Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + x^2 + 1)
gp: K = bnfinit(y^20 + y^18 - 6*y^17 - y^16 - 4*y^15 + 15*y^14 + 7*y^13 + 10*y^12 - 17*y^11 - 11*y^10 - 17*y^9 + 10*y^8 + 7*y^7 + 15*y^6 - 4*y^5 - y^4 - 6*y^3 + y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + x^2 + 1)
\( x^{20} + x^{18} - 6 x^{17} - x^{16} - 4 x^{15} + 15 x^{14} + 7 x^{13} + 10 x^{12} - 17 x^{11} - 11 x^{10} + \cdots + 1 \)
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: |
\(902558723428708305129\)
\(\medspace = 3^{10}\cdot 11119^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.16\) | 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: | $3^{1/2}11119^{1/2}\approx 182.6389881706532$ | ||
| Ramified primes: |
\(3\), \(11119\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{16}-\frac{1}{9}a^{15}-\frac{1}{9}a^{14}-\frac{4}{9}a^{12}+\frac{2}{9}a^{11}+\frac{2}{9}a^{10}-\frac{1}{3}a^{8}-\frac{1}{9}a^{6}-\frac{4}{9}a^{5}+\frac{2}{9}a^{4}-\frac{1}{3}a^{3}-\frac{1}{9}a^{2}-\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{15}-\frac{1}{9}a^{14}-\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{2}{9}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{9}a^{7}+\frac{4}{9}a^{6}+\frac{4}{9}a^{5}+\frac{2}{9}a^{4}-\frac{1}{9}a^{3}-\frac{2}{9}a^{2}-\frac{2}{9}$, $\frac{1}{27}a^{18}+\frac{1}{27}a^{17}+\frac{1}{27}a^{16}+\frac{1}{9}a^{15}+\frac{1}{27}a^{14}+\frac{1}{9}a^{13}+\frac{8}{27}a^{12}+\frac{1}{9}a^{11}-\frac{13}{27}a^{10}+\frac{1}{9}a^{9}-\frac{4}{27}a^{8}+\frac{1}{9}a^{7}-\frac{10}{27}a^{6}+\frac{4}{9}a^{5}-\frac{8}{27}a^{4}+\frac{4}{9}a^{3}+\frac{10}{27}a^{2}+\frac{10}{27}a+\frac{10}{27}$, $\frac{1}{27}a^{19}-\frac{1}{27}a^{16}+\frac{1}{27}a^{15}-\frac{4}{27}a^{14}+\frac{5}{27}a^{13}-\frac{2}{27}a^{12}-\frac{4}{27}a^{11}-\frac{8}{27}a^{10}-\frac{7}{27}a^{9}-\frac{11}{27}a^{8}-\frac{13}{27}a^{7}-\frac{2}{27}a^{6}-\frac{8}{27}a^{5}-\frac{4}{27}a^{4}-\frac{2}{27}a^{3}-\frac{2}{9}a^{2}+\frac{1}{9}a-\frac{13}{27}$
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: |
\( \frac{26}{27} a^{19} + \frac{34}{27} a^{18} + \frac{1}{27} a^{17} - \frac{121}{27} a^{16} - \frac{244}{27} a^{15} + \frac{2}{27} a^{14} + \frac{283}{27} a^{13} + \frac{712}{27} a^{12} + \frac{166}{27} a^{11} - \frac{290}{27} a^{10} - \frac{899}{27} a^{9} - \frac{422}{27} a^{8} - \frac{41}{27} a^{7} + \frac{658}{27} a^{6} + \frac{323}{27} a^{5} + \frac{164}{27} a^{4} - \frac{295}{27} a^{3} - \frac{35}{27} a^{2} - \frac{56}{27} a + \frac{74}{27} \)
(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: |
$a$, $\frac{38}{27}a^{19}-\frac{64}{27}a^{18}+\frac{68}{27}a^{17}-\frac{91}{9}a^{16}+\frac{356}{27}a^{15}-\frac{80}{9}a^{14}+\frac{667}{27}a^{13}-\frac{236}{9}a^{12}+\frac{283}{27}a^{11}-28a^{10}+\frac{793}{27}a^{9}-\frac{35}{3}a^{8}+\frac{775}{27}a^{7}-\frac{236}{9}a^{6}+\frac{257}{27}a^{5}-\frac{194}{9}a^{4}+\frac{491}{27}a^{3}-\frac{133}{27}a^{2}+\frac{176}{27}a-\frac{31}{9}$, $\frac{5}{3}a^{19}+\frac{29}{27}a^{18}+\frac{29}{27}a^{17}-\frac{226}{27}a^{16}-\frac{25}{3}a^{15}-\frac{94}{27}a^{14}+\frac{167}{9}a^{13}+\frac{748}{27}a^{12}+\frac{37}{3}a^{11}-\frac{437}{27}a^{10}-\frac{310}{9}a^{9}-\frac{602}{27}a^{8}+\frac{23}{9}a^{7}+\frac{640}{27}a^{6}+\frac{50}{3}a^{5}+\frac{77}{27}a^{4}-\frac{97}{9}a^{3}-\frac{58}{27}a^{2}-\frac{40}{27}a+\frac{80}{27}$, $\frac{67}{27}a^{19}-\frac{5}{3}a^{18}+\frac{7}{3}a^{17}-\frac{418}{27}a^{16}+\frac{193}{27}a^{15}-\frac{169}{27}a^{14}+\frac{1019}{27}a^{13}-\frac{179}{27}a^{12}+\frac{191}{27}a^{11}-\frac{1211}{27}a^{10}+\frac{71}{27}a^{9}-\frac{368}{27}a^{8}+\frac{1046}{27}a^{7}-\frac{26}{27}a^{6}+\frac{355}{27}a^{5}-\frac{673}{27}a^{4}+\frac{91}{27}a^{3}-\frac{47}{9}a^{2}+\frac{49}{9}a-\frac{7}{27}$, $\frac{64}{27}a^{19}+\frac{2}{3}a^{18}+\frac{5}{3}a^{17}-\frac{346}{27}a^{16}-\frac{176}{27}a^{15}-\frac{154}{27}a^{14}+\frac{806}{27}a^{13}+\frac{676}{27}a^{12}+\frac{467}{27}a^{11}-\frac{806}{27}a^{10}-\frac{826}{27}a^{9}-\frac{848}{27}a^{8}+\frac{302}{27}a^{7}+\frac{478}{27}a^{6}+\frac{607}{27}a^{5}-\frac{82}{27}a^{4}-\frac{92}{27}a^{3}-\frac{31}{9}a^{2}+\frac{8}{9}a+\frac{2}{27}$, $\frac{29}{27}a^{19}-\frac{13}{9}a^{18}+\frac{11}{9}a^{17}-\frac{194}{27}a^{16}+\frac{200}{27}a^{15}-\frac{83}{27}a^{14}+\frac{478}{27}a^{13}-\frac{361}{27}a^{12}+\frac{28}{27}a^{11}-\frac{607}{27}a^{10}+\frac{355}{27}a^{9}-\frac{55}{27}a^{8}+\frac{649}{27}a^{7}-\frac{199}{27}a^{6}+\frac{83}{27}a^{5}-\frac{461}{27}a^{4}+\frac{131}{27}a^{3}-\frac{26}{9}a^{2}+\frac{34}{9}a-\frac{20}{27}$, $\frac{83}{27}a^{19}-\frac{26}{27}a^{18}+\frac{49}{27}a^{17}-\frac{499}{27}a^{16}+\frac{38}{27}a^{15}-\frac{88}{27}a^{14}+\frac{1234}{27}a^{13}+\frac{298}{27}a^{12}+\frac{109}{27}a^{11}-\frac{1532}{27}a^{10}-\frac{614}{27}a^{9}-\frac{485}{27}a^{8}+\frac{1171}{27}a^{7}+\frac{568}{27}a^{6}+\frac{533}{27}a^{5}-\frac{628}{27}a^{4}-\frac{211}{27}a^{3}-\frac{131}{27}a^{2}+\frac{136}{27}a+\frac{56}{27}$, $\frac{4}{27}a^{19}-\frac{38}{27}a^{18}+\frac{19}{27}a^{17}-2a^{16}+\frac{220}{27}a^{15}-2a^{14}+\frac{137}{27}a^{13}-\frac{55}{3}a^{12}-\frac{43}{27}a^{11}-\frac{65}{9}a^{10}+\frac{524}{27}a^{9}+4a^{8}+\frac{344}{27}a^{7}-\frac{40}{3}a^{6}-\frac{41}{27}a^{5}-10a^{4}+\frac{199}{27}a^{3}-\frac{56}{27}a^{2}+\frac{40}{27}a-\frac{2}{3}$, $\frac{2}{3}a^{19}-\frac{11}{27}a^{18}+\frac{4}{27}a^{17}-\frac{104}{27}a^{16}+\frac{10}{9}a^{15}+\frac{31}{27}a^{14}+\frac{80}{9}a^{13}+\frac{11}{27}a^{12}-\frac{38}{9}a^{11}-\frac{310}{27}a^{10}-\frac{14}{9}a^{9}+\frac{53}{27}a^{8}+\frac{95}{9}a^{7}+\frac{47}{27}a^{6}-\frac{11}{9}a^{5}-\frac{176}{27}a^{4}-\frac{1}{9}a^{3}+\frac{43}{27}a^{2}+\frac{55}{27}a-\frac{8}{27}$
| 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: | \( 293.866842819 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 293.866842819 \cdot 1}{6\cdot\sqrt{902558723428708305129}}\cr\approx \mathstrut & 0.156336464484 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + 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^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + 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^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + 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^20 + x^18 - 6*x^17 - x^16 - 4*x^15 + 15*x^14 + 7*x^13 + 10*x^12 - 17*x^11 - 11*x^10 - 17*x^9 + 10*x^8 + 7*x^7 + 15*x^6 - 4*x^5 - x^4 - 6*x^3 + 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
$C_2\wr S_5$ (as 20T279):
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 non-solvable group of order 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 5.3.11119.1, 10.0.30042615123.1, 10.2.1112689449.1, 10.4.3338068347.1 |
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 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.4.3338068347.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(11119\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |