Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 14*x^14 - 10*x^13 - 19*x^12 - 18*x^11 + 140*x^10 - 132*x^9 + 142*x^8 - 336*x^7 + 310*x^6 - 56*x^5 - 84*x^4 + 70*x^3 - 14*x^2 - 2*x + 1)
gp: K = bnfinit(y^16 - 6*y^15 + 14*y^14 - 10*y^13 - 19*y^12 - 18*y^11 + 140*y^10 - 132*y^9 + 142*y^8 - 336*y^7 + 310*y^6 - 56*y^5 - 84*y^4 + 70*y^3 - 14*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 14*x^14 - 10*x^13 - 19*x^12 - 18*x^11 + 140*x^10 - 132*x^9 + 142*x^8 - 336*x^7 + 310*x^6 - 56*x^5 - 84*x^4 + 70*x^3 - 14*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 14*x^14 - 10*x^13 - 19*x^12 - 18*x^11 + 140*x^10 - 132*x^9 + 142*x^8 - 336*x^7 + 310*x^6 - 56*x^5 - 84*x^4 + 70*x^3 - 14*x^2 - 2*x + 1)
\( x^{16} - 6 x^{15} + 14 x^{14} - 10 x^{13} - 19 x^{12} - 18 x^{11} + 140 x^{10} - 132 x^{9} + 142 x^{8} + \cdots + 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2176782336000000000000\)
\(\medspace = 2^{24}\cdot 3^{12}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.56\) | 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: | $2^{3/2}3^{3/4}5^{3/4}\approx 21.558246717785053$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{57}a^{14}+\frac{5}{57}a^{13}+\frac{7}{57}a^{12}-\frac{5}{19}a^{11}+\frac{28}{57}a^{10}+\frac{4}{57}a^{9}-\frac{13}{57}a^{8}+\frac{3}{19}a^{7}-\frac{17}{57}a^{6}+\frac{2}{57}a^{5}-\frac{20}{57}a^{4}-\frac{1}{57}a^{3}+\frac{5}{57}a^{2}-\frac{1}{19}a+\frac{4}{57}$, $\frac{1}{112141910067}a^{15}+\frac{74008139}{112141910067}a^{14}+\frac{5709482033}{37380636689}a^{13}+\frac{8576404487}{112141910067}a^{12}+\frac{3087190156}{37380636689}a^{11}-\frac{12434999681}{37380636689}a^{10}+\frac{2538147808}{10194719097}a^{9}-\frac{4512790090}{10194719097}a^{8}+\frac{959877420}{37380636689}a^{7}+\frac{880981691}{1967401931}a^{6}+\frac{23619260959}{112141910067}a^{5}+\frac{734649610}{1967401931}a^{4}+\frac{11389061512}{37380636689}a^{3}-\frac{13828870841}{112141910067}a^{2}-\frac{18078927105}{37380636689}a-\frac{36095387906}{112141910067}$
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
$C_{2}$, which has order $2$ (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: |
$\frac{67229306}{536564163}a^{15}-\frac{125778328}{178854721}a^{14}+\frac{280079150}{178854721}a^{13}-\frac{179872413}{178854721}a^{12}-\frac{1233979904}{536564163}a^{11}-\frac{1488600320}{536564163}a^{10}+\frac{7990645510}{536564163}a^{9}-\frac{8166631942}{536564163}a^{8}+\frac{9536408684}{536564163}a^{7}-\frac{17345181998}{536564163}a^{6}+\frac{6361214964}{178854721}a^{5}-\frac{1033563709}{178854721}a^{4}-\frac{8856155396}{536564163}a^{3}+\frac{3311852426}{536564163}a^{2}+\frac{88447814}{536564163}a-\frac{177452537}{178854721}$, $\frac{44735207345}{112141910067}a^{15}-\frac{251851868801}{112141910067}a^{14}+\frac{523265959196}{112141910067}a^{13}-\frac{207162096848}{112141910067}a^{12}-\frac{1007397072764}{112141910067}a^{11}-\frac{1172820410074}{112141910067}a^{10}+\frac{546222713266}{10194719097}a^{9}-\frac{298570700692}{10194719097}a^{8}+\frac{4304574330602}{112141910067}a^{7}-\frac{13108654102444}{112141910067}a^{6}+\frac{7883247029261}{112141910067}a^{5}+\frac{2306893608145}{112141910067}a^{4}-\frac{1252848295133}{37380636689}a^{3}+\frac{1446858196654}{112141910067}a^{2}+\frac{380956384295}{112141910067}a-\frac{167020057184}{112141910067}$, $\frac{28471919921}{112141910067}a^{15}-\frac{62895686582}{37380636689}a^{14}+\frac{496354148474}{112141910067}a^{13}-\frac{160201390716}{37380636689}a^{12}-\frac{474807938911}{112141910067}a^{11}-\frac{44808620497}{37380636689}a^{10}+\frac{412960674562}{10194719097}a^{9}-\frac{548507428702}{10194719097}a^{8}+\frac{5032212646423}{112141910067}a^{7}-\frac{3818350821709}{37380636689}a^{6}+\frac{4705614243457}{37380636689}a^{5}-\frac{4321548721019}{112141910067}a^{4}-\frac{2898471336536}{112141910067}a^{3}+\frac{2550570553231}{112141910067}a^{2}-\frac{539583399926}{112141910067}a-\frac{13915666036}{112141910067}$, $\frac{710414658}{37380636689}a^{15}-\frac{29857130806}{112141910067}a^{14}+\frac{36862408191}{37380636689}a^{13}-\frac{162370228192}{112141910067}a^{12}-\frac{20686867649}{112141910067}a^{11}+\frac{244990859842}{112141910067}a^{10}+\frac{30640303907}{3398239699}a^{9}-\frac{50708689513}{3398239699}a^{8}+\frac{871106516927}{112141910067}a^{7}-\frac{3122026811117}{112141910067}a^{6}+\frac{3938695838401}{112141910067}a^{5}-\frac{594600610132}{37380636689}a^{4}+\frac{389101318894}{112141910067}a^{3}+\frac{248919759873}{37380636689}a^{2}-\frac{354190783753}{112141910067}a-\frac{27874206350}{112141910067}$, $\frac{11919680980}{112141910067}a^{15}-\frac{49005880850}{112141910067}a^{14}+\frac{21674134159}{37380636689}a^{13}+\frac{49590225241}{112141910067}a^{12}-\frac{237214932287}{112141910067}a^{11}-\frac{556108326722}{112141910067}a^{10}+\frac{59901352969}{10194719097}a^{9}-\frac{3039938359}{10194719097}a^{8}+\frac{1122002898029}{112141910067}a^{7}-\frac{503116226465}{112141910067}a^{6}+\frac{49785944027}{112141910067}a^{5}+\frac{378585794951}{37380636689}a^{4}-\frac{2240037796658}{112141910067}a^{3}-\frac{54582122585}{112141910067}a^{2}+\frac{372676376879}{112141910067}a-\frac{195530444416}{112141910067}$, $\frac{5781749509}{37380636689}a^{15}-\frac{40008134759}{37380636689}a^{14}+\frac{98339053696}{37380636689}a^{13}-\frac{67926634980}{37380636689}a^{12}-\frac{154100707859}{37380636689}a^{11}-\frac{32740647015}{37380636689}a^{10}+\frac{106181678749}{3398239699}a^{9}-\frac{74804704770}{3398239699}a^{8}+\frac{468311367098}{37380636689}a^{7}-\frac{2665734865363}{37380636689}a^{6}+\frac{1958244752352}{37380636689}a^{5}+\frac{358056135821}{37380636689}a^{4}-\frac{548190091606}{37380636689}a^{3}+\frac{274624146680}{37380636689}a^{2}-\frac{48063178234}{37380636689}a-\frac{6131510466}{37380636689}$, $\frac{102818434921}{112141910067}a^{15}-\frac{448646602354}{112141910067}a^{14}+\frac{645997431323}{112141910067}a^{13}+\frac{305257130681}{112141910067}a^{12}-\frac{1900926835670}{112141910067}a^{11}-\frac{5041320677992}{112141910067}a^{10}+\frac{221978087302}{3398239699}a^{9}+\frac{29465057075}{3398239699}a^{8}+\frac{11058064754711}{112141910067}a^{7}-\frac{15833593023082}{112141910067}a^{6}+\frac{101207403736}{112141910067}a^{5}+\frac{5218104639841}{112141910067}a^{4}-\frac{804498425561}{37380636689}a^{3}-\frac{222425961706}{37380636689}a^{2}+\frac{456276403883}{112141910067}a+\frac{22970418639}{37380636689}$, $\frac{116869359875}{112141910067}a^{15}-\frac{527509614010}{112141910067}a^{14}+\frac{821607058373}{112141910067}a^{13}+\frac{192477127730}{112141910067}a^{12}-\frac{719609545202}{37380636689}a^{11}-\frac{5352438144872}{112141910067}a^{10}+\frac{817756526596}{10194719097}a^{9}-\frac{66770835673}{10194719097}a^{8}+\frac{4350391389889}{37380636689}a^{7}-\frac{19458736060664}{112141910067}a^{6}+\frac{4089689186164}{112141910067}a^{5}+\frac{4570060194298}{112141910067}a^{4}-\frac{4264431754447}{112141910067}a^{3}+\frac{24899271916}{112141910067}a^{2}+\frac{158253999003}{37380636689}a-\frac{51497798283}{37380636689}$, $\frac{185177338915}{112141910067}a^{15}-\frac{970954706116}{112141910067}a^{14}+\frac{629826756553}{37380636689}a^{13}-\frac{566996077357}{112141910067}a^{12}-\frac{1239802137185}{37380636689}a^{11}-\frac{2035489883940}{37380636689}a^{10}+\frac{1885759306438}{10194719097}a^{9}-\frac{930286947142}{10194719097}a^{8}+\frac{7033665094457}{37380636689}a^{7}-\frac{15412656721003}{37380636689}a^{6}+\frac{26381647731316}{112141910067}a^{5}+\frac{1195893392701}{37380636689}a^{4}-\frac{3909532526593}{37380636689}a^{3}+\frac{4211802484399}{112141910067}a^{2}+\frac{96835872212}{37380636689}a-\frac{260408194919}{112141910067}$
| 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: | \( 14037.4430501 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 14037.4430501 \cdot 2}{2\cdot\sqrt{2176782336000000000000}}\cr\approx \mathstrut & 0.296196346860 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 14*x^14 - 10*x^13 - 19*x^12 - 18*x^11 + 140*x^10 - 132*x^9 + 142*x^8 - 336*x^7 + 310*x^6 - 56*x^5 - 84*x^4 + 70*x^3 - 14*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 + 14*x^14 - 10*x^13 - 19*x^12 - 18*x^11 + 140*x^10 - 132*x^9 + 142*x^8 - 336*x^7 + 310*x^6 - 56*x^5 - 84*x^4 + 70*x^3 - 14*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 + 14*x^14 - 10*x^13 - 19*x^12 - 18*x^11 + 140*x^10 - 132*x^9 + 142*x^8 - 336*x^7 + 310*x^6 - 56*x^5 - 84*x^4 + 70*x^3 - 14*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 + 14*x^14 - 10*x^13 - 19*x^12 - 18*x^11 + 140*x^10 - 132*x^9 + 142*x^8 - 336*x^7 + 310*x^6 - 56*x^5 - 84*x^4 + 70*x^3 - 14*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
$C_4^2:C_2$ (as 16T30):
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 $C_4^2:C_2$ |
| Character table for $C_4^2:C_2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), 4.2.2000.1, 4.2.18000.1, \(\Q(\sqrt{3}, \sqrt{5})\), 8.4.5184000000.1, 8.4.46656000000.1, 8.4.116640000.1 |
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 16 sibling: | 16.0.136048896000000000000.1 |
| Minimal sibling: | 16.0.136048896000000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $24$ | |||
|
\(3\)
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |