Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 1)
gp: K = bnfinit(y^20 - y^19 + 5*y^18 - 3*y^17 + 9*y^16 - y^15 + 7*y^14 + 4*y^13 + 4*y^12 + 5*y^11 + 4*y^10 + 3*y^9 + 6*y^8 + 5*y^6 + 2*y^5 - y^4 + 3*y^3 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 1)
\( x^{20} - x^{19} + 5 x^{18} - 3 x^{17} + 9 x^{16} - x^{15} + 7 x^{14} + 4 x^{13} + 4 x^{12} + 5 x^{11} + \cdots + 1 \)
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: |
\(1607377523338966031377\)
\(\medspace = 28753\cdot 236438047^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.49\) | 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: | $28753^{1/2}236438047^{1/2}\approx 2607355.588597574$ | ||
| Ramified primes: |
\(28753\), \(236438047\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{28753}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{503}a^{19}+\frac{201}{503}a^{18}-\frac{136}{503}a^{17}+\frac{190}{503}a^{16}+\frac{161}{503}a^{15}-\frac{174}{503}a^{14}+\frac{69}{503}a^{13}-\frac{142}{503}a^{12}-\frac{9}{503}a^{11}+\frac{199}{503}a^{10}-\frac{38}{503}a^{9}-\frac{128}{503}a^{8}-\frac{197}{503}a^{7}-\frac{57}{503}a^{6}+\frac{60}{503}a^{5}+\frac{50}{503}a^{4}+\frac{39}{503}a^{3}-\frac{167}{503}a^{2}-\frac{33}{503}a-\frac{127}{503}$
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: |
$\frac{30}{503}a^{19}-\frac{509}{503}a^{18}+\frac{447}{503}a^{17}-\frac{2348}{503}a^{16}+\frac{806}{503}a^{15}-\frac{4214}{503}a^{14}-\frac{948}{503}a^{13}-\frac{4260}{503}a^{12}-\frac{3288}{503}a^{11}-\frac{4090}{503}a^{10}-\frac{3655}{503}a^{9}-\frac{3840}{503}a^{8}-\frac{2389}{503}a^{7}-\frac{3722}{503}a^{6}-\frac{715}{503}a^{5}-\frac{2524}{503}a^{4}-\frac{1345}{503}a^{3}+\frac{20}{503}a^{2}-\frac{990}{503}a-\frac{289}{503}$, $\frac{289}{503}a^{19}-\frac{259}{503}a^{18}+\frac{936}{503}a^{17}-\frac{420}{503}a^{16}+\frac{253}{503}a^{15}+\frac{517}{503}a^{14}-\frac{2191}{503}a^{13}+\frac{208}{503}a^{12}-\frac{3104}{503}a^{11}-\frac{1843}{503}a^{10}-\frac{2934}{503}a^{9}-\frac{2788}{503}a^{8}-\frac{2106}{503}a^{7}-\frac{2389}{503}a^{6}-\frac{2277}{503}a^{5}-\frac{137}{503}a^{4}-\frac{2813}{503}a^{3}-\frac{478}{503}a^{2}+\frac{20}{503}a-\frac{990}{503}$, $\frac{776}{503}a^{19}-\frac{960}{503}a^{18}+\frac{3615}{503}a^{17}-\frac{2957}{503}a^{16}+\frac{5725}{503}a^{15}-\frac{2232}{503}a^{14}+\frac{3244}{503}a^{13}-\frac{35}{503}a^{12}+\frac{1064}{503}a^{11}+\frac{3}{503}a^{10}+\frac{692}{503}a^{9}-\frac{740}{503}a^{8}+\frac{2555}{503}a^{7}-\frac{2483}{503}a^{6}+\frac{1793}{503}a^{5}+\frac{69}{503}a^{4}-\frac{1928}{503}a^{3}+\frac{685}{503}a^{2}+\frac{45}{503}a-\frac{467}{503}$, $\frac{442}{503}a^{19}-\frac{692}{503}a^{18}+\frac{2260}{503}a^{17}-\frac{2033}{503}a^{16}+\frac{3760}{503}a^{15}-\frac{955}{503}a^{14}+\frac{1827}{503}a^{13}+\frac{1620}{503}a^{12}+\frac{46}{503}a^{11}+\frac{1442}{503}a^{10}-\frac{197}{503}a^{9}-\frac{240}{503}a^{8}+\frac{951}{503}a^{7}-\frac{2056}{503}a^{6}+\frac{867}{503}a^{5}-\frac{32}{503}a^{4}-\frac{1876}{503}a^{3}+\frac{1133}{503}a^{2}-\frac{502}{503}a-\frac{804}{503}$, $\frac{230}{503}a^{19}-\frac{46}{503}a^{18}+\frac{912}{503}a^{17}+\frac{442}{503}a^{16}+\frac{1317}{503}a^{15}+\frac{2232}{503}a^{14}+\frac{1283}{503}a^{13}+\frac{3556}{503}a^{12}+\frac{1954}{503}a^{11}+\frac{3518}{503}a^{10}+\frac{2326}{503}a^{9}+\frac{2752}{503}a^{8}+\frac{2475}{503}a^{7}+\frac{1980}{503}a^{6}+\frac{1225}{503}a^{5}+\frac{2446}{503}a^{4}+\frac{419}{503}a^{3}+\frac{824}{503}a^{2}+\frac{961}{503}a-\frac{36}{503}$, $\frac{619}{503}a^{19}-\frac{828}{503}a^{18}+\frac{2835}{503}a^{17}-\frac{2607}{503}a^{16}+\frac{4592}{503}a^{15}-\frac{2076}{503}a^{14}+\frac{2974}{503}a^{13}+\frac{127}{503}a^{12}+\frac{1471}{503}a^{11}+\frac{952}{503}a^{10}+\frac{1125}{503}a^{9}+\frac{242}{503}a^{8}+\frac{2298}{503}a^{7}-\frac{1582}{503}a^{6}+\frac{1427}{503}a^{5}+\frac{267}{503}a^{4}-\frac{1512}{503}a^{3}+\frac{748}{503}a^{2}+\frac{196}{503}a-\frac{145}{503}$, $\frac{414}{503}a^{19}-\frac{284}{503}a^{18}+\frac{2044}{503}a^{17}-\frac{311}{503}a^{16}+\frac{3779}{503}a^{15}+\frac{1905}{503}a^{14}+\frac{3919}{503}a^{13}+\frac{4590}{503}a^{12}+\frac{4322}{503}a^{11}+\frac{4924}{503}a^{10}+\frac{4388}{503}a^{9}+\frac{3847}{503}a^{8}+\frac{4455}{503}a^{7}+\frac{2055}{503}a^{6}+\frac{3211}{503}a^{5}+\frac{2592}{503}a^{4}+\frac{553}{503}a^{3}+\frac{1282}{503}a^{2}+\frac{925}{503}a-\frac{266}{503}$, $\frac{413}{503}a^{19}-\frac{485}{503}a^{18}+\frac{1677}{503}a^{17}-\frac{1507}{503}a^{16}+\frac{2109}{503}a^{15}-\frac{1442}{503}a^{14}+\frac{329}{503}a^{13}-\frac{1304}{503}a^{12}-\frac{699}{503}a^{11}-\frac{1814}{503}a^{10}-\frac{604}{503}a^{9}-\frac{2061}{503}a^{8}+\frac{628}{503}a^{7}-\frac{2415}{503}a^{6}+\frac{133}{503}a^{5}-\frac{476}{503}a^{4}-\frac{1498}{503}a^{3}-\frac{60}{503}a^{2}+\frac{455}{503}a-\frac{139}{503}$, $\frac{198}{503}a^{19}-\frac{442}{503}a^{18}+\frac{737}{503}a^{17}-\frac{1111}{503}a^{16}+\frac{189}{503}a^{15}-\frac{248}{503}a^{14}-\frac{1931}{503}a^{13}+\frac{555}{503}a^{12}-\frac{1782}{503}a^{11}-\frac{335}{503}a^{10}-\frac{482}{503}a^{9}-\frac{697}{503}a^{8}+\frac{731}{503}a^{7}-\frac{1226}{503}a^{6}+\frac{311}{503}a^{5}+\frac{846}{503}a^{4}-\frac{1332}{503}a^{3}+\frac{635}{503}a^{2}+\frac{508}{503}a-\frac{499}{503}$
| 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: | \( 140.247818215 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140.247818215 \cdot 1}{2\cdot\sqrt{1607377523338966031377}}\cr\approx \mathstrut & 0.167728111123 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 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^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 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^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 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^19 + 5*x^18 - 3*x^17 + 9*x^16 - x^15 + 7*x^14 + 4*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 3*x^9 + 6*x^8 + 5*x^6 + 2*x^5 - x^4 + 3*x^3 + 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
$C_2^{10}.S_{10}$ (as 20T1110):
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 3715891200 |
| The 481 conjugacy class representatives for $C_2^{10}.S_{10}$ |
| Character table for $C_2^{10}.S_{10}$ |
Intermediate fields
| 10.0.236438047.1 |
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 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | $20$ | $16{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }^{2}$ | $16{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(28753\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(236438047\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |