169910636885}a^{2}+\frac{483602549238}{849553184425}a-\frac{331790129332}{849553184425}$, $\frac{38798143237}{1529195731965}a^{15}-\frac{639140829}{33982127377}a^{14}-\frac{311760607868}{1529195731965}a^{13}-\frac{12454997171}{169910636885}a^{12}+\frac{312373614842}{509731910655}a^{11}+\frac{20327028319}{22162256985}a^{10}-\frac{2502943452617}{1529195731965}a^{9}-\frac{331366176431}{169910636885}a^{8}+\frac{2660050169302}{1529195731965}a^{7}+\frac{2180541654}{507195931}a^{6}-\frac{98883361514}{101946382131}a^{5}-\frac{34570213021}{7607938965}a^{4}+\frac{64539921410}{101946382131}a^{3}-\frac{8855814086}{33982127377}a^{2}+\frac{46210466967}{169910636885}a+\frac{12720008699}{169910636885}$
| 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: | \( 3619.65458774 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 3619.65458774 \cdot 1}{2\cdot\sqrt{44499647686531640625}}\cr\approx \mathstrut & 0.267090253319 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9)
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^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9, 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^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9);
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^14 - 9*x^13 + 18*x^12 + 51*x^11 - 29*x^10 - 108*x^9 - 23*x^8 + 171*x^7 + 99*x^6 - 111*x^5 - 84*x^4 - 45*x^3 + 9*x^2 + 18*x + 9);
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
$C_2\times D_8$ (as 16T29):
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 $C_2\times D_8$ |
| Character table for $C_2\times D_8$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{165}) \), 4.2.2475.1, 4.2.275.1, \(\Q(\sqrt{5}, \sqrt{33})\), 8.2.606436875.2, 8.2.606436875.1, 8.4.741200625.1 |
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 siblings: | 16.0.367765683359765625.1, 16.0.1779985907461265625.1, 16.0.44499647686531640625.2 |
| Minimal sibling: | 16.0.367765683359765625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
| 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(11\)
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |