Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 36*x^12 - 128*x^10 + 418*x^8 - 1632*x^6 + 2808*x^4 + 864*x^2 + 324)
gp: K = bnfinit(y^16 - 8*y^14 + 36*y^12 - 128*y^10 + 418*y^8 - 1632*y^6 + 2808*y^4 + 864*y^2 + 324, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 36*x^12 - 128*x^10 + 418*x^8 - 1632*x^6 + 2808*x^4 + 864*x^2 + 324);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 36*x^12 - 128*x^10 + 418*x^8 - 1632*x^6 + 2808*x^4 + 864*x^2 + 324)
\( x^{16} - 8x^{14} + 36x^{12} - 128x^{10} + 418x^{8} - 1632x^{6} + 2808x^{4} + 864x^{2} + 324 \)
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: |
\(30257271966902092038144\)
\(\medspace = 2^{62}\cdot 3^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.41\) | 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: | $2^{31/8}3^{1/2}\approx 25.412761497225222$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{5}-\frac{2}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{18}a^{8}+\frac{1}{9}a^{6}+\frac{2}{9}a^{4}$, $\frac{1}{18}a^{9}+\frac{1}{3}a^{5}+\frac{2}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{18}a^{10}-\frac{4}{9}a^{4}$, $\frac{1}{18}a^{11}-\frac{4}{9}a^{5}$, $\frac{1}{270}a^{12}+\frac{2}{135}a^{10}-\frac{7}{135}a^{6}+\frac{10}{27}a^{4}-\frac{1}{45}a^{2}+\frac{2}{15}$, $\frac{1}{270}a^{13}+\frac{2}{135}a^{11}-\frac{7}{135}a^{7}+\frac{10}{27}a^{5}-\frac{1}{45}a^{3}+\frac{2}{15}a$, $\frac{1}{10451430}a^{14}-\frac{2302}{5225715}a^{12}+\frac{47011}{3483810}a^{10}-\frac{83009}{10451430}a^{8}+\frac{306611}{5225715}a^{6}-\frac{12304}{193545}a^{4}-\frac{94934}{193545}a^{2}-\frac{49877}{193545}$, $\frac{1}{31354290}a^{15}-\frac{2302}{15677145}a^{13}-\frac{73267}{5225715}a^{11}+\frac{248813}{15677145}a^{9}-\frac{854659}{15677145}a^{7}-\frac{205849}{580635}a^{5}-\frac{159449}{580635}a^{3}-\frac{49877}{580635}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
\( \frac{430}{1045143} a^{14} - \frac{5561}{1045143} a^{12} + \frac{8632}{348381} a^{10} - \frac{201889}{2090286} a^{8} + \frac{309424}{1045143} a^{6} - \frac{43990}{38709} a^{4} + \frac{110168}{38709} a^{2} + \frac{34061}{38709} \)
(order $6$)
| 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: |
$\frac{3104}{1045143}a^{14}-\frac{53099}{2090286}a^{12}+\frac{40826}{348381}a^{10}-\frac{438631}{1045143}a^{8}+\frac{1435664}{1045143}a^{6}-\frac{620362}{116127}a^{4}+\frac{128548}{12903}a^{2}+\frac{48487}{38709}$, $\frac{355}{6270858}a^{15}+\frac{2216}{1741905}a^{14}+\frac{159337}{31354290}a^{13}-\frac{1057}{696762}a^{12}-\frac{129442}{5225715}a^{11}+\frac{8011}{1161270}a^{10}+\frac{668699}{6270858}a^{9}+\frac{33157}{3483810}a^{8}-\frac{5056864}{15677145}a^{7}+\frac{59216}{1741905}a^{6}+\frac{135736}{116127}a^{5}-\frac{16153}{64515}a^{4}-\frac{2663348}{580635}a^{3}-\frac{53765}{12903}a^{2}-\frac{34517}{580635}a-\frac{103892}{64515}$, $\frac{63713}{15677145}a^{15}-\frac{1757}{348381}a^{14}-\frac{115697}{3135429}a^{13}+\frac{132403}{3483810}a^{12}+\frac{1870423}{10451430}a^{11}-\frac{195571}{1161270}a^{10}-\frac{20979899}{31354290}a^{9}+\frac{408401}{696762}a^{8}+\frac{34818998}{15677145}a^{7}-\frac{3339811}{1741905}a^{6}-\frac{4866104}{580635}a^{5}+\frac{870733}{116127}a^{4}+\frac{2114678}{116127}a^{3}-\frac{2182111}{193545}a^{2}-\frac{4292591}{580635}a-\frac{151456}{21505}$, $\frac{63713}{15677145}a^{15}-\frac{1757}{348381}a^{14}+\frac{115697}{3135429}a^{13}+\frac{132403}{3483810}a^{12}-\frac{1870423}{10451430}a^{11}-\frac{195571}{1161270}a^{10}+\frac{20979899}{31354290}a^{9}+\frac{408401}{696762}a^{8}-\frac{34818998}{15677145}a^{7}-\frac{3339811}{1741905}a^{6}+\frac{4866104}{580635}a^{5}+\frac{870733}{116127}a^{4}-\frac{2114678}{116127}a^{3}-\frac{2182111}{193545}a^{2}+\frac{4292591}{580635}a-\frac{151456}{21505}$, $\frac{7162}{15677145}a^{15}+\frac{779}{1741905}a^{14}+\frac{103687}{31354290}a^{13}-\frac{11867}{3483810}a^{12}-\frac{79318}{5225715}a^{11}+\frac{18613}{1161270}a^{10}+\frac{1622341}{31354290}a^{9}-\frac{233507}{3483810}a^{8}-\frac{2411491}{15677145}a^{7}+\frac{417968}{1741905}a^{6}+\frac{310091}{580635}a^{5}-\frac{17229}{21505}a^{4}-\frac{244808}{580635}a^{3}+\frac{47228}{64515}a^{2}-\frac{865463}{580635}a+\frac{96724}{64515}$, $\frac{63713}{15677145}a^{15}+\frac{7885}{2090286}a^{14}-\frac{115697}{3135429}a^{13}-\frac{315871}{10451430}a^{12}+\frac{1870423}{10451430}a^{11}+\frac{460457}{3483810}a^{10}-\frac{20979899}{31354290}a^{9}-\frac{963209}{2090286}a^{8}+\frac{34818998}{15677145}a^{7}+\frac{7756537}{5225715}a^{6}-\frac{4866104}{580635}a^{5}-\frac{677707}{116127}a^{4}+\frac{2114678}{116127}a^{3}+\frac{622298}{64515}a^{2}-\frac{4292591}{580635}a+\frac{1397096}{193545}$, $\frac{7162}{15677145}a^{15}+\frac{388}{580635}a^{14}+\frac{103687}{31354290}a^{13}-\frac{1435}{232254}a^{12}-\frac{79318}{5225715}a^{11}+\frac{28469}{1161270}a^{10}+\frac{1622341}{31354290}a^{9}-\frac{93529}{1161270}a^{8}-\frac{2411491}{15677145}a^{7}+\frac{136448}{580635}a^{6}+\frac{310091}{580635}a^{5}-\frac{615113}{580635}a^{4}-\frac{244808}{580635}a^{3}+\frac{100103}{38709}a^{2}-\frac{865463}{580635}a+\frac{47752}{64515}$
| 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: | \( 200959.74917378288 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 200959.74917378288 \cdot 1}{6\cdot\sqrt{30257271966902092038144}}\cr\approx \mathstrut & 0.467715639395282 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 36*x^12 - 128*x^10 + 418*x^8 - 1632*x^6 + 2808*x^4 + 864*x^2 + 324)
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^14 + 36*x^12 - 128*x^10 + 418*x^8 - 1632*x^6 + 2808*x^4 + 864*x^2 + 324, 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^14 + 36*x^12 - 128*x^10 + 418*x^8 - 1632*x^6 + 2808*x^4 + 864*x^2 + 324);
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^14 + 36*x^12 - 128*x^10 + 418*x^8 - 1632*x^6 + 2808*x^4 + 864*x^2 + 324);
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 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 4.0.6144.1 x2, 4.2.18432.2 x2, 8.0.339738624.10, 8.0.57982058496.7 x4, 8.2.173946175488.2 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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.62.8 | $x^{16} - 16 x^{15} + 288 x^{14} + 144 x^{13} + 704 x^{12} + 672 x^{11} + 928 x^{9} + 500 x^{8} - 32 x^{7} + 2368 x^{6} + 928 x^{5} + 704 x^{4} + 1856 x^{3} + 256 x^{2} + 1856 x + 3364$ | $8$ | $2$ | $62$ | $D_{8}$ | $[3, 4, 5]^{2}$ |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |