Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 94*x^14 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097)
gp: K = bnfinit(y^16 - 3*y^15 + 94*y^14 + 2302*y^13 + 26159*y^12 - 397975*y^11 + 6284898*y^10 - 24995672*y^9 + 142093700*y^8 - 83458080*y^7 + 7911836541*y^6 - 60914865824*y^5 + 494959575758*y^4 - 1703233593999*y^3 + 5636562271845*y^2 - 6761439893945*y + 19724305333097, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 94*x^14 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 94*x^14 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097)
\( x^{16} - 3 x^{15} + 94 x^{14} + 2302 x^{13} + 26159 x^{12} - 397975 x^{11} + 6284898 x^{10} + \cdots + 19724305333097 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{18\!\cdots\!91}a^{15}-\frac{10\!\cdots\!04}{18\!\cdots\!91}a^{14}-\frac{77\!\cdots\!83}{18\!\cdots\!91}a^{13}-\frac{64\!\cdots\!25}{18\!\cdots\!91}a^{12}+\frac{62\!\cdots\!29}{18\!\cdots\!91}a^{11}+\frac{79\!\cdots\!95}{18\!\cdots\!91}a^{10}+\frac{41\!\cdots\!49}{18\!\cdots\!91}a^{9}-\frac{41\!\cdots\!34}{18\!\cdots\!91}a^{8}+\frac{34\!\cdots\!76}{18\!\cdots\!91}a^{7}+\frac{50\!\cdots\!08}{18\!\cdots\!91}a^{6}-\frac{45\!\cdots\!95}{18\!\cdots\!91}a^{5}+\frac{53\!\cdots\!47}{18\!\cdots\!91}a^{4}+\frac{47\!\cdots\!67}{18\!\cdots\!91}a^{3}-\frac{37\!\cdots\!43}{18\!\cdots\!91}a^{2}-\frac{66\!\cdots\!74}{18\!\cdots\!91}a-\frac{20\!\cdots\!42}{31\!\cdots\!49}$
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
$C_{2}\times C_{2}\times C_{3413896}$, which has order $13655584$ (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: |
$\frac{24\!\cdots\!88}{17\!\cdots\!79}a^{15}+\frac{10\!\cdots\!12}{17\!\cdots\!79}a^{14}+\frac{22\!\cdots\!76}{17\!\cdots\!79}a^{13}+\frac{69\!\cdots\!28}{17\!\cdots\!79}a^{12}+\frac{10\!\cdots\!98}{17\!\cdots\!79}a^{11}-\frac{45\!\cdots\!00}{17\!\cdots\!79}a^{10}+\frac{88\!\cdots\!44}{17\!\cdots\!79}a^{9}+\frac{17\!\cdots\!64}{17\!\cdots\!79}a^{8}+\frac{20\!\cdots\!94}{17\!\cdots\!79}a^{7}+\frac{64\!\cdots\!94}{17\!\cdots\!79}a^{6}+\frac{18\!\cdots\!46}{17\!\cdots\!79}a^{5}-\frac{33\!\cdots\!08}{17\!\cdots\!79}a^{4}+\frac{43\!\cdots\!86}{17\!\cdots\!79}a^{3}+\frac{25\!\cdots\!52}{17\!\cdots\!79}a^{2}+\frac{19\!\cdots\!96}{17\!\cdots\!79}a+\frac{78\!\cdots\!61}{17\!\cdots\!79}$, $\frac{11\!\cdots\!25}{72\!\cdots\!23}a^{15}+\frac{11\!\cdots\!56}{72\!\cdots\!23}a^{14}+\frac{10\!\cdots\!56}{72\!\cdots\!23}a^{13}+\frac{36\!\cdots\!69}{72\!\cdots\!23}a^{12}+\frac{66\!\cdots\!24}{72\!\cdots\!23}a^{11}+\frac{17\!\cdots\!62}{72\!\cdots\!23}a^{10}+\frac{16\!\cdots\!03}{72\!\cdots\!23}a^{9}+\frac{35\!\cdots\!91}{72\!\cdots\!23}a^{8}+\frac{86\!\cdots\!68}{72\!\cdots\!23}a^{7}+\frac{45\!\cdots\!58}{72\!\cdots\!23}a^{6}+\frac{82\!\cdots\!66}{72\!\cdots\!23}a^{5}+\frac{31\!\cdots\!15}{72\!\cdots\!23}a^{4}-\frac{56\!\cdots\!59}{72\!\cdots\!23}a^{3}+\frac{10\!\cdots\!86}{72\!\cdots\!23}a^{2}-\frac{17\!\cdots\!95}{72\!\cdots\!23}a+\frac{51\!\cdots\!05}{72\!\cdots\!23}$, $\frac{70\!\cdots\!38}{47\!\cdots\!21}a^{15}+\frac{52\!\cdots\!46}{47\!\cdots\!21}a^{14}+\frac{26\!\cdots\!11}{47\!\cdots\!21}a^{13}+\frac{13\!\cdots\!71}{47\!\cdots\!21}a^{12}+\frac{27\!\cdots\!29}{47\!\cdots\!21}a^{11}-\frac{25\!\cdots\!57}{47\!\cdots\!21}a^{10}-\frac{23\!\cdots\!13}{47\!\cdots\!21}a^{9}+\frac{30\!\cdots\!14}{47\!\cdots\!21}a^{8}-\frac{41\!\cdots\!98}{47\!\cdots\!21}a^{7}-\frac{72\!\cdots\!90}{47\!\cdots\!21}a^{6}-\frac{54\!\cdots\!83}{47\!\cdots\!21}a^{5}+\frac{69\!\cdots\!19}{47\!\cdots\!21}a^{4}-\frac{91\!\cdots\!82}{47\!\cdots\!21}a^{3}+\frac{13\!\cdots\!96}{47\!\cdots\!21}a^{2}-\frac{46\!\cdots\!10}{47\!\cdots\!21}a+\frac{10\!\cdots\!38}{81\!\cdots\!19}$, $\frac{51\!\cdots\!59}{47\!\cdots\!21}a^{15}+\frac{91\!\cdots\!51}{47\!\cdots\!21}a^{14}+\frac{45\!\cdots\!82}{47\!\cdots\!21}a^{13}+\frac{13\!\cdots\!66}{47\!\cdots\!21}a^{12}+\frac{19\!\cdots\!34}{47\!\cdots\!21}a^{11}-\frac{13\!\cdots\!35}{47\!\cdots\!21}a^{10}+\frac{22\!\cdots\!13}{47\!\cdots\!21}a^{9}-\frac{28\!\cdots\!58}{47\!\cdots\!21}a^{8}+\frac{38\!\cdots\!26}{47\!\cdots\!21}a^{7}+\frac{11\!\cdots\!44}{47\!\cdots\!21}a^{6}+\frac{40\!\cdots\!57}{47\!\cdots\!21}a^{5}-\frac{14\!\cdots\!35}{47\!\cdots\!21}a^{4}+\frac{12\!\cdots\!63}{47\!\cdots\!21}a^{3}-\frac{13\!\cdots\!45}{47\!\cdots\!21}a^{2}+\frac{54\!\cdots\!65}{47\!\cdots\!21}a-\frac{29\!\cdots\!36}{81\!\cdots\!19}$, $\frac{19\!\cdots\!93}{72\!\cdots\!23}a^{15}+\frac{62\!\cdots\!16}{72\!\cdots\!23}a^{14}+\frac{17\!\cdots\!51}{72\!\cdots\!23}a^{13}+\frac{54\!\cdots\!82}{72\!\cdots\!23}a^{12}+\frac{81\!\cdots\!46}{72\!\cdots\!23}a^{11}-\frac{41\!\cdots\!92}{72\!\cdots\!23}a^{10}+\frac{79\!\cdots\!81}{72\!\cdots\!23}a^{9}+\frac{70\!\cdots\!31}{72\!\cdots\!23}a^{8}+\frac{17\!\cdots\!27}{72\!\cdots\!23}a^{7}+\frac{50\!\cdots\!96}{72\!\cdots\!23}a^{6}+\frac{14\!\cdots\!50}{72\!\cdots\!23}a^{5}-\frac{39\!\cdots\!50}{72\!\cdots\!23}a^{4}+\frac{42\!\cdots\!18}{72\!\cdots\!23}a^{3}-\frac{28\!\cdots\!93}{72\!\cdots\!23}a^{2}+\frac{18\!\cdots\!79}{72\!\cdots\!23}a+\frac{22\!\cdots\!47}{72\!\cdots\!23}$, $\frac{53\!\cdots\!62}{47\!\cdots\!21}a^{15}+\frac{10\!\cdots\!57}{47\!\cdots\!21}a^{14}+\frac{46\!\cdots\!89}{47\!\cdots\!21}a^{13}+\frac{14\!\cdots\!99}{47\!\cdots\!21}a^{12}+\frac{20\!\cdots\!69}{47\!\cdots\!21}a^{11}-\frac{13\!\cdots\!50}{47\!\cdots\!21}a^{10}+\frac{23\!\cdots\!73}{47\!\cdots\!21}a^{9}+\frac{13\!\cdots\!54}{47\!\cdots\!21}a^{8}+\frac{41\!\cdots\!46}{47\!\cdots\!21}a^{7}+\frac{11\!\cdots\!40}{47\!\cdots\!21}a^{6}+\frac{42\!\cdots\!04}{47\!\cdots\!21}a^{5}-\frac{14\!\cdots\!25}{47\!\cdots\!21}a^{4}+\frac{12\!\cdots\!55}{47\!\cdots\!21}a^{3}-\frac{12\!\cdots\!90}{47\!\cdots\!21}a^{2}+\frac{55\!\cdots\!64}{47\!\cdots\!21}a-\frac{62\!\cdots\!03}{81\!\cdots\!19}$, $\frac{21\!\cdots\!42}{47\!\cdots\!21}a^{15}+\frac{32\!\cdots\!12}{47\!\cdots\!21}a^{14}+\frac{20\!\cdots\!61}{47\!\cdots\!21}a^{13}+\frac{74\!\cdots\!86}{47\!\cdots\!21}a^{12}+\frac{15\!\cdots\!96}{47\!\cdots\!21}a^{11}+\frac{36\!\cdots\!48}{47\!\cdots\!21}a^{10}-\frac{11\!\cdots\!90}{47\!\cdots\!21}a^{9}+\frac{10\!\cdots\!59}{47\!\cdots\!21}a^{8}+\frac{23\!\cdots\!33}{47\!\cdots\!21}a^{7}-\frac{43\!\cdots\!80}{47\!\cdots\!21}a^{6}+\frac{22\!\cdots\!10}{47\!\cdots\!21}a^{5}+\frac{96\!\cdots\!93}{47\!\cdots\!21}a^{4}-\frac{41\!\cdots\!72}{47\!\cdots\!21}a^{3}+\frac{34\!\cdots\!08}{47\!\cdots\!21}a^{2}-\frac{16\!\cdots\!78}{47\!\cdots\!21}a+\frac{28\!\cdots\!30}{81\!\cdots\!19}$
| 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: | \( 921838016.564 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 921838016.564 \cdot 13655584}{2\cdot\sqrt{5600075906386661768959437042812533816731958913}}\cr\approx \mathstrut & 0.204304008115 \end{aligned}\] (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 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097)
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 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097, 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 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097);
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 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097);
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, 8.8.203930643438763634753.1 |
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 |
| 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 | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.15.3 | $x^{16} + 68$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
|
\(89\)
| 89.8.7.4 | $x^{8} + 801$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 89.8.7.4 | $x^{8} + 801$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |