Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 + 216*x^12 - 936*x^10 + 2094*x^8 - 2376*x^6 + 1224*x^4 - 216*x^2 + 9)
gp: K = bnfinit(y^16 - 24*y^14 + 216*y^12 - 936*y^10 + 2094*y^8 - 2376*y^6 + 1224*y^4 - 216*y^2 + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 24*x^14 + 216*x^12 - 936*x^10 + 2094*x^8 - 2376*x^6 + 1224*x^4 - 216*x^2 + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 24*x^14 + 216*x^12 - 936*x^10 + 2094*x^8 - 2376*x^6 + 1224*x^4 - 216*x^2 + 9)
\( x^{16} - 24x^{14} + 216x^{12} - 936x^{10} + 2094x^{8} - 2376x^{6} + 1224x^{4} - 216x^{2} + 9 \)
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: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1378596953991976568487936\)
\(\medspace = 2^{58}\cdot 3^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.26\) | 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: | $2^{29/8}3^{7/8}\approx 32.263749133641326$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{8}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a$, $\frac{1}{30}a^{10}+\frac{1}{30}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{4}-\frac{1}{10}a^{2}-\frac{1}{10}$, $\frac{1}{60}a^{11}-\frac{1}{60}a^{10}+\frac{1}{60}a^{9}-\frac{1}{60}a^{8}+\frac{1}{10}a^{7}-\frac{1}{10}a^{6}-\frac{3}{10}a^{5}+\frac{3}{10}a^{4}-\frac{1}{20}a^{3}+\frac{1}{20}a^{2}-\frac{1}{20}a+\frac{1}{20}$, $\frac{1}{60}a^{12}-\frac{1}{12}a^{8}+\frac{1}{10}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}+\frac{1}{20}$, $\frac{1}{60}a^{13}-\frac{1}{12}a^{9}+\frac{1}{10}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}+\frac{1}{20}a$, $\frac{1}{180}a^{14}-\frac{1}{60}a^{10}-\frac{1}{15}a^{8}-\frac{11}{60}a^{6}-\frac{1}{5}a^{4}+\frac{3}{20}a^{2}-\frac{1}{5}$, $\frac{1}{180}a^{15}-\frac{1}{60}a^{10}-\frac{1}{20}a^{9}-\frac{1}{60}a^{8}-\frac{1}{12}a^{7}-\frac{1}{10}a^{6}-\frac{1}{2}a^{5}+\frac{3}{10}a^{4}+\frac{1}{10}a^{3}+\frac{1}{20}a^{2}-\frac{1}{4}a+\frac{1}{20}$
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: | $15$ | 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: |
$\frac{1}{45}a^{14}-\frac{11}{20}a^{12}+\frac{77}{15}a^{10}-\frac{1379}{60}a^{8}+\frac{155}{3}a^{6}-\frac{1093}{20}a^{4}+21a^{2}-\frac{11}{20}$, $\frac{1}{18}a^{14}-\frac{77}{60}a^{12}+\frac{109}{10}a^{10}-\frac{867}{20}a^{8}+\frac{1288}{15}a^{6}-\frac{1639}{20}a^{4}+\frac{164}{5}a^{2}-\frac{71}{20}$, $\frac{1}{90}a^{14}-\frac{17}{60}a^{12}+\frac{83}{30}a^{10}-\frac{53}{4}a^{8}+\frac{491}{15}a^{6}-\frac{771}{20}a^{4}+\frac{77}{5}a^{2}+\frac{7}{20}$, $\frac{2}{45}a^{14}-\frac{31}{30}a^{12}+\frac{133}{15}a^{10}-\frac{538}{15}a^{8}+\frac{220}{3}a^{6}-\frac{751}{10}a^{4}+\frac{171}{5}a^{2}-\frac{21}{5}$, $\frac{1}{45}a^{15}+\frac{1}{36}a^{14}-\frac{3}{5}a^{13}-\frac{2}{3}a^{12}+\frac{25}{4}a^{11}+\frac{179}{30}a^{10}-\frac{639}{20}a^{9}-\frac{1517}{60}a^{8}+\frac{1261}{15}a^{7}+\frac{3173}{60}a^{6}-109a^{5}-\frac{252}{5}a^{4}+\frac{1183}{20}a^{3}+\frac{181}{10}a^{2}-\frac{141}{20}a-\frac{53}{20}$, $\frac{1}{6}a^{15}+\frac{13}{180}a^{14}-\frac{119}{30}a^{13}-\frac{17}{10}a^{12}+\frac{141}{4}a^{11}+\frac{89}{6}a^{10}-\frac{1799}{12}a^{9}-\frac{1223}{20}a^{8}+\frac{3267}{10}a^{7}+\frac{7573}{60}a^{6}-\frac{715}{2}a^{5}-\frac{251}{2}a^{4}+\frac{695}{4}a^{3}+\frac{523}{10}a^{2}-\frac{503}{20}a-\frac{117}{20}$, $\frac{1}{90}a^{15}+\frac{1}{90}a^{14}-\frac{7}{30}a^{13}-\frac{7}{30}a^{12}+\frac{97}{60}a^{11}+\frac{97}{60}a^{10}-\frac{73}{20}a^{9}-\frac{73}{20}a^{8}-\frac{101}{30}a^{7}-\frac{101}{30}a^{6}+\frac{219}{10}a^{5}+\frac{219}{10}a^{4}-\frac{383}{20}a^{3}-\frac{383}{20}a^{2}-\frac{11}{20}a+\frac{9}{20}$, $\frac{1}{12}a^{15}-\frac{1}{18}a^{14}-\frac{119}{60}a^{13}+\frac{13}{10}a^{12}+\frac{88}{5}a^{11}-\frac{673}{60}a^{10}-\frac{372}{5}a^{9}+\frac{2717}{60}a^{8}+\frac{3179}{20}a^{7}-\frac{541}{6}a^{6}-\frac{3281}{20}a^{5}+\frac{849}{10}a^{4}+\frac{677}{10}a^{3}-\frac{637}{20}a^{2}-\frac{11}{2}a+\frac{41}{20}$, $\frac{13}{90}a^{15}-\frac{1}{45}a^{14}-\frac{209}{60}a^{13}+\frac{3}{5}a^{12}+\frac{1891}{60}a^{11}-\frac{25}{4}a^{10}-\frac{4111}{30}a^{9}+\frac{639}{20}a^{8}+\frac{9151}{30}a^{7}-\frac{1261}{15}a^{6}-\frac{6741}{20}a^{5}+109a^{4}+\frac{3211}{20}a^{3}-\frac{1183}{20}a^{2}-\frac{45}{2}a+\frac{141}{20}$, $\frac{7}{45}a^{15}+\frac{13}{180}a^{14}-\frac{11}{3}a^{13}-\frac{26}{15}a^{12}+\frac{641}{20}a^{11}+\frac{467}{30}a^{10}-\frac{2647}{20}a^{9}-\frac{803}{12}a^{8}+\frac{8189}{30}a^{7}+\frac{1757}{12}a^{6}-\frac{2669}{10}a^{5}-\frac{786}{5}a^{4}+\frac{1973}{20}a^{3}+\frac{691}{10}a^{2}-\frac{83}{20}a-\frac{103}{20}$, $\frac{13}{90}a^{15}+\frac{1}{45}a^{14}-\frac{209}{60}a^{13}-\frac{3}{5}a^{12}+\frac{1891}{60}a^{11}+\frac{25}{4}a^{10}-\frac{4111}{30}a^{9}-\frac{639}{20}a^{8}+\frac{9151}{30}a^{7}+\frac{1261}{15}a^{6}-\frac{6741}{20}a^{5}-109a^{4}+\frac{3211}{20}a^{3}+\frac{1183}{20}a^{2}-\frac{45}{2}a-\frac{141}{20}$, $\frac{1}{12}a^{15}+\frac{1}{18}a^{14}-\frac{119}{60}a^{13}-\frac{13}{10}a^{12}+\frac{88}{5}a^{11}+\frac{673}{60}a^{10}-\frac{372}{5}a^{9}-\frac{2717}{60}a^{8}+\frac{3179}{20}a^{7}+\frac{541}{6}a^{6}-\frac{3281}{20}a^{5}-\frac{849}{10}a^{4}+\frac{677}{10}a^{3}+\frac{637}{20}a^{2}-\frac{11}{2}a-\frac{41}{20}$, $\frac{1}{45}a^{15}+\frac{1}{36}a^{14}-\frac{29}{60}a^{13}-\frac{7}{12}a^{12}+\frac{56}{15}a^{11}+\frac{251}{60}a^{10}-\frac{773}{60}a^{9}-\frac{719}{60}a^{8}+\frac{65}{3}a^{7}+\frac{641}{60}a^{6}-\frac{409}{20}a^{5}+\frac{129}{20}a^{4}+\frac{66}{5}a^{3}-\frac{221}{20}a^{2}-\frac{73}{20}a+\frac{29}{20}$, $\frac{1}{9}a^{15}-\frac{1}{45}a^{14}-\frac{53}{20}a^{13}+\frac{13}{30}a^{12}+\frac{473}{20}a^{11}-\frac{157}{60}a^{10}-\frac{3043}{30}a^{9}+\frac{47}{12}a^{8}+\frac{673}{3}a^{7}+\frac{161}{15}a^{6}-\frac{4999}{20}a^{5}-\frac{339}{10}a^{4}+\frac{2411}{20}a^{3}+\frac{499}{20}a^{2}-\frac{149}{10}a-\frac{77}{20}$, $\frac{1}{6}a^{15}-\frac{1}{45}a^{14}-\frac{119}{30}a^{13}+\frac{3}{5}a^{12}+\frac{141}{4}a^{11}-\frac{25}{4}a^{10}-\frac{1799}{12}a^{9}+\frac{639}{20}a^{8}+\frac{3267}{10}a^{7}-\frac{1261}{15}a^{6}-\frac{715}{2}a^{5}+109a^{4}+\frac{695}{4}a^{3}-\frac{1183}{20}a^{2}-\frac{503}{20}a+\frac{161}{20}$
| 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: | \( 12887977.1035 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 12887977.1035 \cdot 1}{2\cdot\sqrt{1378596953991976568487936}}\cr\approx \mathstrut & 0.359679787955 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 + 216*x^12 - 936*x^10 + 2094*x^8 - 2376*x^6 + 1224*x^4 - 216*x^2 + 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 - 24*x^14 + 216*x^12 - 936*x^10 + 2094*x^8 - 2376*x^6 + 1224*x^4 - 216*x^2 + 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 - 24*x^14 + 216*x^12 - 936*x^10 + 2094*x^8 - 2376*x^6 + 1224*x^4 - 216*x^2 + 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 - 24*x^14 + 216*x^12 - 936*x^10 + 2094*x^8 - 2376*x^6 + 1224*x^4 - 216*x^2 + 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
$\SD_{16}$ (as 16T12):
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 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.27648.1 x2, 4.4.13824.1 x2, 8.8.3057647616.1, 8.8.587068342272.1 x4 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.58.3 | $x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 6 x^{8} + 12 x^{4} + 8 x^{2} + 14$ | $16$ | $1$ | $58$ | $QD_{16}$ | $[2, 3, 7/2, 9/2]$ |
|
\(3\)
| 3.16.14.2 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34188 x^{9} + 53458 x^{8} + 68592 x^{7} + 71008 x^{6} + 56896 x^{5} + 33488 x^{4} + 14784 x^{3} + 6308 x^{2} + 2732 x + 661$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |