Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 5*x^14 + 27*x^13 - 94*x^12 + 110*x^11 + 40*x^10 - 378*x^9 + 805*x^8 - 1131*x^7 + 1165*x^6 - 887*x^5 + 497*x^4 - 201*x^3 + 56*x^2 - 10*x + 1)
gp: K = bnfinit(y^16 - 5*y^15 + 5*y^14 + 27*y^13 - 94*y^12 + 110*y^11 + 40*y^10 - 378*y^9 + 805*y^8 - 1131*y^7 + 1165*y^6 - 887*y^5 + 497*y^4 - 201*y^3 + 56*y^2 - 10*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 5*x^14 + 27*x^13 - 94*x^12 + 110*x^11 + 40*x^10 - 378*x^9 + 805*x^8 - 1131*x^7 + 1165*x^6 - 887*x^5 + 497*x^4 - 201*x^3 + 56*x^2 - 10*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 5*x^14 + 27*x^13 - 94*x^12 + 110*x^11 + 40*x^10 - 378*x^9 + 805*x^8 - 1131*x^7 + 1165*x^6 - 887*x^5 + 497*x^4 - 201*x^3 + 56*x^2 - 10*x + 1)
\( x^{16} - 5 x^{15} + 5 x^{14} + 27 x^{13} - 94 x^{12} + 110 x^{11} + 40 x^{10} - 378 x^{9} + 805 x^{8} + \cdots + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(141027526969876265625\)
\(\medspace = 3^{12}\cdot 5^{6}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.17\) | 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: | $3^{3/4}5^{1/2}19^{1/2}\approx 22.21788649167895$ | ||
| Ramified primes: |
\(3\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{33}a^{14}-\frac{1}{11}a^{13}-\frac{1}{11}a^{12}+\frac{5}{33}a^{11}-\frac{1}{33}a^{10}-\frac{1}{33}a^{9}-\frac{4}{33}a^{8}+\frac{4}{11}a^{7}+\frac{4}{11}a^{6}-\frac{3}{11}a^{5}-\frac{10}{33}a^{4}+\frac{2}{33}a^{3}+\frac{5}{11}a^{2}+\frac{1}{33}a-\frac{5}{33}$, $\frac{1}{20559}a^{15}-\frac{10}{20559}a^{14}+\frac{226}{6853}a^{13}+\frac{1621}{20559}a^{12}-\frac{3215}{20559}a^{11}-\frac{1259}{20559}a^{10}-\frac{163}{2937}a^{9}-\frac{307}{2937}a^{8}+\frac{1294}{2937}a^{7}-\frac{1560}{6853}a^{6}-\frac{2224}{20559}a^{5}+\frac{1542}{6853}a^{4}-\frac{2768}{6853}a^{3}+\frac{8300}{20559}a^{2}+\frac{9019}{20559}a-\frac{3364}{20559}$
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
$C_{2}$, which has order $2$
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: |
\( -\frac{34411}{20559} a^{15} + \frac{144127}{20559} a^{14} - \frac{18044}{6853} a^{13} - \frac{973052}{20559} a^{12} + \frac{2429971}{20559} a^{11} - \frac{593917}{6853} a^{10} - \frac{132005}{979} a^{9} + \frac{1504244}{2937} a^{8} - \frac{901058}{979} a^{7} + \frac{23687555}{20559} a^{6} - \frac{1961816}{1869} a^{5} + \frac{4779427}{6853} a^{4} - \frac{2374452}{6853} a^{3} + \frac{2524954}{20559} a^{2} - \frac{18135}{623} a + \frac{93047}{20559} \)
(order $6$)
| 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: |
$\frac{1912}{2937}a^{15}-\frac{565}{267}a^{14}-\frac{1763}{979}a^{13}+\frac{60002}{2937}a^{12}-\frac{28074}{979}a^{11}-\frac{17304}{979}a^{10}+\frac{98693}{979}a^{9}-\frac{41185}{267}a^{8}+\frac{406849}{2937}a^{7}-\frac{35218}{979}a^{6}-\frac{356491}{2937}a^{5}+\frac{651839}{2937}a^{4}-\frac{555542}{2937}a^{3}+\frac{96509}{979}a^{2}-\frac{89668}{2937}a+\frac{12878}{2937}$, $\frac{38077}{20559}a^{15}-\frac{183902}{20559}a^{14}+\frac{54727}{6853}a^{13}+\frac{1038256}{20559}a^{12}-\frac{3410849}{20559}a^{11}+\frac{1253986}{6853}a^{10}+\frac{272738}{2937}a^{9}-\frac{2014093}{2937}a^{8}+\frac{1376234}{979}a^{7}-\frac{39374155}{20559}a^{6}+\frac{39180989}{20559}a^{5}-\frac{9459533}{6853}a^{4}+\frac{14564899}{20559}a^{3}-\frac{4921778}{20559}a^{2}+\frac{311973}{6853}a-\frac{17979}{6853}$, $\frac{1636}{20559}a^{15}+\frac{2953}{20559}a^{14}-\frac{31505}{20559}a^{13}+\frac{30995}{20559}a^{12}+\frac{175312}{20559}a^{11}-\frac{400052}{20559}a^{10}+\frac{5154}{979}a^{9}+\frac{93692}{2937}a^{8}-\frac{204403}{2937}a^{7}+\frac{1957033}{20559}a^{6}-\frac{159671}{1869}a^{5}+\frac{720743}{20559}a^{4}+\frac{87359}{6853}a^{3}-\frac{426863}{20559}a^{2}+\frac{20498}{1869}a-\frac{49139}{20559}$, $\frac{27928}{20559}a^{15}-\frac{117923}{20559}a^{14}+\frac{43955}{20559}a^{13}+\frac{268221}{6853}a^{12}-\frac{2003459}{20559}a^{11}+\frac{1395665}{20559}a^{10}+\frac{353950}{2937}a^{9}-\frac{1254091}{2937}a^{8}+\frac{2175898}{2937}a^{7}-\frac{6169613}{6853}a^{6}+\frac{16184276}{20559}a^{5}-\frac{9978911}{20559}a^{4}+\frac{4391657}{20559}a^{3}-\frac{1328561}{20559}a^{2}+\frac{251063}{20559}a-\frac{34411}{20559}$, $\frac{4776}{6853}a^{15}-\frac{75373}{20559}a^{14}+\frac{31547}{6853}a^{13}+\frac{352243}{20559}a^{12}-\frac{482584}{6853}a^{11}+\frac{2047814}{20559}a^{10}-\frac{13187}{2937}a^{9}-\frac{777019}{2937}a^{8}+\frac{636690}{979}a^{7}-\frac{20441557}{20559}a^{6}+\frac{22813369}{20559}a^{5}-\frac{19093969}{20559}a^{4}+\frac{11918195}{20559}a^{3}-\frac{1774994}{6853}a^{2}+\frac{1531660}{20559}a-\frac{211571}{20559}$, $\frac{82}{2937}a^{15}-\frac{303}{979}a^{14}+\frac{999}{979}a^{13}+\frac{15}{979}a^{12}-\frac{7747}{979}a^{11}+\frac{57407}{2937}a^{10}-\frac{39628}{2937}a^{9}-\frac{25661}{979}a^{8}+\frac{269810}{2937}a^{7}-\frac{153730}{979}a^{6}+\frac{184554}{979}a^{5}-\frac{158801}{979}a^{4}+\frac{95382}{979}a^{3}-\frac{39540}{979}a^{2}+\frac{34589}{2937}a-\frac{7157}{2937}$, $\frac{12625}{6853}a^{15}-\frac{5134}{623}a^{14}+\frac{106346}{20559}a^{13}+\frac{1050349}{20559}a^{12}-\frac{2993798}{20559}a^{11}+\frac{927994}{6853}a^{10}+\frac{118264}{979}a^{9}-\frac{165026}{267}a^{8}+\frac{1167323}{979}a^{7}-\frac{32070457}{20559}a^{6}+\frac{30691433}{20559}a^{5}-\frac{7137596}{6853}a^{4}+\frac{10702591}{20559}a^{3}-\frac{1196014}{6853}a^{2}+\frac{743849}{20559}a-\frac{94597}{20559}$
| 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: | \( 6365.05426835 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 6365.05426835 \cdot 2}{6\cdot\sqrt{141027526969876265625}}\cr\approx \mathstrut & 0.433978143669 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 5*x^14 + 27*x^13 - 94*x^12 + 110*x^11 + 40*x^10 - 378*x^9 + 805*x^8 - 1131*x^7 + 1165*x^6 - 887*x^5 + 497*x^4 - 201*x^3 + 56*x^2 - 10*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 - 5*x^15 + 5*x^14 + 27*x^13 - 94*x^12 + 110*x^11 + 40*x^10 - 378*x^9 + 805*x^8 - 1131*x^7 + 1165*x^6 - 887*x^5 + 497*x^4 - 201*x^3 + 56*x^2 - 10*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 - 5*x^15 + 5*x^14 + 27*x^13 - 94*x^12 + 110*x^11 + 40*x^10 - 378*x^9 + 805*x^8 - 1131*x^7 + 1165*x^6 - 887*x^5 + 497*x^4 - 201*x^3 + 56*x^2 - 10*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 - 5*x^15 + 5*x^14 + 27*x^13 - 94*x^12 + 110*x^11 + 40*x^10 - 378*x^9 + 805*x^8 - 1131*x^7 + 1165*x^6 - 887*x^5 + 497*x^4 - 201*x^3 + 56*x^2 - 10*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
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 $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-3}) \), 4.0.1805.1, 4.0.16245.1, \(\Q(\sqrt{-3}, \sqrt{-19})\), 8.0.263900025.1 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.16.12.2 | $x^{16} + 12 x^{12} + 36 x^{8} + 324$ | $4$ | $4$ | $12$ | $C_8: C_2$ | $[\ ]_{4}^{4}$ |
|
\(5\)
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(19\)
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |