Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*x^2 + 2*x + 1)
gp: K = bnfinit(y^16 - 6*y^15 + 17*y^14 - 28*y^13 + 26*y^12 - 10*y^11 - 3*y^10 + 2*y^9 + y^8 + 6*y^7 - 10*y^6 + 2*y^5 + 6*y^4 - 4*y^3 - 2*y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*x^2 + 2*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*x^2 + 2*x + 1)
\( x^{16} - 6 x^{15} + 17 x^{14} - 28 x^{13} + 26 x^{12} - 10 x^{11} - 3 x^{10} + 2 x^{9} + x^{8} + \cdots + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(7771775074107392\)
\(\medspace = 2^{16}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.84\) | 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: | $2\cdot 17^{7/8}\approx 23.86012979730628$ | ||
| Ramified primes: |
\(2\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{67}a^{15}-\frac{29}{67}a^{14}+\frac{14}{67}a^{13}-\frac{15}{67}a^{12}-\frac{31}{67}a^{11}+\frac{33}{67}a^{10}-\frac{25}{67}a^{9}-\frac{26}{67}a^{8}-\frac{4}{67}a^{7}+\frac{31}{67}a^{6}+\frac{14}{67}a^{5}+\frac{15}{67}a^{4}-\frac{4}{67}a^{3}+\frac{21}{67}a^{2}-\frac{16}{67}a-\frac{32}{67}$
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{44}{67} a^{15} - \frac{137}{67} a^{14} + \frac{80}{67} a^{13} + \frac{546}{67} a^{12} - \frac{1699}{67} a^{11} + \frac{2390}{67} a^{10} - \frac{1971}{67} a^{9} + \frac{1067}{67} a^{8} - \frac{779}{67} a^{7} + \frac{694}{67} a^{6} - \frac{54}{67} a^{5} - \frac{546}{67} a^{4} + \frac{427}{67} a^{3} + \frac{53}{67} a^{2} - \frac{168}{67} a - \frac{68}{67} \)
(order $4$)
| 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: |
$\frac{69}{67}a^{15}-\frac{393}{67}a^{14}+\frac{1100}{67}a^{13}-\frac{1839}{67}a^{12}+\frac{1881}{67}a^{11}-\frac{1140}{67}a^{10}+\frac{419}{67}a^{9}-\frac{253}{67}a^{8}+\frac{126}{67}a^{7}+\frac{330}{67}a^{6}-\frac{508}{67}a^{5}+\frac{97}{67}a^{4}+\frac{260}{67}a^{3}-\frac{226}{67}a^{2}-\frac{32}{67}a+\frac{3}{67}$, $\frac{115}{67}a^{15}-\frac{655}{67}a^{14}+\frac{1878}{67}a^{13}-\frac{3266}{67}a^{12}+\frac{3604}{67}a^{11}-\frac{2570}{67}a^{10}+\frac{1346}{67}a^{9}-\frac{980}{67}a^{8}+\frac{679}{67}a^{7}+\frac{215}{67}a^{6}-\frac{802}{67}a^{5}+\frac{318}{67}a^{4}+\frac{277}{67}a^{3}-\frac{265}{67}a^{2}-\frac{98}{67}a+\frac{72}{67}$, $\frac{9}{67}a^{15}+\frac{7}{67}a^{14}-\frac{142}{67}a^{13}+\frac{468}{67}a^{12}-\frac{748}{67}a^{11}+\frac{565}{67}a^{10}-\frac{24}{67}a^{9}-\frac{301}{67}a^{8}+\frac{98}{67}a^{7}+\frac{11}{67}a^{6}+\frac{193}{67}a^{5}-\frac{200}{67}a^{4}-\frac{103}{67}a^{3}+\frac{256}{67}a^{2}-\frac{77}{67}a-\frac{87}{67}$, $\frac{111}{67}a^{15}-\frac{606}{67}a^{14}+\frac{1621}{67}a^{13}-\frac{2536}{67}a^{12}+\frac{2254}{67}a^{11}-\frac{893}{67}a^{10}-\frac{162}{67}a^{9}+\frac{129}{67}a^{8}-\frac{42}{67}a^{7}+\frac{560}{67}a^{6}-\frac{657}{67}a^{5}+\frac{57}{67}a^{4}+\frac{494}{67}a^{3}-\frac{349}{67}a^{2}-\frac{101}{67}a+\frac{66}{67}$, $\frac{84}{67}a^{15}-\frac{426}{67}a^{14}+\frac{1042}{67}a^{13}-\frac{1394}{67}a^{12}+\frac{813}{67}a^{11}+\frac{293}{67}a^{10}-\frac{827}{67}a^{9}+\frac{563}{67}a^{8}-\frac{470}{67}a^{7}+\frac{728}{67}a^{6}-\frac{499}{67}a^{5}-\frac{147}{67}a^{4}+\frac{401}{67}a^{3}-\frac{179}{67}a^{2}-\frac{138}{67}a+\frac{59}{67}$, $\frac{108}{67}a^{15}-\frac{720}{67}a^{14}+\frac{2316}{67}a^{13}-\frac{4568}{67}a^{12}+\frac{5831}{67}a^{11}-\frac{4878}{67}a^{10}+\frac{2727}{67}a^{9}-\frac{1334}{67}a^{8}+\frac{774}{67}a^{7}+\frac{266}{67}a^{6}-\frac{1302}{67}a^{5}+\frac{1084}{67}a^{4}-\frac{30}{67}a^{3}-\frac{479}{67}a^{2}+\frac{148}{67}a+\frac{95}{67}$, $\frac{59}{67}a^{15}-\frac{438}{67}a^{14}+\frac{1496}{67}a^{13}-\frac{3096}{67}a^{12}+\frac{4067}{67}a^{11}-\frac{3346}{67}a^{10}+\frac{1607}{67}a^{9}-\frac{529}{67}a^{8}+\frac{367}{67}a^{7}+\frac{221}{67}a^{6}-\frac{1050}{67}a^{5}+\frac{952}{67}a^{4}+\frac{32}{67}a^{3}-\frac{436}{67}a^{2}+\frac{128}{67}a+\frac{122}{67}$
| 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: | \( 25.2889722607 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 25.2889722607 \cdot 1}{4\cdot\sqrt{7771775074107392}}\cr\approx \mathstrut & 0.174200751443 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*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 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*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 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*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 + 17*x^14 - 28*x^13 + 26*x^12 - 10*x^11 - 3*x^10 + 2*x^9 + x^8 + 6*x^7 - 10*x^6 + 2*x^5 + 6*x^4 - 4*x^3 - 2*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 128 |
| The 32 conjugacy class representatives for $D_8:C_8$ |
| Character table for $D_8:C_8$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.0.272.1, 8.0.1257728.1 |
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
| Degree 16 sibling: | data not computed |
| Degree 32 siblings: | data not computed |
| 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 | $16$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | $16$ | $16$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $2$ | $8$ | $16$ | |||
|
\(17\)
| 17.4.2.2 | $x^{4} - 272 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.8.7.4 | $x^{8} + 34$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |