Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 - y^14 + 21*y^13 - 33*y^12 + 30*y^11 - y^10 - 30*y^9 - 33*y^8 - 21*y^7 + 220*y^6 - 47*y^5 - 135*y^4 - 38*y^3 - y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*x + 1)
\( x^{16} - 4 x^{15} - x^{14} + 21 x^{13} - 33 x^{12} + 30 x^{11} - x^{10} - 30 x^{9} - 33 x^{8} - 21 x^{7} + \cdots + 1 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(732780301186512843008\)
\(\medspace = 2^{8}\cdot 17^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.14\) | 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: | $2\cdot 17^{15/16}\approx 28.48235266104985$ | ||
| Ramified primes: |
\(2\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{101}a^{13}-\frac{19}{101}a^{12}+\frac{20}{101}a^{11}-\frac{21}{101}a^{10}-\frac{42}{101}a^{9}+\frac{25}{101}a^{8}+\frac{35}{101}a^{7}-\frac{27}{101}a^{6}+\frac{48}{101}a^{5}+\frac{43}{101}a^{4}+\frac{27}{101}a^{3}-\frac{10}{101}a^{2}-\frac{11}{101}a+\frac{46}{101}$, $\frac{1}{101}a^{14}-\frac{38}{101}a^{12}-\frac{45}{101}a^{11}-\frac{37}{101}a^{10}+\frac{35}{101}a^{9}+\frac{5}{101}a^{8}+\frac{32}{101}a^{7}+\frac{40}{101}a^{6}+\frac{46}{101}a^{5}+\frac{36}{101}a^{4}-\frac{2}{101}a^{3}+\frac{1}{101}a^{2}+\frac{39}{101}a-\frac{35}{101}$, $\frac{1}{4874751769}a^{15}+\frac{46136}{48264869}a^{14}+\frac{287264}{4874751769}a^{13}-\frac{243649810}{4874751769}a^{12}+\frac{173049271}{4874751769}a^{11}+\frac{161655478}{4874751769}a^{10}-\frac{2381238526}{4874751769}a^{9}+\frac{233598726}{4874751769}a^{8}+\frac{1003789055}{4874751769}a^{7}-\frac{2301253998}{4874751769}a^{6}-\frac{61689798}{4874751769}a^{5}-\frac{1402450380}{4874751769}a^{4}-\frac{1682431991}{4874751769}a^{3}+\frac{2211017123}{4874751769}a^{2}-\frac{2325954418}{4874751769}a-\frac{267146776}{4874751769}$
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: | $11$ | 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: |
$\frac{589059786}{4874751769}a^{15}-\frac{23619886}{48264869}a^{14}+\frac{101478622}{4874751769}a^{13}+\frac{10577161134}{4874751769}a^{12}-\frac{21388611594}{4874751769}a^{11}+\frac{27561119515}{4874751769}a^{10}-\frac{15798122172}{4874751769}a^{9}-\frac{1573586490}{4874751769}a^{8}-\frac{18944401904}{4874751769}a^{7}-\frac{16047402189}{4874751769}a^{6}+\frac{114429790876}{4874751769}a^{5}-\frac{62805093660}{4874751769}a^{4}-\frac{3016976322}{4874751769}a^{3}-\frac{28850754177}{4874751769}a^{2}-\frac{5175579308}{4874751769}a+\frac{7182644758}{4874751769}$, $\frac{1056762832}{4874751769}a^{15}-\frac{4775080619}{4874751769}a^{14}+\frac{929660405}{4874751769}a^{13}+\frac{23503738717}{4874751769}a^{12}-\frac{46238306962}{4874751769}a^{11}+\frac{46288696305}{4874751769}a^{10}-\frac{10733831631}{4874751769}a^{9}-\frac{39397688064}{4874751769}a^{8}-\frac{15322157157}{4874751769}a^{7}-\frac{2403553125}{4874751769}a^{6}+\frac{249818674085}{4874751769}a^{5}-\frac{162916358059}{4874751769}a^{4}-\frac{151264246795}{4874751769}a^{3}+\frac{49655485937}{4874751769}a^{2}+\frac{20184023627}{4874751769}a+\frac{2567492762}{4874751769}$, $\frac{2158147061}{4874751769}a^{15}-\frac{8938306762}{4874751769}a^{14}-\frac{230724900}{4874751769}a^{13}+\frac{43449533265}{4874751769}a^{12}-\frac{79562659829}{4874751769}a^{11}+\frac{85813368466}{4874751769}a^{10}-\frac{27415102290}{4874751769}a^{9}-\frac{47273508303}{4874751769}a^{8}-\frac{60630510525}{4874751769}a^{7}-\frac{40622008296}{4874751769}a^{6}+\frac{459236786421}{4874751769}a^{5}-\frac{207301058733}{4874751769}a^{4}-\frac{180541008492}{4874751769}a^{3}-\frac{41505522589}{4874751769}a^{2}-\frac{3858464455}{4874751769}a+\frac{4620072269}{4874751769}$, $\frac{889470650}{4874751769}a^{15}-\frac{4067833714}{4874751769}a^{14}+\frac{1141823919}{4874751769}a^{13}+\frac{19295746642}{4874751769}a^{12}-\frac{40319658376}{4874751769}a^{11}+\frac{43298761599}{4874751769}a^{10}-\frac{14792396397}{4874751769}a^{9}-\frac{28671596239}{4874751769}a^{8}-\frac{10837287347}{4874751769}a^{7}-\frac{4062088766}{4874751769}a^{6}+\frac{207994451943}{4874751769}a^{5}-\frac{157460443447}{4874751769}a^{4}-\frac{99832650697}{4874751769}a^{3}+\frac{46463843428}{4874751769}a^{2}+\frac{14812483444}{4874751769}a+\frac{8024949655}{4874751769}$, $a$, $\frac{137669810}{4874751769}a^{15}-\frac{538137335}{4874751769}a^{14}-\frac{942947974}{4874751769}a^{13}+\frac{4831782447}{4874751769}a^{12}-\frac{1237574929}{4874751769}a^{11}-\frac{6553496751}{4874751769}a^{10}+\frac{12349518261}{4874751769}a^{9}-\frac{16081918608}{4874751769}a^{8}-\frac{13072583395}{4874751769}a^{7}-\frac{1583734468}{4874751769}a^{6}+\frac{57073501441}{4874751769}a^{5}+\frac{46169766869}{4874751769}a^{4}-\frac{93693217406}{4874751769}a^{3}-\frac{45726792627}{4874751769}a^{2}-\frac{5166022070}{4874751769}a-\frac{2047457497}{4874751769}$, $\frac{1489099014}{4874751769}a^{15}-\frac{5860383300}{4874751769}a^{14}-\frac{768621203}{4874751769}a^{13}+\frac{28134547474}{4874751769}a^{12}-\frac{51221732120}{4874751769}a^{11}+\frac{57624794638}{4874751769}a^{10}-\frac{18561537955}{4874751769}a^{9}-\frac{25753430036}{4874751769}a^{8}-\frac{40461068672}{4874751769}a^{7}-\frac{43379727301}{4874751769}a^{6}+\frac{290271893381}{4874751769}a^{5}-\frac{120224635340}{4874751769}a^{4}-\frac{76859705453}{4874751769}a^{3}-\frac{23368259647}{4874751769}a^{2}-\frac{19136299627}{4874751769}a+\frac{1827552994}{4874751769}$, $\frac{3397012542}{4874751769}a^{15}-\frac{14428733556}{4874751769}a^{14}+\frac{452561929}{4874751769}a^{13}+\frac{70735578053}{4874751769}a^{12}-\frac{131377045543}{4874751769}a^{11}+\frac{137321325453}{4874751769}a^{10}-\frac{38297441049}{4874751769}a^{9}-\frac{91939884707}{4874751769}a^{8}-\frac{83185963527}{4874751769}a^{7}-\frac{48954962766}{4874751769}a^{6}+\frac{748298271960}{4874751769}a^{5}-\frac{372442673369}{4874751769}a^{4}-\frac{353943088088}{4874751769}a^{3}+\frac{1607974989}{4874751769}a^{2}+\frac{14376397799}{4874751769}a+\frac{4787683795}{4874751769}$, $\frac{859345945}{4874751769}a^{15}-\frac{3453887649}{4874751769}a^{14}-\frac{1075400082}{4874751769}a^{13}+\frac{18957425072}{4874751769}a^{12}-\frac{28109369437}{4874751769}a^{11}+\frac{22112121135}{4874751769}a^{10}+\frac{5897534377}{4874751769}a^{9}-\frac{34510024268}{4874751769}a^{8}-\frac{26334960539}{4874751769}a^{7}-\frac{16581774647}{4874751769}a^{6}+\frac{196722356493}{4874751769}a^{5}-\frac{27927058242}{4874751769}a^{4}-\frac{153833157027}{4874751769}a^{3}-\frac{19052299944}{4874751769}a^{2}-\frac{5919440014}{4874751769}a+\frac{2781636059}{4874751769}$, $\frac{1999364766}{4874751769}a^{15}-\frac{8337971735}{4874751769}a^{14}+\frac{239592146}{4874751769}a^{13}+\frac{39455285853}{4874751769}a^{12}-\frac{75039618916}{4874751769}a^{11}+\frac{85320861613}{4874751769}a^{10}-\frac{34690377739}{4874751769}a^{9}-\frac{34335824600}{4874751769}a^{8}-\frac{57747576264}{4874751769}a^{7}-\frac{38814466120}{4874751769}a^{6}+\frac{421134379584}{4874751769}a^{5}-\frac{213265471859}{4874751769}a^{4}-\frac{125433224156}{4874751769}a^{3}-\frac{48654281039}{4874751769}a^{2}-\frac{630834906}{4874751769}a+\frac{4692826279}{4874751769}$, $\frac{1851851945}{4874751769}a^{15}-\frac{7430999058}{4874751769}a^{14}-\frac{1803508166}{4874751769}a^{13}+\frac{38785998874}{4874751769}a^{12}-\frac{60827963372}{4874751769}a^{11}+\frac{56743289529}{4874751769}a^{10}-\frac{4875423658}{4874751769}a^{9}-\frac{52593927909}{4874751769}a^{8}-\frac{64676302781}{4874751769}a^{7}-\frac{40299950756}{4874751769}a^{6}+\frac{408795965620}{4874751769}a^{5}-\frac{82152871113}{4874751769}a^{4}-\frac{236396749884}{4874751769}a^{3}-\frac{92092533310}{4874751769}a^{2}-\frac{15476179930}{4874751769}a+\frac{1718848970}{4874751769}$
| 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: | \( 23552.8271774 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 23552.8271774 \cdot 1}{2\cdot\sqrt{732780301186512843008}}\cr\approx \mathstrut & 0.173574369138 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*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 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*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 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*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 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*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
$\OD_{32}$ (as 16T22):
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 20 conjugacy class representatives for $C_{16} : C_2$ |
| Character table for $C_{16} : C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(17\)
| 17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |