Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 54*x^13 + 113*x^12 - 196*x^11 + 274*x^10 - 312*x^9 + 298*x^8 - 256*x^7 + 186*x^6 - 92*x^5 - 7*x^4 + 38*x^3 - 11*x^2 - 2*x + 1)
gp: K = bnfinit(y^16 - 6*y^15 + 21*y^14 - 54*y^13 + 113*y^12 - 196*y^11 + 274*y^10 - 312*y^9 + 298*y^8 - 256*y^7 + 186*y^6 - 92*y^5 - 7*y^4 + 38*y^3 - 11*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 21*x^14 - 54*x^13 + 113*x^12 - 196*x^11 + 274*x^10 - 312*x^9 + 298*x^8 - 256*x^7 + 186*x^6 - 92*x^5 - 7*x^4 + 38*x^3 - 11*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 21*x^14 - 54*x^13 + 113*x^12 - 196*x^11 + 274*x^10 - 312*x^9 + 298*x^8 - 256*x^7 + 186*x^6 - 92*x^5 - 7*x^4 + 38*x^3 - 11*x^2 - 2*x + 1)
\( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 196 x^{11} + 274 x^{10} - 312 x^{9} + 298 x^{8} + \cdots + 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(74649600000000000000\)
\(\medspace = 2^{24}\cdot 3^{6}\cdot 5^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.46\) | 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: | $2^{3/2}3^{1/2}5^{7/8}\approx 20.031080424826147$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{3}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{3}{8}a+\frac{1}{8}$, $\frac{1}{16}a^{13}+\frac{1}{16}a^{11}-\frac{1}{16}a^{10}+\frac{1}{16}a^{9}+\frac{3}{16}a^{8}-\frac{3}{16}a^{7}+\frac{3}{16}a^{6}+\frac{5}{16}a^{5}-\frac{5}{16}a^{4}-\frac{7}{16}a^{3}-\frac{5}{16}a^{2}-\frac{1}{4}a+\frac{7}{16}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{13}-\frac{1}{32}a^{12}-\frac{1}{16}a^{10}+\frac{1}{16}a^{8}-\frac{1}{4}a^{7}-\frac{1}{16}a^{6}-\frac{3}{8}a^{5}-\frac{5}{16}a^{4}+\frac{3}{8}a^{3}-\frac{3}{32}a^{2}-\frac{7}{32}a-\frac{5}{32}$, $\frac{1}{5536}a^{15}+\frac{7}{692}a^{14}-\frac{35}{1384}a^{13}-\frac{257}{5536}a^{12}+\frac{67}{1384}a^{11}+\frac{63}{692}a^{10}-\frac{41}{346}a^{9}+\frac{307}{1384}a^{8}-\frac{181}{1384}a^{7}+\frac{305}{1384}a^{6}-\frac{20}{173}a^{5}+\frac{437}{1384}a^{4}-\frac{2697}{5536}a^{3}-\frac{155}{346}a^{2}-\frac{599}{1384}a+\frac{53}{5536}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
$\frac{1177}{2768}a^{15}-\frac{6575}{2768}a^{14}+\frac{11203}{1384}a^{13}-\frac{3541}{173}a^{12}+\frac{117697}{2768}a^{11}-\frac{202765}{2768}a^{10}+\frac{281977}{2768}a^{9}-\frac{325645}{2768}a^{8}+\frac{320965}{2768}a^{7}-\frac{289389}{2768}a^{6}+\frac{217769}{2768}a^{5}-\frac{122577}{2768}a^{4}+\frac{1125}{173}a^{3}+\frac{5311}{1384}a^{2}-\frac{3825}{2768}a+\frac{1831}{2768}$, $\frac{2893}{5536}a^{15}-\frac{14279}{5536}a^{14}+\frac{43569}{5536}a^{13}-\frac{49019}{2768}a^{12}+\frac{90621}{2768}a^{11}-\frac{8368}{173}a^{10}+\frac{142207}{2768}a^{9}-\frac{13199}{346}a^{8}+\frac{45229}{2768}a^{7}-\frac{3395}{1384}a^{6}-\frac{45363}{2768}a^{5}+\frac{44591}{1384}a^{4}-\frac{234363}{5536}a^{3}+\frac{79423}{5536}a^{2}+\frac{33525}{5536}a-\frac{9057}{2768}$, $\frac{3029}{5536}a^{15}-\frac{8435}{2768}a^{14}+\frac{27921}{2768}a^{13}-\frac{136623}{5536}a^{12}+\frac{68349}{1384}a^{11}-\frac{112607}{1384}a^{10}+\frac{72883}{692}a^{9}-\frac{37707}{346}a^{8}+\frac{129565}{1384}a^{7}-\frac{12907}{173}a^{6}+\frac{16029}{346}a^{5}-\frac{3707}{346}a^{4}-\frac{122637}{5536}a^{3}+\frac{50213}{2768}a^{2}+\frac{7199}{2768}a-\frac{15577}{5536}$, $a$, $\frac{7065}{5536}a^{15}-\frac{38763}{5536}a^{14}+\frac{127961}{5536}a^{13}-\frac{156427}{2768}a^{12}+\frac{313703}{2768}a^{11}-\frac{259049}{1384}a^{10}+\frac{674369}{2768}a^{9}-\frac{354077}{1384}a^{8}+\frac{622383}{2768}a^{7}-\frac{31775}{173}a^{6}+\frac{322263}{2768}a^{5}-\frac{23591}{692}a^{4}-\frac{244707}{5536}a^{3}+\frac{193135}{5536}a^{2}+\frac{18129}{5536}a-\frac{9219}{2768}$, $\frac{200}{173}a^{15}-\frac{36213}{5536}a^{14}+\frac{122105}{5536}a^{13}-\frac{304223}{5536}a^{12}+\frac{77557}{692}a^{11}-\frac{522885}{2768}a^{10}+\frac{43876}{173}a^{9}-\frac{766139}{2768}a^{8}+\frac{88059}{346}a^{7}-\frac{596941}{2768}a^{6}+\frac{206895}{1384}a^{5}-\frac{176469}{2768}a^{4}-\frac{31893}{1384}a^{3}+\frac{175307}{5536}a^{2}-\frac{545}{5536}a-\frac{14239}{5536}$, $\frac{1129}{1384}a^{15}-\frac{26845}{5536}a^{14}+\frac{93841}{5536}a^{13}-\frac{241463}{5536}a^{12}+\frac{126631}{1384}a^{11}-\frac{440593}{2768}a^{10}+\frac{309659}{1384}a^{9}-\frac{713299}{2768}a^{8}+\frac{346721}{1384}a^{7}-\frac{609225}{2768}a^{6}+\frac{113951}{692}a^{5}-\frac{248093}{2768}a^{4}+\frac{10267}{1384}a^{3}+\frac{117115}{5536}a^{2}-\frac{41917}{5536}a+\frac{2857}{5536}$, $\frac{35}{5536}a^{15}+\frac{57}{5536}a^{14}-\frac{1267}{5536}a^{13}+\frac{1341}{1384}a^{12}-\frac{7593}{2768}a^{11}+\frac{8389}{1384}a^{10}-\frac{30683}{2768}a^{9}+\frac{22163}{1384}a^{8}-\frac{49865}{2768}a^{7}+\frac{2913}{173}a^{6}-\frac{37669}{2768}a^{5}+\frac{6869}{692}a^{4}-\frac{22773}{5536}a^{3}-\frac{3933}{5536}a^{2}+\frac{23573}{5536}a-\frac{3}{346}$, $\frac{659}{5536}a^{15}-\frac{2135}{2768}a^{14}+\frac{8019}{2768}a^{13}-\frac{43765}{5536}a^{12}+\frac{12129}{692}a^{11}-\frac{5580}{173}a^{10}+\frac{67865}{1384}a^{9}-\frac{85559}{1384}a^{8}+\frac{45631}{692}a^{7}-\frac{86531}{1384}a^{6}+\frac{70847}{1384}a^{5}-\frac{48675}{1384}a^{4}+\frac{76545}{5536}a^{3}-\frac{2849}{2768}a^{2}-\frac{7349}{2768}a+\frac{6209}{5536}$
| 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: | \( 8932.80253984 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 8932.80253984 \cdot 1}{2\cdot\sqrt{74649600000000000000}}\cr\approx \mathstrut & 0.508912582527 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 54*x^13 + 113*x^12 - 196*x^11 + 274*x^10 - 312*x^9 + 298*x^8 - 256*x^7 + 186*x^6 - 92*x^5 - 7*x^4 + 38*x^3 - 11*x^2 - 2*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 - 6*x^15 + 21*x^14 - 54*x^13 + 113*x^12 - 196*x^11 + 274*x^10 - 312*x^9 + 298*x^8 - 256*x^7 + 186*x^6 - 92*x^5 - 7*x^4 + 38*x^3 - 11*x^2 - 2*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 - 6*x^15 + 21*x^14 - 54*x^13 + 113*x^12 - 196*x^11 + 274*x^10 - 312*x^9 + 298*x^8 - 256*x^7 + 186*x^6 - 92*x^5 - 7*x^4 + 38*x^3 - 11*x^2 - 2*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 - 6*x^15 + 21*x^14 - 54*x^13 + 113*x^12 - 196*x^11 + 274*x^10 - 312*x^9 + 298*x^8 - 256*x^7 + 186*x^6 - 92*x^5 - 7*x^4 + 38*x^3 - 11*x^2 - 2*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
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
| \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), 4.2.24000.1, 4.2.24000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.576000000.1 |
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 | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(3\)
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.14.2 | $x^{16} + 32 x^{15} + 464 x^{14} + 4032 x^{13} + 23408 x^{12} + 95872 x^{11} + 285376 x^{10} + 627456 x^{9} + 1027188 x^{8} + 1255232 x^{7} + 1144864 x^{6} + 789376 x^{5} + 469728 x^{4} + 388864 x^{3} + 473216 x^{2} + 436736 x + 184996$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |