Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 220*x^10 - 120*x^9 - 56*x^8 + 240*x^7 - 320*x^6 + 256*x^5 - 136*x^4 + 48*x^3 - 16*x^2 + 8*x - 2)
gp: K = bnfinit(y^16 - 8*y^15 + 32*y^14 - 84*y^13 + 158*y^12 - 220*y^11 + 220*y^10 - 120*y^9 - 56*y^8 + 240*y^7 - 320*y^6 + 256*y^5 - 136*y^4 + 48*y^3 - 16*y^2 + 8*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 220*x^10 - 120*x^9 - 56*x^8 + 240*x^7 - 320*x^6 + 256*x^5 - 136*x^4 + 48*x^3 - 16*x^2 + 8*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 220*x^10 - 120*x^9 - 56*x^8 + 240*x^7 - 320*x^6 + 256*x^5 - 136*x^4 + 48*x^3 - 16*x^2 + 8*x - 2)
\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 158 x^{12} - 220 x^{11} + 220 x^{10} - 120 x^{9} - 56 x^{8} + \cdots - 2 \)
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: |
\(21035720123168587776\)
\(\medspace = 2^{42}\cdot 3^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.13\) | 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: | $2^{21/8}3^{7/8}\approx 16.131874566820663$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{1}{7}a^{11}-\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{2}-\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{13}-\frac{1}{7}a^{11}-\frac{3}{7}a^{10}-\frac{2}{7}a^{9}-\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{3}-\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{14}+\frac{3}{7}a^{11}-\frac{2}{7}a^{10}+\frac{3}{7}a^{9}-\frac{2}{7}a^{8}-\frac{1}{7}a^{7}+\frac{2}{7}a^{6}+\frac{3}{7}a^{5}+\frac{2}{7}a^{4}-\frac{1}{7}a^{3}+\frac{3}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{9674539}a^{15}-\frac{43287}{1382077}a^{14}+\frac{654931}{9674539}a^{13}-\frac{604047}{9674539}a^{12}+\frac{164607}{1382077}a^{11}-\frac{4746422}{9674539}a^{10}+\frac{2017597}{9674539}a^{9}-\frac{1171384}{9674539}a^{8}-\frac{2053119}{9674539}a^{7}+\frac{757350}{9674539}a^{6}-\frac{4652975}{9674539}a^{5}-\frac{669469}{9674539}a^{4}+\frac{136138}{1382077}a^{3}-\frac{3359924}{9674539}a^{2}-\frac{3845755}{9674539}a+\frac{3940138}{9674539}$
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 $1$
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: |
$\frac{12287481}{9674539}a^{15}-\frac{85708570}{9674539}a^{14}+\frac{303358003}{9674539}a^{13}-\frac{101024748}{1382077}a^{12}+\frac{1166898359}{9674539}a^{11}-\frac{1392520406}{9674539}a^{10}+\frac{1090786477}{9674539}a^{9}-\frac{144788997}{9674539}a^{8}-\frac{985147183}{9674539}a^{7}+\frac{1923725281}{9674539}a^{6}-\frac{1762869558}{9674539}a^{5}+\frac{144347539}{1382077}a^{4}-\frac{369415100}{9674539}a^{3}+\frac{99157693}{9674539}a^{2}-\frac{95158850}{9674539}a+\frac{24157829}{9674539}$, $\frac{2860993}{9674539}a^{15}-\frac{23707073}{9674539}a^{14}+\frac{96094969}{9674539}a^{13}-\frac{253200576}{9674539}a^{12}+\frac{476088804}{9674539}a^{11}-\frac{660662897}{9674539}a^{10}+\frac{94266468}{1382077}a^{9}-\frac{363789575}{9674539}a^{8}-\frac{159478707}{9674539}a^{7}+\frac{694903299}{9674539}a^{6}-\frac{948661076}{9674539}a^{5}+\frac{747463082}{9674539}a^{4}-\frac{430978994}{9674539}a^{3}+\frac{26999401}{1382077}a^{2}-\frac{81996337}{9674539}a+\frac{29757547}{9674539}$, $\frac{18916993}{9674539}a^{15}-\frac{136718145}{9674539}a^{14}+\frac{500244797}{9674539}a^{13}-\frac{1206618368}{9674539}a^{12}+\frac{295995457}{1382077}a^{11}-\frac{2597704805}{9674539}a^{10}+\frac{2216161053}{9674539}a^{9}-\frac{631482104}{9674539}a^{8}-\frac{212464027}{1382077}a^{7}+\frac{3375356882}{9674539}a^{6}-\frac{3487429833}{9674539}a^{5}+\frac{2236379428}{9674539}a^{4}-\frac{928869244}{9674539}a^{3}+\frac{239481905}{9674539}a^{2}-\frac{143977165}{9674539}a+\frac{57755875}{9674539}$, $\frac{23521730}{9674539}a^{15}-\frac{168042122}{9674539}a^{14}+\frac{607059714}{9674539}a^{13}-\frac{206026997}{1382077}a^{12}+\frac{2427487962}{9674539}a^{11}-\frac{2958721770}{9674539}a^{10}+\frac{2393977811}{9674539}a^{9}-\frac{64406215}{1382077}a^{8}-\frac{1986361575}{9674539}a^{7}+\frac{4031752153}{9674539}a^{6}-\frac{554087341}{1382077}a^{5}+\frac{2253948584}{9674539}a^{4}-\frac{785383836}{9674539}a^{3}+\frac{159956715}{9674539}a^{2}-\frac{164241055}{9674539}a+\frac{74832385}{9674539}$, $\frac{27984}{22657}a^{15}-\frac{1374720}{158599}a^{14}+\frac{4884898}{158599}a^{13}-\frac{11407103}{158599}a^{12}+\frac{18804582}{158599}a^{11}-\frac{22300058}{158599}a^{10}+\frac{17126078}{158599}a^{9}-\frac{1528062}{158599}a^{8}-\frac{16905832}{158599}a^{7}+\frac{31726300}{158599}a^{6}-\frac{28541522}{158599}a^{5}+\frac{15331703}{158599}a^{4}-\frac{4439770}{158599}a^{3}+\frac{517878}{158599}a^{2}-\frac{1068240}{158599}a+\frac{116581}{158599}$, $\frac{7344431}{9674539}a^{15}-\frac{7504849}{1382077}a^{14}+\frac{27309123}{1382077}a^{13}-\frac{460262672}{9674539}a^{12}+\frac{791559739}{9674539}a^{11}-\frac{999286573}{9674539}a^{10}+\frac{866874000}{9674539}a^{9}-\frac{39121483}{1382077}a^{8}-\frac{76020338}{1382077}a^{7}+\frac{1278909029}{9674539}a^{6}-\frac{1349851288}{9674539}a^{5}+\frac{131094961}{1382077}a^{4}-\frac{414311378}{9674539}a^{3}+\frac{111115748}{9674539}a^{2}-\frac{44898750}{9674539}a+\frac{9384009}{9674539}$, $\frac{3634674}{9674539}a^{15}-\frac{35022847}{9674539}a^{14}+\frac{22152705}{1382077}a^{13}-\frac{434276659}{9674539}a^{12}+\frac{856483700}{9674539}a^{11}-\frac{1230692924}{9674539}a^{10}+\frac{1263104371}{9674539}a^{9}-\frac{724412504}{9674539}a^{8}-\frac{308109806}{9674539}a^{7}+\frac{1349184073}{9674539}a^{6}-\frac{1908098357}{9674539}a^{5}+\frac{205701036}{1382077}a^{4}-\frac{706795430}{9674539}a^{3}+\frac{188184630}{9674539}a^{2}-\frac{36077425}{9674539}a+\frac{6420793}{1382077}$, $\frac{14832906}{9674539}a^{15}-\frac{112615007}{9674539}a^{14}+\frac{429005361}{9674539}a^{13}-\frac{1071860270}{9674539}a^{12}+\frac{1906071445}{9674539}a^{11}-\frac{2477621611}{9674539}a^{10}+\frac{2226740197}{9674539}a^{9}-\frac{825464590}{9674539}a^{8}-\frac{1221714132}{9674539}a^{7}+\frac{443030554}{1382077}a^{6}-\frac{3475465831}{9674539}a^{5}+\frac{2331543059}{9674539}a^{4}-\frac{139731755}{1382077}a^{3}+\frac{258378779}{9674539}a^{2}-\frac{126471967}{9674539}a+\frac{64930427}{9674539}$, $\frac{1819}{1382077}a^{15}-\frac{2218767}{9674539}a^{14}+\frac{13604575}{9674539}a^{13}-\frac{42916333}{9674539}a^{12}+\frac{89292090}{9674539}a^{11}-\frac{129261431}{9674539}a^{10}+\frac{131312640}{9674539}a^{9}-\frac{10641378}{1382077}a^{8}-\frac{5779715}{1382077}a^{7}+\frac{19037959}{1382077}a^{6}-\frac{201237081}{9674539}a^{5}+\frac{116396627}{9674539}a^{4}-\frac{47510600}{9674539}a^{3}+\frac{26540706}{9674539}a^{2}-\frac{6538613}{9674539}a+\frac{10056593}{9674539}$
| 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: | \( 3695.54224592 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 3695.54224592 \cdot 1}{2\cdot\sqrt{21035720123168587776}}\cr\approx \mathstrut & 0.396614581059 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 220*x^10 - 120*x^9 - 56*x^8 + 240*x^7 - 320*x^6 + 256*x^5 - 136*x^4 + 48*x^3 - 16*x^2 + 8*x - 2)
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 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 220*x^10 - 120*x^9 - 56*x^8 + 240*x^7 - 320*x^6 + 256*x^5 - 136*x^4 + 48*x^3 - 16*x^2 + 8*x - 2, 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 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 220*x^10 - 120*x^9 - 56*x^8 + 240*x^7 - 320*x^6 + 256*x^5 - 136*x^4 + 48*x^3 - 16*x^2 + 8*x - 2);
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 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 220*x^10 - 120*x^9 - 56*x^8 + 240*x^7 - 320*x^6 + 256*x^5 - 136*x^4 + 48*x^3 - 16*x^2 + 8*x - 2);
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
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 $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), 4.2.6912.1, 4.2.1728.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.4.191102976.1 |
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
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.42.2 | $x^{16} + 4 x^{15} + 4 x^{13} + 2 x^{12} + 4 x^{11} + 4 x^{10} + 14$ | $16$ | $1$ | $42$ | 16T45 | $[2, 2, 3, 3]^{2}$ |
|
\(3\)
| 3.16.14.2 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34188 x^{9} + 53458 x^{8} + 68592 x^{7} + 71008 x^{6} + 56896 x^{5} + 33488 x^{4} + 14784 x^{3} + 6308 x^{2} + 2732 x + 661$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |