Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 4*x^18 - 5*x^17 + 9*x^16 - 8*x^15 + 9*x^14 - 16*x^13 + 11*x^12 - 10*x^11 + 17*x^10 - 10*x^9 + 11*x^8 - 16*x^7 + 9*x^6 - 8*x^5 + 9*x^4 - 5*x^3 + 4*x^2 - 3*x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 4*y^18 - 5*y^17 + 9*y^16 - 8*y^15 + 9*y^14 - 16*y^13 + 11*y^12 - 10*y^11 + 17*y^10 - 10*y^9 + 11*y^8 - 16*y^7 + 9*y^6 - 8*y^5 + 9*y^4 - 5*y^3 + 4*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 4*x^18 - 5*x^17 + 9*x^16 - 8*x^15 + 9*x^14 - 16*x^13 + 11*x^12 - 10*x^11 + 17*x^10 - 10*x^9 + 11*x^8 - 16*x^7 + 9*x^6 - 8*x^5 + 9*x^4 - 5*x^3 + 4*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 4*x^18 - 5*x^17 + 9*x^16 - 8*x^15 + 9*x^14 - 16*x^13 + 11*x^12 - 10*x^11 + 17*x^10 - 10*x^9 + 11*x^8 - 16*x^7 + 9*x^6 - 8*x^5 + 9*x^4 - 5*x^3 + 4*x^2 - 3*x + 1)
\( x^{20} - 3 x^{19} + 4 x^{18} - 5 x^{17} + 9 x^{16} - 8 x^{15} + 9 x^{14} - 16 x^{13} + 11 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(602207464968018231001\)
\(\medspace = 47^{10}\cdot 107^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.94\) | 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: | $47^{1/2}107^{1/2}\approx 70.9154426059656$ | ||
| Ramified primes: |
\(47\), \(107\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29}a^{18}+\frac{10}{29}a^{17}-\frac{12}{29}a^{16}+\frac{3}{29}a^{15}+\frac{2}{29}a^{14}-\frac{14}{29}a^{13}-\frac{1}{29}a^{12}+\frac{14}{29}a^{11}-\frac{9}{29}a^{10}+\frac{4}{29}a^{9}-\frac{9}{29}a^{8}+\frac{14}{29}a^{7}-\frac{1}{29}a^{6}-\frac{14}{29}a^{5}+\frac{2}{29}a^{4}+\frac{3}{29}a^{3}-\frac{12}{29}a^{2}+\frac{10}{29}a+\frac{1}{29}$, $\frac{1}{377}a^{19}-\frac{4}{377}a^{18}+\frac{138}{377}a^{17}+\frac{2}{29}a^{16}-\frac{69}{377}a^{15}+\frac{74}{377}a^{14}-\frac{14}{29}a^{13}-\frac{146}{377}a^{12}+\frac{27}{377}a^{11}-\frac{102}{377}a^{10}+\frac{80}{377}a^{9}+\frac{53}{377}a^{8}-\frac{81}{377}a^{7}-\frac{121}{377}a^{5}+\frac{178}{377}a^{4}+\frac{7}{29}a^{3}-\frac{83}{377}a^{2}+\frac{35}{377}a+\frac{131}{377}$
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
Trivial group, which has order $1$
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: | $9$ | 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: |
$\frac{245}{377}a^{19}-\frac{343}{377}a^{18}-\frac{159}{377}a^{17}-\frac{11}{29}a^{16}+\frac{1217}{377}a^{15}+\frac{554}{377}a^{14}+\frac{31}{29}a^{13}-\frac{2854}{377}a^{12}-\frac{2186}{377}a^{11}-\frac{940}{377}a^{10}+\frac{3675}{377}a^{9}+\frac{2728}{377}a^{8}+\frac{1514}{377}a^{7}-\frac{194}{29}a^{6}-\frac{1994}{377}a^{5}-\frac{1487}{377}a^{4}+\frac{93}{29}a^{3}+\frac{296}{377}a^{2}+\frac{619}{377}a-\frac{444}{377}$, $\frac{96}{377}a^{19}-\frac{202}{377}a^{18}+\frac{365}{377}a^{17}-\frac{63}{29}a^{16}+\frac{1085}{377}a^{15}-\frac{449}{377}a^{14}+\frac{113}{29}a^{13}-\frac{1757}{377}a^{12}+\frac{239}{377}a^{11}-\frac{2382}{377}a^{10}+\frac{1999}{377}a^{9}+\frac{434}{377}a^{8}+\frac{2689}{377}a^{7}-\frac{159}{29}a^{6}-\frac{215}{377}a^{5}-\frac{2152}{377}a^{4}+\frac{76}{29}a^{3}+\frac{27}{377}a^{2}+\frac{1033}{377}a-\frac{437}{377}$, $a$, $\frac{269}{377}a^{19}-\frac{582}{377}a^{18}+\frac{215}{377}a^{17}-\frac{5}{29}a^{16}+\frac{1017}{377}a^{15}+\frac{159}{377}a^{14}-\frac{6}{29}a^{13}-\frac{2445}{377}a^{12}-\frac{901}{377}a^{11}+\frac{538}{377}a^{10}+\frac{3138}{377}a^{9}+\frac{1517}{377}a^{8}+\frac{207}{377}a^{7}-\frac{183}{29}a^{6}-\frac{1765}{377}a^{5}-\frac{140}{377}a^{4}+\frac{83}{29}a^{3}+\frac{774}{377}a^{2}+\frac{14}{13}a-\frac{82}{377}$, $\frac{254}{377}a^{19}-\frac{600}{377}a^{18}+\frac{758}{377}a^{17}-\frac{50}{29}a^{16}+\frac{1064}{377}a^{15}-\frac{730}{377}a^{14}+\frac{143}{29}a^{13}-\frac{1308}{377}a^{12}+\frac{1749}{377}a^{11}-\frac{1754}{377}a^{10}+\frac{118}{377}a^{9}-\frac{2346}{377}a^{8}+\frac{2215}{377}a^{7}-\frac{3}{29}a^{6}+\frac{2273}{377}a^{5}-\frac{704}{377}a^{4}+\frac{18}{29}a^{3}-\frac{1192}{377}a^{2}+\frac{21}{13}a-\frac{240}{377}$, $\frac{210}{377}a^{19}-\frac{190}{377}a^{18}-\frac{335}{377}a^{17}+\frac{23}{29}a^{16}+\frac{278}{377}a^{15}+\frac{1383}{377}a^{14}-\frac{15}{29}a^{13}-\frac{773}{377}a^{12}-\frac{2572}{377}a^{11}-\frac{126}{377}a^{10}+\frac{550}{377}a^{9}+\frac{2264}{377}a^{8}+\frac{384}{377}a^{7}+\frac{37}{29}a^{6}-\frac{46}{13}a^{5}+\frac{226}{377}a^{4}-\frac{62}{29}a^{3}+\frac{1}{13}a^{2}-\frac{99}{377}a+\frac{262}{377}$, $\frac{266}{377}a^{19}-\frac{63}{377}a^{18}-\frac{407}{377}a^{17}+\frac{14}{29}a^{16}+\frac{106}{377}a^{15}+\frac{2082}{377}a^{14}+\frac{99}{29}a^{13}+\frac{502}{377}a^{12}-\frac{2932}{377}a^{11}-\frac{2211}{377}a^{10}-\frac{1860}{377}a^{9}+\frac{2073}{377}a^{8}+\frac{2647}{377}a^{7}+\frac{213}{29}a^{6}-\frac{583}{377}a^{5}-\frac{414}{377}a^{4}-\frac{111}{29}a^{3}-\frac{160}{377}a^{2}+\frac{93}{377}a+\frac{409}{377}$, $\frac{100}{377}a^{19}-\frac{309}{377}a^{18}+\frac{384}{377}a^{17}-a^{16}+\frac{536}{377}a^{15}-\frac{335}{377}a^{14}+\frac{39}{29}a^{13}-\frac{1119}{377}a^{12}+\frac{204}{377}a^{11}-\frac{840}{377}a^{10}+\frac{1201}{377}a^{9}+\frac{334}{377}a^{8}+\frac{1468}{377}a^{7}-\frac{94}{29}a^{6}-\frac{556}{377}a^{5}-\frac{1245}{377}a^{4}+\frac{25}{29}a^{3}+\frac{410}{377}a^{2}+\frac{640}{377}a-\frac{4}{377}$, $\frac{45}{377}a^{19}-\frac{193}{377}a^{18}+\frac{48}{377}a^{17}+\frac{15}{29}a^{16}+\frac{249}{377}a^{15}-\frac{466}{377}a^{14}-\frac{36}{29}a^{13}-\frac{902}{377}a^{12}+\frac{279}{377}a^{11}+\frac{1559}{377}a^{10}+\frac{1286}{377}a^{9}-\frac{137}{377}a^{8}-\frac{1188}{377}a^{7}-\frac{86}{29}a^{6}-\frac{362}{377}a^{5}+\frac{444}{377}a^{4}+\frac{51}{29}a^{3}-\frac{186}{377}a^{2}-\frac{63}{377}a+\frac{227}{377}$
| 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: | \( 79.7167288943 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 79.7167288943 \cdot 1}{2\cdot\sqrt{602207464968018231001}}\cr\approx \mathstrut & 0.155756109243 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 4*x^18 - 5*x^17 + 9*x^16 - 8*x^15 + 9*x^14 - 16*x^13 + 11*x^12 - 10*x^11 + 17*x^10 - 10*x^9 + 11*x^8 - 16*x^7 + 9*x^6 - 8*x^5 + 9*x^4 - 5*x^3 + 4*x^2 - 3*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^20 - 3*x^19 + 4*x^18 - 5*x^17 + 9*x^16 - 8*x^15 + 9*x^14 - 16*x^13 + 11*x^12 - 10*x^11 + 17*x^10 - 10*x^9 + 11*x^8 - 16*x^7 + 9*x^6 - 8*x^5 + 9*x^4 - 5*x^3 + 4*x^2 - 3*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^20 - 3*x^19 + 4*x^18 - 5*x^17 + 9*x^16 - 8*x^15 + 9*x^14 - 16*x^13 + 11*x^12 - 10*x^11 + 17*x^10 - 10*x^9 + 11*x^8 - 16*x^7 + 9*x^6 - 8*x^5 + 9*x^4 - 5*x^3 + 4*x^2 - 3*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^20 - 3*x^19 + 4*x^18 - 5*x^17 + 9*x^16 - 8*x^15 + 9*x^14 - 16*x^13 + 11*x^12 - 10*x^11 + 17*x^10 - 10*x^9 + 11*x^8 - 16*x^7 + 9*x^6 - 8*x^5 + 9*x^4 - 5*x^3 + 4*x^2 - 3*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
$C_2\wr D_5$ (as 20T81):
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 320 |
| The 20 conjugacy class representatives for $C_2\wr D_5$ |
| Character table for $C_2\wr D_5$ |
Intermediate fields
| \(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.2.24539915749.1, 10.0.522125867.1, 10.0.229345007.1 |
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
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.0.522125867.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(47\)
| 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 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(107\)
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.4.0.1 | $x^{4} + 13 x^{2} + 79 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 107.4.0.1 | $x^{4} + 13 x^{2} + 79 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |