Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 + 499*x^12 - 3216*x^10 + 64721*x^8 - 216560*x^6 + 1770675*x^4 + 478*x^2 + 1)
gp: K = bnfinit(y^16 - 2*y^14 + 499*y^12 - 3216*y^10 + 64721*y^8 - 216560*y^6 + 1770675*y^4 + 478*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^14 + 499*x^12 - 3216*x^10 + 64721*x^8 - 216560*x^6 + 1770675*x^4 + 478*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^14 + 499*x^12 - 3216*x^10 + 64721*x^8 - 216560*x^6 + 1770675*x^4 + 478*x^2 + 1)
\( x^{16} - 2x^{14} + 499x^{12} - 3216x^{10} + 64721x^{8} - 216560x^{6} + 1770675x^{4} + 478x^{2} + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4009292695690170390860412175969\)
\(\medspace = 17^{14}\cdot 47^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(81.79\) | 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: | $17^{7/8}47^{1/2}\approx 81.7884043055344$ | ||
| Ramified primes: |
\(17\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{1}{4}a^{2}+\frac{1}{8}a-\frac{1}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}-\frac{1}{8}$, $\frac{1}{10\!\cdots\!76}a^{14}-\frac{61\!\cdots\!05}{10\!\cdots\!76}a^{12}-\frac{61\!\cdots\!87}{10\!\cdots\!76}a^{10}-\frac{1}{8}a^{9}+\frac{60\!\cdots\!25}{10\!\cdots\!76}a^{8}-\frac{1}{8}a^{7}+\frac{24\!\cdots\!05}{10\!\cdots\!76}a^{6}-\frac{1}{8}a^{5}+\frac{55\!\cdots\!29}{25\!\cdots\!94}a^{4}+\frac{3}{8}a^{3}-\frac{11\!\cdots\!25}{51\!\cdots\!88}a^{2}+\frac{3}{8}a-\frac{86\!\cdots\!63}{25\!\cdots\!94}$, $\frac{1}{10\!\cdots\!76}a^{15}-\frac{61\!\cdots\!05}{10\!\cdots\!76}a^{13}-\frac{61\!\cdots\!87}{10\!\cdots\!76}a^{11}-\frac{1}{8}a^{10}+\frac{60\!\cdots\!25}{10\!\cdots\!76}a^{9}-\frac{1}{8}a^{8}+\frac{24\!\cdots\!05}{10\!\cdots\!76}a^{7}-\frac{1}{8}a^{6}+\frac{55\!\cdots\!29}{25\!\cdots\!94}a^{5}-\frac{1}{8}a^{4}+\frac{14\!\cdots\!69}{51\!\cdots\!88}a^{3}-\frac{1}{8}a^{2}+\frac{21\!\cdots\!67}{12\!\cdots\!97}a-\frac{1}{2}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{4}\times C_{160}$, which has order $1280$ (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: |
$\frac{270978780282205}{99\!\cdots\!92}a^{15}+\frac{6020905}{41760813714728}a^{14}-\frac{271086622641305}{49\!\cdots\!96}a^{13}+\frac{36519201}{41760813714728}a^{12}+\frac{16\!\cdots\!12}{12\!\cdots\!99}a^{11}+\frac{359701679}{5220101714341}a^{10}-\frac{87\!\cdots\!23}{99\!\cdots\!92}a^{9}+\frac{5166309919}{41760813714728}a^{8}+\frac{17\!\cdots\!97}{99\!\cdots\!92}a^{7}+\frac{228557809215}{41760813714728}a^{6}-\frac{73\!\cdots\!40}{12\!\cdots\!99}a^{5}+\frac{930432162791}{20880406857364}a^{4}+\frac{47\!\cdots\!23}{99\!\cdots\!92}a^{3}+\frac{251285647}{20880406857364}a^{2}-\frac{23\!\cdots\!99}{99\!\cdots\!92}a+\frac{32605859111575}{41760813714728}$, $\frac{347063108831827}{99\!\cdots\!92}a^{15}+\frac{6020905}{41760813714728}a^{14}-\frac{347188882692873}{49\!\cdots\!96}a^{13}+\frac{36519201}{41760813714728}a^{12}+\frac{86\!\cdots\!47}{49\!\cdots\!96}a^{11}+\frac{359701679}{5220101714341}a^{10}-\frac{11\!\cdots\!51}{99\!\cdots\!92}a^{9}+\frac{5166309919}{41760813714728}a^{8}+\frac{22\!\cdots\!25}{99\!\cdots\!92}a^{7}+\frac{228557809215}{41760813714728}a^{6}-\frac{37\!\cdots\!91}{49\!\cdots\!96}a^{5}+\frac{930432162791}{20880406857364}a^{4}+\frac{61\!\cdots\!11}{99\!\cdots\!92}a^{3}+\frac{251285647}{20880406857364}a^{2}-\frac{29\!\cdots\!17}{99\!\cdots\!92}a+\frac{53486265968939}{41760813714728}$, $\frac{38042164274811}{49\!\cdots\!96}a^{15}-\frac{19025565012892}{12\!\cdots\!99}a^{13}+\frac{18\!\cdots\!99}{49\!\cdots\!96}a^{11}-\frac{61\!\cdots\!57}{24\!\cdots\!98}a^{9}+\frac{12\!\cdots\!57}{24\!\cdots\!98}a^{7}-\frac{82\!\cdots\!31}{49\!\cdots\!96}a^{5}+\frac{33\!\cdots\!97}{24\!\cdots\!98}a^{3}-\frac{32\!\cdots\!09}{49\!\cdots\!96}a-\frac{1}{2}$, $\frac{6020905}{20880406857364}a^{15}-\frac{116808110073}{24\!\cdots\!98}a^{14}+\frac{36519201}{20880406857364}a^{13}+\frac{461857667711}{49\!\cdots\!96}a^{12}+\frac{719403358}{5220101714341}a^{11}-\frac{116499754671181}{49\!\cdots\!96}a^{10}+\frac{5166309919}{20880406857364}a^{9}+\frac{748705333689975}{49\!\cdots\!96}a^{8}+\frac{228557809215}{20880406857364}a^{7}-\frac{15\!\cdots\!91}{49\!\cdots\!96}a^{6}+\frac{930432162791}{10440203428682}a^{5}+\frac{50\!\cdots\!17}{49\!\cdots\!96}a^{4}+\frac{251285647}{10440203428682}a^{3}-\frac{10\!\cdots\!39}{12\!\cdots\!99}a^{2}+\frac{84806876254985}{20880406857364}a-\frac{77255236139254}{12\!\cdots\!99}$, $\frac{34\!\cdots\!71}{51\!\cdots\!88}a^{15}-\frac{16749689300551}{25\!\cdots\!94}a^{14}-\frac{68\!\cdots\!05}{51\!\cdots\!88}a^{13}+\frac{139749041765755}{10\!\cdots\!76}a^{12}+\frac{34\!\cdots\!09}{10\!\cdots\!76}a^{11}-\frac{41\!\cdots\!96}{12\!\cdots\!97}a^{10}-\frac{11\!\cdots\!81}{51\!\cdots\!88}a^{9}+\frac{21\!\cdots\!73}{10\!\cdots\!76}a^{8}+\frac{22\!\cdots\!09}{51\!\cdots\!88}a^{7}-\frac{43\!\cdots\!75}{10\!\cdots\!76}a^{6}-\frac{37\!\cdots\!63}{25\!\cdots\!94}a^{5}+\frac{14\!\cdots\!61}{10\!\cdots\!76}a^{4}+\frac{30\!\cdots\!47}{25\!\cdots\!94}a^{3}-\frac{59\!\cdots\!49}{51\!\cdots\!88}a^{2}-\frac{10\!\cdots\!15}{10\!\cdots\!76}a-\frac{19\!\cdots\!99}{10\!\cdots\!76}$, $\frac{13\!\cdots\!63}{10\!\cdots\!76}a^{15}-\frac{1801221586046}{576247730501939}a^{14}-\frac{12\!\cdots\!51}{51\!\cdots\!88}a^{13}+\frac{14410510718709}{23\!\cdots\!56}a^{12}+\frac{33\!\cdots\!29}{51\!\cdots\!88}a^{11}-\frac{71\!\cdots\!37}{46\!\cdots\!12}a^{10}-\frac{21\!\cdots\!05}{51\!\cdots\!88}a^{9}+\frac{23\!\cdots\!85}{23\!\cdots\!56}a^{8}+\frac{42\!\cdots\!19}{51\!\cdots\!88}a^{7}-\frac{46\!\cdots\!21}{23\!\cdots\!56}a^{6}-\frac{27\!\cdots\!13}{10\!\cdots\!76}a^{5}+\frac{15\!\cdots\!35}{23\!\cdots\!56}a^{4}+\frac{29\!\cdots\!22}{12\!\cdots\!97}a^{3}-\frac{63\!\cdots\!71}{11\!\cdots\!78}a^{2}+\frac{50\!\cdots\!73}{25\!\cdots\!94}a-\frac{19\!\cdots\!41}{46\!\cdots\!12}$, $\frac{12\!\cdots\!75}{10\!\cdots\!76}a^{15}+\frac{196640268495609}{10\!\cdots\!76}a^{14}-\frac{12\!\cdots\!17}{51\!\cdots\!88}a^{13}-\frac{380899440253225}{10\!\cdots\!76}a^{12}+\frac{15\!\cdots\!99}{25\!\cdots\!94}a^{11}+\frac{12\!\cdots\!85}{12\!\cdots\!97}a^{10}-\frac{40\!\cdots\!45}{10\!\cdots\!76}a^{9}-\frac{62\!\cdots\!83}{10\!\cdots\!76}a^{8}+\frac{81\!\cdots\!91}{10\!\cdots\!76}a^{7}+\frac{12\!\cdots\!93}{10\!\cdots\!76}a^{6}-\frac{68\!\cdots\!49}{25\!\cdots\!94}a^{5}-\frac{52\!\cdots\!02}{12\!\cdots\!97}a^{4}+\frac{22\!\cdots\!33}{10\!\cdots\!76}a^{3}+\frac{17\!\cdots\!93}{51\!\cdots\!88}a^{2}-\frac{22\!\cdots\!73}{10\!\cdots\!76}a+\frac{92\!\cdots\!65}{10\!\cdots\!76}$
| 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: | \( 417637.331377 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 417637.331377 \cdot 1280}{2\cdot\sqrt{4009292695690170390860412175969}}\cr\approx \mathstrut & 0.324253260284 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 + 499*x^12 - 3216*x^10 + 64721*x^8 - 216560*x^6 + 1770675*x^4 + 478*x^2 + 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 - 2*x^14 + 499*x^12 - 3216*x^10 + 64721*x^8 - 216560*x^6 + 1770675*x^4 + 478*x^2 + 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 - 2*x^14 + 499*x^12 - 3216*x^10 + 64721*x^8 - 216560*x^6 + 1770675*x^4 + 478*x^2 + 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 - 2*x^14 + 499*x^12 - 3216*x^10 + 64721*x^8 - 216560*x^6 + 1770675*x^4 + 478*x^2 + 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
$\OD_{16}:C_2$ (as 16T36):
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 11 conjugacy class representatives for $\OD_{16}:C_2$ |
| Character table for $\OD_{16}:C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-799}) \), \(\Q(\sqrt{-47}) \), 4.0.10852817.2, 4.4.4913.1, \(\Q(\sqrt{17}, \sqrt{-47})\), 8.0.2002321826203313.2 x2, 8.4.906438128657.2 x2, 8.0.117783636835489.4 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
|
\(47\)
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |