Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723)
gp: K = bnfinit(y^16 - 3*y^15 + 94*y^14 - 724*y^13 + 26159*y^12 - 167999*y^11 + 2096914*y^10 - 10821888*y^9 - 25156346*y^8 - 651196200*y^7 + 18347527091*y^6 + 172806626196*y^5 + 741026109860*y^4 + 1097105545547*y^3 - 4848874441183*y^2 - 7650125420549*y + 20600166208723, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723)
\( x^{16} - 3 x^{15} + 94 x^{14} - 724 x^{13} + 26159 x^{12} - 167999 x^{11} + 2096914 x^{10} + \cdots + 20600166208723 \)
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: |
\(5600075906386661768959437042812533816731958913\)
\(\medspace = 17^{15}\cdot 89^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(723.21\) | 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: | $17^{15/16}89^{7/8}\approx 723.2061649377035$ | ||
| Ramified primes: |
\(17\), \(89\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{47}a^{13}+\frac{16}{47}a^{12}-\frac{15}{47}a^{11}+\frac{17}{47}a^{10}+\frac{3}{47}a^{9}+\frac{3}{47}a^{7}-\frac{21}{47}a^{6}-\frac{12}{47}a^{5}-\frac{16}{47}a^{4}-\frac{6}{47}a^{3}-\frac{17}{47}a^{2}-\frac{10}{47}a-\frac{17}{47}$, $\frac{1}{47}a^{14}+\frac{11}{47}a^{12}+\frac{22}{47}a^{11}+\frac{13}{47}a^{10}-\frac{1}{47}a^{9}+\frac{3}{47}a^{8}-\frac{22}{47}a^{7}-\frac{5}{47}a^{6}-\frac{12}{47}a^{5}+\frac{15}{47}a^{4}-\frac{15}{47}a^{3}-\frac{20}{47}a^{2}+\frac{2}{47}a-\frac{10}{47}$, $\frac{1}{10\!\cdots\!91}a^{15}-\frac{83\!\cdots\!24}{10\!\cdots\!91}a^{14}+\frac{77\!\cdots\!63}{10\!\cdots\!91}a^{13}-\frac{15\!\cdots\!74}{10\!\cdots\!91}a^{12}-\frac{14\!\cdots\!79}{10\!\cdots\!91}a^{11}+\frac{66\!\cdots\!11}{15\!\cdots\!73}a^{10}-\frac{51\!\cdots\!61}{10\!\cdots\!91}a^{9}-\frac{56\!\cdots\!84}{10\!\cdots\!91}a^{8}+\frac{46\!\cdots\!53}{10\!\cdots\!91}a^{7}-\frac{28\!\cdots\!56}{10\!\cdots\!91}a^{6}-\frac{20\!\cdots\!91}{10\!\cdots\!91}a^{5}+\frac{17\!\cdots\!62}{10\!\cdots\!91}a^{4}-\frac{20\!\cdots\!59}{10\!\cdots\!91}a^{3}-\frac{38\!\cdots\!27}{10\!\cdots\!91}a^{2}-\frac{31\!\cdots\!60}{10\!\cdots\!91}a-\frac{95\!\cdots\!36}{24\!\cdots\!37}$
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
$C_{2}\times C_{6}\times C_{1876632}$, which has order $22519584$ (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: | $7$ | 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: |
$\frac{62\!\cdots\!44}{27\!\cdots\!91}a^{15}-\frac{87\!\cdots\!36}{13\!\cdots\!77}a^{14}+\frac{27\!\cdots\!52}{13\!\cdots\!77}a^{13}-\frac{22\!\cdots\!20}{13\!\cdots\!77}a^{12}+\frac{77\!\cdots\!74}{13\!\cdots\!77}a^{11}-\frac{74\!\cdots\!48}{19\!\cdots\!31}a^{10}+\frac{61\!\cdots\!48}{13\!\cdots\!77}a^{9}-\frac{33\!\cdots\!68}{13\!\cdots\!77}a^{8}-\frac{54\!\cdots\!10}{13\!\cdots\!77}a^{7}-\frac{21\!\cdots\!14}{13\!\cdots\!77}a^{6}+\frac{54\!\cdots\!66}{13\!\cdots\!77}a^{5}+\frac{52\!\cdots\!84}{13\!\cdots\!77}a^{4}+\frac{20\!\cdots\!42}{13\!\cdots\!77}a^{3}+\frac{13\!\cdots\!00}{13\!\cdots\!77}a^{2}-\frac{28\!\cdots\!28}{13\!\cdots\!77}a-\frac{68\!\cdots\!41}{13\!\cdots\!77}$, $\frac{31\!\cdots\!51}{64\!\cdots\!77}a^{15}-\frac{93\!\cdots\!22}{64\!\cdots\!77}a^{14}+\frac{29\!\cdots\!52}{64\!\cdots\!77}a^{13}-\frac{23\!\cdots\!85}{64\!\cdots\!77}a^{12}+\frac{82\!\cdots\!93}{64\!\cdots\!77}a^{11}-\frac{79\!\cdots\!94}{95\!\cdots\!31}a^{10}+\frac{65\!\cdots\!83}{64\!\cdots\!77}a^{9}-\frac{35\!\cdots\!05}{64\!\cdots\!77}a^{8}-\frac{67\!\cdots\!47}{64\!\cdots\!77}a^{7}-\frac{21\!\cdots\!45}{64\!\cdots\!77}a^{6}+\frac{58\!\cdots\!17}{64\!\cdots\!77}a^{5}+\frac{55\!\cdots\!83}{64\!\cdots\!77}a^{4}+\frac{22\!\cdots\!38}{64\!\cdots\!77}a^{3}+\frac{13\!\cdots\!04}{64\!\cdots\!77}a^{2}-\frac{29\!\cdots\!89}{64\!\cdots\!77}a+\frac{11\!\cdots\!02}{64\!\cdots\!77}$, $\frac{24\!\cdots\!29}{64\!\cdots\!77}a^{15}-\frac{74\!\cdots\!94}{64\!\cdots\!77}a^{14}+\frac{22\!\cdots\!63}{64\!\cdots\!77}a^{13}-\frac{17\!\cdots\!18}{64\!\cdots\!77}a^{12}+\frac{13\!\cdots\!77}{13\!\cdots\!91}a^{11}-\frac{61\!\cdots\!80}{95\!\cdots\!31}a^{10}+\frac{49\!\cdots\!47}{64\!\cdots\!77}a^{9}-\frac{25\!\cdots\!35}{64\!\cdots\!77}a^{8}-\frac{74\!\cdots\!64}{64\!\cdots\!77}a^{7}-\frac{15\!\cdots\!19}{64\!\cdots\!77}a^{6}+\frac{45\!\cdots\!53}{64\!\cdots\!77}a^{5}+\frac{42\!\cdots\!92}{64\!\cdots\!77}a^{4}+\frac{16\!\cdots\!21}{64\!\cdots\!77}a^{3}+\frac{95\!\cdots\!23}{64\!\cdots\!77}a^{2}-\frac{22\!\cdots\!75}{64\!\cdots\!77}a-\frac{55\!\cdots\!53}{64\!\cdots\!77}$, $\frac{96\!\cdots\!46}{15\!\cdots\!41}a^{15}-\frac{29\!\cdots\!33}{15\!\cdots\!41}a^{14}+\frac{87\!\cdots\!69}{15\!\cdots\!41}a^{13}-\frac{14\!\cdots\!22}{32\!\cdots\!03}a^{12}+\frac{25\!\cdots\!28}{15\!\cdots\!41}a^{11}-\frac{24\!\cdots\!99}{22\!\cdots\!23}a^{10}+\frac{19\!\cdots\!92}{15\!\cdots\!41}a^{9}-\frac{10\!\cdots\!57}{15\!\cdots\!41}a^{8}-\frac{27\!\cdots\!03}{15\!\cdots\!41}a^{7}-\frac{61\!\cdots\!92}{15\!\cdots\!41}a^{6}+\frac{17\!\cdots\!37}{15\!\cdots\!41}a^{5}+\frac{16\!\cdots\!19}{15\!\cdots\!41}a^{4}+\frac{66\!\cdots\!53}{15\!\cdots\!41}a^{3}+\frac{37\!\cdots\!98}{15\!\cdots\!41}a^{2}-\frac{88\!\cdots\!04}{15\!\cdots\!41}a-\frac{21\!\cdots\!13}{15\!\cdots\!41}$, $\frac{11\!\cdots\!57}{15\!\cdots\!41}a^{15}-\frac{57\!\cdots\!72}{15\!\cdots\!41}a^{14}+\frac{12\!\cdots\!12}{15\!\cdots\!41}a^{13}-\frac{10\!\cdots\!84}{15\!\cdots\!41}a^{12}+\frac{32\!\cdots\!84}{15\!\cdots\!41}a^{11}-\frac{38\!\cdots\!30}{22\!\cdots\!23}a^{10}+\frac{29\!\cdots\!21}{15\!\cdots\!41}a^{9}-\frac{17\!\cdots\!62}{15\!\cdots\!41}a^{8}+\frac{22\!\cdots\!84}{15\!\cdots\!41}a^{7}-\frac{70\!\cdots\!54}{15\!\cdots\!41}a^{6}+\frac{22\!\cdots\!94}{15\!\cdots\!41}a^{5}+\frac{16\!\cdots\!57}{15\!\cdots\!41}a^{4}+\frac{47\!\cdots\!14}{15\!\cdots\!41}a^{3}+\frac{37\!\cdots\!83}{15\!\cdots\!41}a^{2}-\frac{75\!\cdots\!05}{15\!\cdots\!41}a+\frac{10\!\cdots\!35}{15\!\cdots\!41}$, $\frac{11\!\cdots\!82}{15\!\cdots\!41}a^{15}-\frac{32\!\cdots\!94}{15\!\cdots\!41}a^{14}+\frac{99\!\cdots\!90}{15\!\cdots\!41}a^{13}-\frac{78\!\cdots\!34}{15\!\cdots\!41}a^{12}+\frac{28\!\cdots\!22}{15\!\cdots\!41}a^{11}-\frac{27\!\cdots\!18}{22\!\cdots\!23}a^{10}+\frac{22\!\cdots\!03}{15\!\cdots\!41}a^{9}-\frac{11\!\cdots\!11}{15\!\cdots\!41}a^{8}-\frac{75\!\cdots\!93}{32\!\cdots\!03}a^{7}-\frac{70\!\cdots\!64}{15\!\cdots\!41}a^{6}+\frac{20\!\cdots\!57}{15\!\cdots\!41}a^{5}+\frac{19\!\cdots\!47}{15\!\cdots\!41}a^{4}+\frac{78\!\cdots\!57}{15\!\cdots\!41}a^{3}+\frac{43\!\cdots\!67}{15\!\cdots\!41}a^{2}-\frac{10\!\cdots\!95}{15\!\cdots\!41}a-\frac{27\!\cdots\!65}{15\!\cdots\!41}$, $\frac{27\!\cdots\!67}{15\!\cdots\!41}a^{15}-\frac{33\!\cdots\!31}{15\!\cdots\!41}a^{14}+\frac{36\!\cdots\!39}{15\!\cdots\!41}a^{13}-\frac{39\!\cdots\!60}{15\!\cdots\!41}a^{12}+\frac{88\!\cdots\!13}{15\!\cdots\!41}a^{11}-\frac{16\!\cdots\!18}{22\!\cdots\!23}a^{10}+\frac{10\!\cdots\!70}{15\!\cdots\!41}a^{9}-\frac{71\!\cdots\!57}{15\!\cdots\!41}a^{8}+\frac{99\!\cdots\!10}{15\!\cdots\!41}a^{7}+\frac{93\!\cdots\!86}{15\!\cdots\!41}a^{6}+\frac{56\!\cdots\!52}{15\!\cdots\!41}a^{5}-\frac{35\!\cdots\!04}{15\!\cdots\!41}a^{4}-\frac{19\!\cdots\!84}{15\!\cdots\!41}a^{3}-\frac{70\!\cdots\!94}{15\!\cdots\!41}a^{2}+\frac{15\!\cdots\!81}{15\!\cdots\!41}a+\frac{12\!\cdots\!40}{15\!\cdots\!41}$
| 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: | \( 1263629880.47 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1263629880.47 \cdot 22519584}{2\cdot\sqrt{5600075906386661768959437042812533816731958913}}\cr\approx \mathstrut & 0.461840744811 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723)
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 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723, 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 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723);
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 - 3*x^15 + 94*x^14 - 724*x^13 + 26159*x^12 - 167999*x^11 + 2096914*x^10 - 10821888*x^9 - 25156346*x^8 - 651196200*x^7 + 18347527091*x^6 + 172806626196*x^5 + 741026109860*x^4 + 1097105545547*x^3 - 4848874441183*x^2 - 7650125420549*x + 20600166208723);
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
| \(\Q(\sqrt{17}) \), 4.4.38915873.1, 8.8.203930643438763634753.2 |
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
| 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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.1.0.1}{1} }^{16}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.15.7 | $x^{16} + 221$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
|
\(89\)
| 89.8.7.2 | $x^{8} + 445$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 89.8.7.2 | $x^{8} + 445$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |