Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 70*x^10 - 108*x^9 + 155*x^8 - 148*x^7 + 78*x^6 - 4*x^5 - 6*x^4 - 20*x^3 + 22*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 8*y^15 + 28*y^14 - 56*y^13 + 72*y^12 - 68*y^11 + 70*y^10 - 108*y^9 + 155*y^8 - 148*y^7 + 78*y^6 - 4*y^5 - 6*y^4 - 20*y^3 + 22*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 70*x^10 - 108*x^9 + 155*x^8 - 148*x^7 + 78*x^6 - 4*x^5 - 6*x^4 - 20*x^3 + 22*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 70*x^10 - 108*x^9 + 155*x^8 - 148*x^7 + 78*x^6 - 4*x^5 - 6*x^4 - 20*x^3 + 22*x^2 - 8*x + 1)
\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 72 x^{12} - 68 x^{11} + 70 x^{10} - 108 x^{9} + 155 x^{8} + \cdots + 1 \)
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: |
\(41943040000000000\)
\(\medspace = 2^{32}\cdot 5^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.94\) | 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: | $2^{2}5^{3/4}\approx 13.37480609952844$ | ||
| Ramified primes: |
\(2\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{247}a^{14}-\frac{7}{247}a^{13}-\frac{8}{247}a^{12}-\frac{108}{247}a^{11}-\frac{51}{247}a^{10}+\frac{49}{247}a^{9}+\frac{116}{247}a^{8}+\frac{69}{247}a^{7}+\frac{71}{247}a^{6}-\frac{102}{247}a^{5}-\frac{107}{247}a^{4}-\frac{9}{19}a^{3}+\frac{16}{247}a^{2}-\frac{69}{247}a-\frac{17}{247}$, $\frac{1}{4199}a^{15}+\frac{1}{4199}a^{14}-\frac{558}{4199}a^{13}-\frac{913}{4199}a^{12}+\frac{814}{4199}a^{11}+\frac{135}{4199}a^{10}-\frac{1962}{4199}a^{9}-\frac{732}{4199}a^{8}+\frac{129}{4199}a^{7}+\frac{1948}{4199}a^{6}+\frac{43}{323}a^{5}-\frac{1467}{4199}a^{4}+\frac{4}{247}a^{3}-\frac{1176}{4199}a^{2}+\frac{913}{4199}a-\frac{1124}{4199}$
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: |
\( -\frac{146}{19} a^{14} + \frac{1022}{19} a^{13} - \frac{3031}{19} a^{12} + \frac{4900}{19} a^{11} - \frac{4885}{19} a^{10} + \frac{3866}{19} a^{9} - \frac{5175}{19} a^{8} + \frac{9648}{19} a^{7} - \frac{11715}{19} a^{6} + \frac{7558}{19} a^{5} - \frac{1003}{19} a^{4} - \frac{2260}{19} a^{3} - \frac{1101}{19} a^{2} + \frac{2322}{19} a - \frac{615}{19} \)
(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: |
$\frac{62778}{4199}a^{15}-\frac{458272}{4199}a^{14}+\frac{1436610}{4199}a^{13}-\frac{2507974}{4199}a^{12}+\frac{2760806}{4199}a^{11}-\frac{2335117}{4199}a^{10}+\frac{2764442}{4199}a^{9}-\frac{4850879}{4199}a^{8}+\frac{6334170}{4199}a^{7}-\frac{4851337}{4199}a^{6}+\frac{1505458}{4199}a^{5}+\frac{784582}{4199}a^{4}+\frac{11132}{247}a^{3}-\frac{1127227}{4199}a^{2}+\frac{580038}{4199}a-\frac{92695}{4199}$, $\frac{52140}{4199}a^{15}-\frac{403086}{4199}a^{14}+\frac{1348171}{4199}a^{13}-\frac{2534678}{4199}a^{12}+\frac{3007468}{4199}a^{11}-\frac{2635341}{4199}a^{10}+\frac{2843294}{4199}a^{9}-\frac{4788221}{4199}a^{8}+\frac{351684}{221}a^{7}-\frac{5712874}{4199}a^{6}+\frac{2289966}{4199}a^{5}+\frac{554654}{4199}a^{4}-\frac{10698}{247}a^{3}-\frac{1130724}{4199}a^{2}+\frac{808039}{4199}a-\frac{150996}{4199}$, $\frac{171602}{4199}a^{15}-\frac{1295481}{4199}a^{14}+\frac{4223101}{4199}a^{13}-\frac{7721040}{4199}a^{12}+\frac{8918668}{4199}a^{11}-\frac{406217}{221}a^{10}+\frac{8600459}{4199}a^{9}-\frac{14709033}{4199}a^{8}+\frac{20042122}{4199}a^{7}-\frac{16502064}{4199}a^{6}+\frac{6111611}{4199}a^{5}+\frac{103469}{221}a^{4}-\frac{7792}{247}a^{3}-\frac{3460906}{4199}a^{2}+\frac{2236840}{4199}a-\frac{400137}{4199}$, $\frac{62778}{4199}a^{15}-\frac{475204}{4199}a^{14}+\frac{1555134}{4199}a^{13}-\frac{2859602}{4199}a^{12}+\frac{3329762}{4199}a^{11}-\frac{2903444}{4199}a^{10}+\frac{3215469}{4199}a^{9}-\frac{5454515}{4199}a^{8}+\frac{573732}{323}a^{7}-\frac{6217270}{4199}a^{6}+\frac{2388523}{4199}a^{5}+\frac{660567}{4199}a^{4}-\frac{3740}{247}a^{3}-\frac{1263771}{4199}a^{2}+\frac{858158}{4199}a-\frac{8735}{221}$, $\frac{1686}{247}a^{15}-\frac{11555}{247}a^{14}+\frac{33195}{247}a^{13}-\frac{50931}{247}a^{12}+\frac{46653}{247}a^{11}-\frac{34217}{247}a^{10}+\frac{51328}{247}a^{9}-\frac{101273}{247}a^{8}+\frac{116742}{247}a^{7}-\frac{63045}{247}a^{6}-\frac{6020}{247}a^{5}+\frac{2325}{19}a^{4}+\frac{16849}{247}a^{3}-\frac{26179}{247}a^{2}+\frac{2213}{247}a+8$, $\frac{77335}{4199}a^{15}-\frac{595202}{4199}a^{14}+\frac{1982801}{4199}a^{13}-\frac{3715408}{4199}a^{12}+\frac{4399528}{4199}a^{11}-\frac{3855415}{4199}a^{10}+\frac{4172840}{4199}a^{9}-\frac{369921}{221}a^{8}+\frac{9781157}{4199}a^{7}-\frac{8345654}{4199}a^{6}+\frac{3335277}{4199}a^{5}+\frac{815939}{4199}a^{4}-\frac{14126}{247}a^{3}-\frac{1648581}{4199}a^{2}+\frac{91930}{323}a-\frac{228535}{4199}$, $\frac{72613}{4199}a^{15}-\frac{552766}{4199}a^{14}+\frac{1818559}{4199}a^{13}-\frac{3359034}{4199}a^{12}+\frac{3919542}{4199}a^{11}-\frac{262198}{323}a^{10}+\frac{3751603}{4199}a^{9}-\frac{6377896}{4199}a^{8}+\frac{8768592}{4199}a^{7}-\frac{7326328}{4199}a^{6}+\frac{2801316}{4199}a^{5}+\frac{811856}{4199}a^{4}-\frac{7573}{247}a^{3}-\frac{1500914}{4199}a^{2}+\frac{1011844}{4199}a-\frac{186230}{4199}$
| 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: | \( 78.6198421644 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 78.6198421644 \cdot 1}{4\cdot\sqrt{41943040000000000}}\cr\approx \mathstrut & 0.233120878392 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 70*x^10 - 108*x^9 + 155*x^8 - 148*x^7 + 78*x^6 - 4*x^5 - 6*x^4 - 20*x^3 + 22*x^2 - 8*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 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 70*x^10 - 108*x^9 + 155*x^8 - 148*x^7 + 78*x^6 - 4*x^5 - 6*x^4 - 20*x^3 + 22*x^2 - 8*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 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 70*x^10 - 108*x^9 + 155*x^8 - 148*x^7 + 78*x^6 - 4*x^5 - 6*x^4 - 20*x^3 + 22*x^2 - 8*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 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 70*x^10 - 108*x^9 + 155*x^8 - 148*x^7 + 78*x^6 - 4*x^5 - 6*x^4 - 20*x^3 + 22*x^2 - 8*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
$C_4\wr C_2$ (as 16T42):
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 $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.0.320.1 x2, 4.2.400.1 x2, \(\Q(i, \sqrt{5})\), 8.0.204800000.1, 8.0.8192000.1, 8.0.2560000.1 |
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 8 siblings: | 8.0.204800000.1, 8.0.8192000.1 |
| Degree 16 sibling: | 16.4.1048576000000000000.1 |
| Minimal sibling: | 8.0.8192000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 $16$ | $8$ | $2$ | $32$ | |||
|
\(5\)
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |