Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*x^2 - 3*x + 1)
gp: K = bnfinit(y^20 - 2*y^19 + 2*y^18 - 4*y^17 + 6*y^16 - 4*y^15 + 4*y^14 - 7*y^13 + 5*y^12 - 6*y^11 + 13*y^10 - 14*y^9 + 16*y^8 - 22*y^7 + 21*y^6 - 17*y^5 + 16*y^4 - 11*y^3 + 6*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*x^2 - 3*x + 1)
\( x^{20} - 2 x^{19} + 2 x^{18} - 4 x^{17} + 6 x^{16} - 4 x^{15} + 4 x^{14} - 7 x^{13} + 5 x^{12} + \cdots + 1 \)
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: |
\(3407822241034569802929\)
\(\medspace = 3^{10}\cdot 467^{2}\cdot 514417^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.93\) | 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: | $3^{1/2}467^{1/2}514417^{1/2}\approx 26845.823082930423$ | ||
| Ramified primes: |
\(3\), \(467\), \(514417\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6197}a^{19}+\frac{117}{6197}a^{18}+\frac{1531}{6197}a^{17}+\frac{2472}{6197}a^{16}+\frac{2915}{6197}a^{15}-\frac{151}{6197}a^{14}+\frac{626}{6197}a^{13}+\frac{123}{6197}a^{12}+\frac{2248}{6197}a^{11}+\frac{1035}{6197}a^{10}-\frac{762}{6197}a^{9}+\frac{2263}{6197}a^{8}+\frac{2842}{6197}a^{7}-\frac{2659}{6197}a^{6}-\frac{353}{6197}a^{5}+\frac{1355}{6197}a^{4}+\frac{139}{6197}a^{3}-\frac{2061}{6197}a^{2}+\frac{2627}{6197}a+\frac{2760}{6197}$
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 $1$ (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: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{5521}{6197} a^{19} + \frac{10925}{6197} a^{18} - \frac{12337}{6197} a^{17} + \frac{22670}{6197} a^{16} - \frac{31091}{6197} a^{15} + \frac{21864}{6197} a^{14} - \frac{23008}{6197} a^{13} + \frac{33572}{6197} a^{12} - \frac{23405}{6197} a^{11} + \frac{36581}{6197} a^{10} - \frac{68928}{6197} a^{9} + \frac{73493}{6197} a^{8} - \frac{92833}{6197} a^{7} + \frac{117389}{6197} a^{6} - \frac{108491}{6197} a^{5} + \frac{91779}{6197} a^{4} - \frac{79552}{6197} a^{3} + \frac{44468}{6197} a^{2} - \frac{27475}{6197} a + \frac{12857}{6197} \)
(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: |
$\frac{1127}{6197}a^{19}-\frac{4475}{6197}a^{18}+\frac{2671}{6197}a^{17}-\frac{2706}{6197}a^{16}+\frac{13189}{6197}a^{15}-\frac{9055}{6197}a^{14}-\frac{956}{6197}a^{13}-\frac{16304}{6197}a^{12}+\frac{17514}{6197}a^{11}+\frac{1409}{6197}a^{10}+\frac{21200}{6197}a^{9}-\frac{33748}{6197}a^{8}+\frac{11479}{6197}a^{7}-\frac{34527}{6197}a^{6}+\frac{48353}{6197}a^{5}-\frac{22165}{6197}a^{4}+\frac{26516}{6197}a^{3}-\frac{29857}{6197}a^{2}+\frac{10857}{6197}a-\frac{6571}{6197}$, $a$, $\frac{432}{6197}a^{19}+\frac{968}{6197}a^{18}-\frac{1687}{6197}a^{17}-\frac{4177}{6197}a^{16}+\frac{1289}{6197}a^{15}+\frac{2935}{6197}a^{14}+\frac{10158}{6197}a^{13}-\frac{8834}{6197}a^{12}-\frac{7990}{6197}a^{11}-\frac{5261}{6197}a^{10}+\frac{11651}{6197}a^{9}+\frac{10884}{6197}a^{8}+\frac{6935}{6197}a^{7}-\frac{20834}{6197}a^{6}+\frac{2429}{6197}a^{5}-\frac{15749}{6197}a^{4}+\frac{22866}{6197}a^{3}-\frac{10378}{6197}a^{2}+\frac{7010}{6197}a-\frac{9898}{6197}$, $\frac{3887}{6197}a^{19}-\frac{3799}{6197}a^{18}+\frac{1877}{6197}a^{17}-\frac{9080}{6197}a^{16}+\frac{8686}{6197}a^{15}+\frac{1778}{6197}a^{14}+\frac{4038}{6197}a^{13}-\frac{11462}{6197}a^{12}+\frac{206}{6197}a^{11}-\frac{11202}{6197}a^{10}+\frac{25060}{6197}a^{9}-\frac{15853}{6197}a^{8}+\frac{22391}{6197}a^{7}-\frac{23725}{6197}a^{6}+\frac{16017}{6197}a^{5}-\frac{6762}{6197}a^{4}+\frac{7351}{6197}a^{3}+\frac{1614}{6197}a^{2}-\frac{1507}{6197}a+\frac{7310}{6197}$, $\frac{1795}{6197}a^{19}+\frac{5514}{6197}a^{18}-\frac{9520}{6197}a^{17}+\frac{188}{6197}a^{16}-\frac{10237}{6197}a^{15}+\frac{20214}{6197}a^{14}+\frac{2013}{6197}a^{13}-\frac{2307}{6197}a^{12}-\frac{23875}{6197}a^{11}+\frac{4922}{6197}a^{10}-\frac{10647}{6197}a^{9}+\frac{46429}{6197}a^{8}-\frac{23529}{6197}a^{7}+\frac{29770}{6197}a^{6}-\frac{51117}{6197}a^{5}+\frac{27789}{6197}a^{4}-\frac{10769}{6197}a^{3}+\frac{18705}{6197}a^{2}-\frac{452}{6197}a+\frac{2797}{6197}$, $\frac{2816}{6197}a^{19}-\frac{11363}{6197}a^{18}+\frac{10578}{6197}a^{17}-\frac{10473}{6197}a^{16}+\frac{28600}{6197}a^{15}-\frac{22411}{6197}a^{14}+\frac{2868}{6197}a^{13}-\frac{25452}{6197}a^{12}+\frac{34216}{6197}a^{11}-\frac{10424}{6197}a^{10}+\frac{47946}{6197}a^{9}-\frac{78469}{6197}a^{8}+\frac{52321}{6197}a^{7}-\frac{82329}{6197}a^{6}+\frac{96624}{6197}a^{5}-\frac{57445}{6197}a^{4}+\frac{44392}{6197}a^{3}-\frac{40566}{6197}a^{2}+\frac{10808}{6197}a+\frac{1122}{6197}$, $\frac{5768}{6197}a^{19}-\frac{6814}{6197}a^{18}+\frac{6280}{6197}a^{17}-\frac{19392}{6197}a^{16}+\frac{19850}{6197}a^{15}-\frac{9585}{6197}a^{14}+\frac{22705}{6197}a^{13}-\frac{27979}{6197}a^{12}+\frac{8537}{6197}a^{11}-\frac{35013}{6197}a^{10}+\frac{54230}{6197}a^{9}-\frac{35080}{6197}a^{8}+\frac{69758}{6197}a^{7}-\frac{86295}{6197}a^{6}+\frac{64679}{6197}a^{5}-\frac{66944}{6197}a^{4}+\frac{64309}{6197}a^{3}-\frac{32987}{6197}a^{2}+\frac{25659}{6197}a-\frac{6610}{6197}$, $\frac{8412}{6197}a^{19}-\frac{13513}{6197}a^{18}+\frac{13800}{6197}a^{17}-\frac{33653}{6197}a^{16}+\frac{42830}{6197}a^{15}-\frac{24615}{6197}a^{14}+\frac{35644}{6197}a^{13}-\frac{49799}{6197}a^{12}+\frac{21720}{6197}a^{11}-\frac{49941}{6197}a^{10}+\frac{96906}{6197}a^{9}-\frac{87586}{6197}a^{8}+\frac{129015}{6197}a^{7}-\frac{169854}{6197}a^{6}+\frac{135261}{6197}a^{5}-\frac{121963}{6197}a^{4}+\frac{115778}{6197}a^{3}-\frac{53699}{6197}a^{2}+\frac{37004}{6197}a-\frac{15433}{6197}$, $\frac{2066}{6197}a^{19}+\frac{39}{6197}a^{18}-\frac{3621}{6197}a^{17}+\frac{824}{6197}a^{16}-\frac{7291}{6197}a^{15}+\frac{16475}{6197}a^{14}-\frac{8054}{6197}a^{13}+\frac{6238}{6197}a^{12}-\frac{15776}{6197}a^{11}+\frac{345}{6197}a^{10}-\frac{254}{6197}a^{9}+\frac{27608}{6197}a^{8}-\frac{21775}{6197}a^{7}+\frac{28033}{6197}a^{6}-\frac{53825}{6197}a^{5}+\frac{47962}{6197}a^{4}-\frac{35070}{6197}a^{3}+\frac{36495}{6197}a^{2}-\frac{19781}{6197}a+\frac{7117}{6197}$
| 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: | \( 658.849285992 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 658.849285992 \cdot 1}{6\cdot\sqrt{3407822241034569802929}}\cr\approx \mathstrut & 0.180382767736 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*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 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*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 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*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 - 2*x^19 + 2*x^18 - 4*x^17 + 6*x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 5*x^12 - 6*x^11 + 13*x^10 - 14*x^9 + 16*x^8 - 22*x^7 + 21*x^6 - 17*x^5 + 16*x^4 - 11*x^3 + 6*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
$C_2\times S_{10}$ (as 20T1021):
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 7257600 |
| The 84 conjugacy class representatives for $C_2\times S_{10}$ are not computed |
| Character table for $C_2\times S_{10}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 10.0.240232739.1 |
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 sibling: | data not computed |
| Degree 40 sibling: | 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(467\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
|
\(514417\)
| 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |