Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983)
gp: K = bnfinit(y^16 - 3*y^15 + 26*y^14 - 129*y^13 + 302*y^12 - 1025*y^11 + 4282*y^10 + 16326*y^9 + 95981*y^8 + 338337*y^7 + 1003331*y^6 + 2604733*y^5 + 6573615*y^4 + 13657839*y^3 + 25464387*y^2 + 29036657*y + 45417983, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983)
\( x^{16} - 3 x^{15} + 26 x^{14} - 129 x^{13} + 302 x^{12} - 1025 x^{11} + 4282 x^{10} + 16326 x^{9} + \cdots + 45417983 \)
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: |
\(66688975910627504451630153142433\)
\(\medspace = 13^{12}\cdot 17^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(97.50\) | 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: | $13^{3/4}17^{15/16}\approx 97.49972213953866$ | ||
| Ramified primes: |
\(13\), \(17\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}+\frac{6}{13}a^{11}-\frac{4}{13}a^{10}+\frac{3}{13}a^{9}+\frac{3}{13}a^{8}-\frac{5}{13}a^{7}-\frac{1}{13}a^{6}+\frac{4}{13}a^{5}-\frac{4}{13}a^{4}$, $\frac{1}{13}a^{13}-\frac{1}{13}a^{11}+\frac{1}{13}a^{10}-\frac{2}{13}a^{9}+\frac{3}{13}a^{8}+\frac{3}{13}a^{7}-\frac{3}{13}a^{6}-\frac{2}{13}a^{5}-\frac{2}{13}a^{4}$, $\frac{1}{13}a^{14}-\frac{6}{13}a^{11}-\frac{6}{13}a^{10}+\frac{6}{13}a^{9}+\frac{6}{13}a^{8}+\frac{5}{13}a^{7}-\frac{3}{13}a^{6}+\frac{2}{13}a^{5}-\frac{4}{13}a^{4}$, $\frac{1}{63\!\cdots\!13}a^{15}-\frac{56\!\cdots\!43}{63\!\cdots\!13}a^{14}+\frac{17\!\cdots\!39}{63\!\cdots\!13}a^{13}-\frac{49\!\cdots\!21}{63\!\cdots\!13}a^{12}+\frac{24\!\cdots\!39}{63\!\cdots\!13}a^{11}-\frac{15\!\cdots\!42}{63\!\cdots\!13}a^{10}+\frac{30\!\cdots\!40}{63\!\cdots\!13}a^{9}+\frac{41\!\cdots\!30}{63\!\cdots\!13}a^{8}+\frac{10\!\cdots\!62}{63\!\cdots\!13}a^{7}-\frac{25\!\cdots\!48}{63\!\cdots\!13}a^{6}-\frac{37\!\cdots\!22}{42\!\cdots\!37}a^{5}-\frac{42\!\cdots\!57}{49\!\cdots\!01}a^{4}+\frac{15\!\cdots\!42}{49\!\cdots\!01}a^{3}+\frac{12\!\cdots\!19}{49\!\cdots\!01}a^{2}+\frac{18\!\cdots\!81}{49\!\cdots\!01}a+\frac{87\!\cdots\!67}{48\!\cdots\!01}$
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
$C_{2}\times C_{2}\times C_{2}\times C_{388}$, which has order $3104$ (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: |
$\frac{92\!\cdots\!48}{10\!\cdots\!83}a^{15}-\frac{17\!\cdots\!69}{10\!\cdots\!83}a^{14}+\frac{17\!\cdots\!97}{10\!\cdots\!83}a^{13}+\frac{10\!\cdots\!03}{10\!\cdots\!83}a^{12}-\frac{29\!\cdots\!80}{10\!\cdots\!83}a^{11}+\frac{20\!\cdots\!09}{10\!\cdots\!83}a^{10}-\frac{11\!\cdots\!24}{10\!\cdots\!83}a^{9}+\frac{52\!\cdots\!89}{10\!\cdots\!83}a^{8}+\frac{15\!\cdots\!08}{10\!\cdots\!83}a^{7}+\frac{54\!\cdots\!64}{77\!\cdots\!91}a^{6}+\frac{25\!\cdots\!62}{10\!\cdots\!83}a^{5}+\frac{58\!\cdots\!82}{77\!\cdots\!91}a^{4}+\frac{19\!\cdots\!19}{77\!\cdots\!91}a^{3}+\frac{31\!\cdots\!96}{77\!\cdots\!91}a^{2}+\frac{47\!\cdots\!13}{77\!\cdots\!91}a+\frac{92\!\cdots\!60}{76\!\cdots\!91}$, $\frac{27\!\cdots\!69}{10\!\cdots\!83}a^{15}-\frac{43\!\cdots\!08}{10\!\cdots\!83}a^{14}+\frac{42\!\cdots\!01}{10\!\cdots\!83}a^{13}-\frac{23\!\cdots\!33}{10\!\cdots\!83}a^{12}+\frac{13\!\cdots\!68}{10\!\cdots\!83}a^{11}-\frac{56\!\cdots\!31}{10\!\cdots\!83}a^{10}+\frac{18\!\cdots\!43}{10\!\cdots\!83}a^{9}-\frac{56\!\cdots\!37}{10\!\cdots\!83}a^{8}+\frac{11\!\cdots\!16}{10\!\cdots\!83}a^{7}+\frac{45\!\cdots\!56}{77\!\cdots\!91}a^{6}+\frac{11\!\cdots\!45}{10\!\cdots\!83}a^{5}+\frac{44\!\cdots\!77}{77\!\cdots\!91}a^{4}+\frac{77\!\cdots\!36}{77\!\cdots\!91}a^{3}+\frac{24\!\cdots\!63}{77\!\cdots\!91}a^{2}+\frac{19\!\cdots\!60}{77\!\cdots\!91}a+\frac{77\!\cdots\!47}{76\!\cdots\!91}$, $\frac{11\!\cdots\!17}{10\!\cdots\!83}a^{15}-\frac{60\!\cdots\!77}{10\!\cdots\!83}a^{14}+\frac{60\!\cdots\!98}{10\!\cdots\!83}a^{13}-\frac{22\!\cdots\!30}{10\!\cdots\!83}a^{12}+\frac{10\!\cdots\!88}{10\!\cdots\!83}a^{11}-\frac{36\!\cdots\!22}{10\!\cdots\!83}a^{10}+\frac{71\!\cdots\!19}{10\!\cdots\!83}a^{9}-\frac{39\!\cdots\!48}{10\!\cdots\!83}a^{8}+\frac{12\!\cdots\!24}{10\!\cdots\!83}a^{7}+\frac{59\!\cdots\!20}{77\!\cdots\!91}a^{6}+\frac{37\!\cdots\!07}{10\!\cdots\!83}a^{5}+\frac{10\!\cdots\!59}{77\!\cdots\!91}a^{4}+\frac{27\!\cdots\!55}{77\!\cdots\!91}a^{3}+\frac{56\!\cdots\!59}{77\!\cdots\!91}a^{2}+\frac{67\!\cdots\!73}{77\!\cdots\!91}a+\frac{17\!\cdots\!25}{76\!\cdots\!91}$, $\frac{74\!\cdots\!70}{42\!\cdots\!37}a^{15}-\frac{25\!\cdots\!15}{42\!\cdots\!37}a^{14}+\frac{17\!\cdots\!96}{42\!\cdots\!37}a^{13}-\frac{96\!\cdots\!07}{42\!\cdots\!37}a^{12}+\frac{21\!\cdots\!49}{42\!\cdots\!37}a^{11}-\frac{58\!\cdots\!55}{42\!\cdots\!37}a^{10}+\frac{30\!\cdots\!15}{42\!\cdots\!37}a^{9}+\frac{11\!\cdots\!13}{42\!\cdots\!37}a^{8}+\frac{44\!\cdots\!32}{32\!\cdots\!49}a^{7}+\frac{16\!\cdots\!62}{42\!\cdots\!37}a^{6}+\frac{41\!\cdots\!86}{42\!\cdots\!37}a^{5}+\frac{91\!\cdots\!09}{42\!\cdots\!37}a^{4}+\frac{17\!\cdots\!77}{32\!\cdots\!49}a^{3}+\frac{32\!\cdots\!38}{32\!\cdots\!49}a^{2}+\frac{35\!\cdots\!26}{32\!\cdots\!49}a-\frac{24\!\cdots\!36}{32\!\cdots\!49}$, $\frac{93\!\cdots\!84}{42\!\cdots\!37}a^{15}+\frac{35\!\cdots\!72}{42\!\cdots\!37}a^{14}+\frac{29\!\cdots\!65}{42\!\cdots\!37}a^{13}-\frac{77\!\cdots\!19}{42\!\cdots\!37}a^{12}-\frac{44\!\cdots\!71}{42\!\cdots\!37}a^{11}+\frac{18\!\cdots\!17}{42\!\cdots\!37}a^{10}-\frac{36\!\cdots\!38}{42\!\cdots\!37}a^{9}+\frac{32\!\cdots\!26}{32\!\cdots\!49}a^{8}+\frac{86\!\cdots\!78}{42\!\cdots\!37}a^{7}+\frac{38\!\cdots\!65}{42\!\cdots\!37}a^{6}+\frac{83\!\cdots\!14}{42\!\cdots\!37}a^{5}+\frac{17\!\cdots\!81}{42\!\cdots\!37}a^{4}+\frac{44\!\cdots\!61}{32\!\cdots\!49}a^{3}+\frac{52\!\cdots\!85}{32\!\cdots\!49}a^{2}+\frac{11\!\cdots\!81}{32\!\cdots\!49}a-\frac{98\!\cdots\!66}{32\!\cdots\!49}$, $\frac{11\!\cdots\!10}{42\!\cdots\!37}a^{15}+\frac{13\!\cdots\!42}{42\!\cdots\!37}a^{14}-\frac{36\!\cdots\!42}{42\!\cdots\!37}a^{13}+\frac{23\!\cdots\!78}{42\!\cdots\!37}a^{12}-\frac{55\!\cdots\!23}{42\!\cdots\!37}a^{11}+\frac{17\!\cdots\!45}{42\!\cdots\!37}a^{10}-\frac{52\!\cdots\!94}{42\!\cdots\!37}a^{9}+\frac{35\!\cdots\!22}{42\!\cdots\!37}a^{8}+\frac{15\!\cdots\!03}{42\!\cdots\!37}a^{7}+\frac{44\!\cdots\!23}{42\!\cdots\!37}a^{6}+\frac{11\!\cdots\!57}{42\!\cdots\!37}a^{5}+\frac{32\!\cdots\!48}{42\!\cdots\!37}a^{4}+\frac{58\!\cdots\!96}{32\!\cdots\!49}a^{3}+\frac{10\!\cdots\!31}{32\!\cdots\!49}a^{2}+\frac{15\!\cdots\!38}{32\!\cdots\!49}a+\frac{20\!\cdots\!28}{32\!\cdots\!49}$, $\frac{23\!\cdots\!43}{42\!\cdots\!37}a^{15}-\frac{10\!\cdots\!03}{42\!\cdots\!37}a^{14}+\frac{65\!\cdots\!41}{42\!\cdots\!37}a^{13}-\frac{32\!\cdots\!58}{42\!\cdots\!37}a^{12}+\frac{86\!\cdots\!32}{42\!\cdots\!37}a^{11}-\frac{17\!\cdots\!47}{32\!\cdots\!49}a^{10}+\frac{79\!\cdots\!00}{42\!\cdots\!37}a^{9}+\frac{36\!\cdots\!32}{42\!\cdots\!37}a^{8}+\frac{16\!\cdots\!16}{42\!\cdots\!37}a^{7}+\frac{54\!\cdots\!35}{42\!\cdots\!37}a^{6}+\frac{12\!\cdots\!15}{42\!\cdots\!37}a^{5}+\frac{26\!\cdots\!71}{42\!\cdots\!37}a^{4}+\frac{57\!\cdots\!21}{32\!\cdots\!49}a^{3}+\frac{10\!\cdots\!52}{32\!\cdots\!49}a^{2}+\frac{11\!\cdots\!30}{32\!\cdots\!49}a-\frac{18\!\cdots\!67}{32\!\cdots\!49}$
| 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: | \( 464752.47625 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 464752.47625 \cdot 3104}{2\cdot\sqrt{66688975910627504451630153142433}}\cr\approx \mathstrut & 0.21454843552 \end{aligned}\] (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 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983)
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 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983, 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 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983);
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 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983);
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, 8.8.11719682839553.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | R | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.4.3.3 | $x^{4} + 26$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 13.4.3.4 | $x^{4} + 91$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.4.3.3 | $x^{4} + 26$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.4.3.4 | $x^{4} + 91$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(17\)
| 17.16.15.1 | $x^{16} + 272$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |