Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 144*x^12 + 6408*x^8 - 10368*x^6 - 95904*x^4 + 378432*x^2 + 187272)
gp: K = bnfinit(y^16 - 144*y^12 + 6408*y^8 - 10368*y^6 - 95904*y^4 + 378432*y^2 + 187272, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 144*x^12 + 6408*x^8 - 10368*x^6 - 95904*x^4 + 378432*x^2 + 187272);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 144*x^12 + 6408*x^8 - 10368*x^6 - 95904*x^4 + 378432*x^2 + 187272)
\( x^{16} - 144x^{12} + 6408x^{8} - 10368x^{6} - 95904x^{4} + 378432x^{2} + 187272 \)
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: |
\(321236373250909071617512439808\)
\(\medspace = 2^{79}\cdot 3^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(69.85\) | 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: | $2^{2849/512}3^{3/4}\approx 107.8718783637264$ | ||
| 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: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{6}a^{6}$, $\frac{1}{6}a^{7}$, $\frac{1}{18}a^{8}$, $\frac{1}{18}a^{9}$, $\frac{1}{18}a^{10}$, $\frac{1}{36}a^{11}$, $\frac{1}{756}a^{12}-\frac{1}{126}a^{10}+\frac{1}{126}a^{8}+\frac{1}{21}a^{6}+\frac{2}{21}a^{4}+\frac{1}{7}a^{2}+\frac{1}{7}$, $\frac{1}{756}a^{13}-\frac{1}{126}a^{11}+\frac{1}{126}a^{9}+\frac{1}{21}a^{7}+\frac{2}{21}a^{5}+\frac{1}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{1709605010484}a^{14}+\frac{79858843}{244229287212}a^{12}-\frac{2306461387}{142467084207}a^{10}+\frac{1686632519}{284934168414}a^{8}-\frac{614295035}{31659352046}a^{6}-\frac{6086962328}{47489028069}a^{4}-\frac{447831428}{2261382289}a^{2}+\frac{7660378998}{15829676023}$, $\frac{1}{29063285178228}a^{15}+\frac{4697579}{244229287212}a^{13}+\frac{2201276825}{3229253908692}a^{11}-\frac{253142966}{15829676023}a^{9}+\frac{6993395459}{807313477173}a^{7}+\frac{24353805929}{269104492391}a^{5}-\frac{447831428}{38443498913}a^{3}+\frac{118468111159}{269104492391}a$
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: | $9$ | 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: |
$\frac{2286761}{854802505242}a^{14}-\frac{8371697}{1709605010484}a^{12}-\frac{7718342}{47489028069}a^{10}+\frac{9190610}{47489028069}a^{8}-\frac{125810014}{15829676023}a^{6}-\frac{905194844}{47489028069}a^{4}+\frac{4540043244}{15829676023}a^{2}+\frac{827687443}{2261382289}$, $\frac{40}{84449961}a^{14}-\frac{337}{9383329}a^{12}+\frac{10366}{84449961}a^{10}+\frac{633671}{168899922}a^{8}-\frac{56848}{9383329}a^{6}-\frac{511088}{9383329}a^{4}+\frac{2959640}{9383329}a^{2}-\frac{2467113}{9383329}$, $\frac{3030655}{854802505242}a^{14}-\frac{11544871}{1709605010484}a^{12}+\frac{9809232}{15829676023}a^{10}+\frac{37532729}{47489028069}a^{8}-\frac{570855660}{15829676023}a^{6}+\frac{35645011}{6784146867}a^{4}+\frac{13926157068}{15829676023}a^{2}-\frac{7258801031}{15829676023}$, $\frac{62409167}{1709605010484}a^{14}+\frac{13753621}{27136587468}a^{12}-\frac{374902823}{94978056138}a^{10}-\frac{2593305935}{47489028069}a^{8}+\frac{4402890242}{47489028069}a^{6}+\frac{42183779731}{47489028069}a^{4}-\frac{12153894681}{2261382289}a^{2}-\frac{46210952867}{15829676023}$, $\frac{176118785}{244229287212}a^{14}-\frac{441810601}{1709605010484}a^{12}-\frac{4914583850}{47489028069}a^{10}+\frac{11103031421}{284934168414}a^{8}+\frac{217343207087}{47489028069}a^{6}-\frac{441668007923}{47489028069}a^{4}-\frac{1027209233037}{15829676023}a^{2}+\frac{4695991122983}{15829676023}$, $\frac{503228099}{7265821294557}a^{15}+\frac{119995681}{427401252621}a^{14}+\frac{14612434}{142467084207}a^{13}+\frac{318605041}{1709605010484}a^{12}-\frac{22299697340}{2421940431519}a^{11}-\frac{10986041099}{284934168414}a^{10}-\frac{165426111}{15829676023}a^{9}-\frac{250302511}{15829676023}a^{8}+\frac{292607676118}{807313477173}a^{7}+\frac{75993315499}{47489028069}a^{6}-\frac{417649531202}{807313477173}a^{5}-\frac{44988973670}{15829676023}a^{4}-\frac{1540944766721}{269104492391}a^{3}-\frac{382067495706}{15829676023}a^{2}+\frac{5224799488852}{269104492391}a+\frac{1505632008057}{15829676023}$, $\frac{5864549}{14531642589114}a^{15}+\frac{1822951}{1709605010484}a^{14}-\frac{189064522}{427401252621}a^{13}+\frac{195138791}{142467084207}a^{12}-\frac{3418944353}{2421940431519}a^{11}+\frac{1029825835}{284934168414}a^{10}+\frac{4469306326}{142467084207}a^{9}-\frac{8807933849}{94978056138}a^{8}+\frac{19399820597}{807313477173}a^{7}-\frac{11491176}{15829676023}a^{6}-\frac{725633976604}{807313477173}a^{5}+\frac{4964171679}{2261382289}a^{4}+\frac{825706562365}{269104492391}a^{3}-\frac{123770518930}{15829676023}a^{2}+\frac{420717026088}{269104492391}a-\frac{62459100661}{15829676023}$, $\frac{44394523625}{29063285178228}a^{15}-\frac{6445562897}{1709605010484}a^{14}+\frac{154776068}{20352440601}a^{13}-\frac{25430826041}{1709605010484}a^{12}-\frac{483487735580}{2421940431519}a^{11}+\frac{16240881981}{31659352046}a^{10}-\frac{158332219366}{142467084207}a^{9}+\frac{104099566589}{47489028069}a^{8}+\frac{1693569863432}{269104492391}a^{7}-\frac{912993304939}{47489028069}a^{6}+\frac{8352697441388}{269104492391}a^{5}-\frac{427161663469}{6784146867}a^{4}-\frac{1467238360646}{38443498913}a^{3}+\frac{3197734429332}{15829676023}a^{2}-\frac{6001194000764}{269104492391}a+\frac{1610616058785}{15829676023}$, $\frac{18539460665}{9687761726076}a^{15}-\frac{963574613}{284934168414}a^{14}+\frac{16711901539}{854802505242}a^{13}+\frac{6803875147}{569868336828}a^{12}+\frac{540331890107}{4843880863038}a^{11}+\frac{16058607263}{31659352046}a^{10}-\frac{444325084157}{284934168414}a^{9}-\frac{240477297743}{94978056138}a^{8}+\frac{1104455349496}{807313477173}a^{7}-\frac{168959564847}{15829676023}a^{6}+\frac{8978500977632}{269104492391}a^{5}+\frac{3709009054039}{47489028069}a^{4}-\frac{28393394169417}{269104492391}a^{3}-\frac{1937342205767}{15829676023}a^{2}-\frac{24277663416310}{269104492391}a-\frac{838681908439}{15829676023}$
| 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: | \( 521695451.5673963 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 521695451.5673963 \cdot 1}{2\cdot\sqrt{321236373250909071617512439808}}\cr\approx \mathstrut & 0.453078955257900 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 144*x^12 + 6408*x^8 - 10368*x^6 - 95904*x^4 + 378432*x^2 + 187272)
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 - 144*x^12 + 6408*x^8 - 10368*x^6 - 95904*x^4 + 378432*x^2 + 187272, 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 - 144*x^12 + 6408*x^8 - 10368*x^6 - 95904*x^4 + 378432*x^2 + 187272);
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 - 144*x^12 + 6408*x^8 - 10368*x^6 - 95904*x^4 + 378432*x^2 + 187272);
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
$C_2^7.C_8$ (as 16T1155):
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 1024 |
| The 40 conjugacy class representatives for $C_2^7.C_8$ |
| Character table for $C_2^7.C_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.173946175488.2 |
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 | $16$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | $16$ | $16$ | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $16$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.79.2546 | $x^{16} + 56 x^{12} + 16 x^{10} + 32 x^{9} + 20 x^{8} + 32 x^{7} + 16 x^{6} + 32 x^{5} + 66$ | $16$ | $1$ | $79$ | 16T1155 | $[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$ |
|
\(3\)
| 3.16.12.3 | $x^{16} - 6 x^{12} + 162$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |