Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - x^19 + 72*x^18 - 100*x^17 - 299*x^16 + 733*x^15 + 386*x^14 - 2195*x^13 + 634*x^12 + 3126*x^11 - 2357*x^10 - 2090*x^9 + 2868*x^8 + 87*x^7 - 1535*x^6 + 673*x^5 + 170*x^4 - 243*x^3 + 89*x^2 - 15*x + 1)
gp: K = bnfinit(y^21 - 6*y^20 - y^19 + 72*y^18 - 100*y^17 - 299*y^16 + 733*y^15 + 386*y^14 - 2195*y^13 + 634*y^12 + 3126*y^11 - 2357*y^10 - 2090*y^9 + 2868*y^8 + 87*y^7 - 1535*y^6 + 673*y^5 + 170*y^4 - 243*y^3 + 89*y^2 - 15*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 6*x^20 - x^19 + 72*x^18 - 100*x^17 - 299*x^16 + 733*x^15 + 386*x^14 - 2195*x^13 + 634*x^12 + 3126*x^11 - 2357*x^10 - 2090*x^9 + 2868*x^8 + 87*x^7 - 1535*x^6 + 673*x^5 + 170*x^4 - 243*x^3 + 89*x^2 - 15*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 6*x^20 - x^19 + 72*x^18 - 100*x^17 - 299*x^16 + 733*x^15 + 386*x^14 - 2195*x^13 + 634*x^12 + 3126*x^11 - 2357*x^10 - 2090*x^9 + 2868*x^8 + 87*x^7 - 1535*x^6 + 673*x^5 + 170*x^4 - 243*x^3 + 89*x^2 - 15*x + 1)
\( x^{21} - 6 x^{20} - x^{19} + 72 x^{18} - 100 x^{17} - 299 x^{16} + 733 x^{15} + 386 x^{14} - 2195 x^{13} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-10367263422675536780303271503\)
\(\medspace = -\,11^{4}\cdot 103^{4}\cdot 184607^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.58\) | 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: | $11^{2/3}103^{2/3}184607^{1/2}\approx 46695.75004699004$ | ||
| Ramified primes: |
\(11\), \(103\), \(184607\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-184607}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: | $11$ | 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: |
$1919a^{20}-10660a^{19}-6663a^{18}+135203a^{17}-131731a^{16}-632407a^{15}+1125190a^{14}+1241484a^{13}-3659720a^{12}-412053a^{11}+5815447a^{10}-1935020a^{9}-4871890a^{8}+3335549a^{7}+1651402a^{6}-2210746a^{5}+307627a^{4}+463138a^{3}-260205a^{2}+54990a-4312$, $437a^{20}-2669a^{19}-282a^{18}+32167a^{17}-46393a^{16}-134615a^{15}+339760a^{14}+177824a^{13}-1034724a^{12}+277590a^{11}+1536634a^{10}-1076395a^{9}-1141307a^{8}+1353540a^{7}+214213a^{6}-777263a^{5}+228752a^{4}+133824a^{3}-104655a^{2}+25852a-2281$, $437a^{20}-2460a^{19}-1327a^{18}+30861a^{17}-32391a^{16}-141188a^{15}+267212a^{14}+260108a^{13}-853713a^{12}-20628a^{11}+1325876a^{10}-559604a^{9}-1064688a^{8}+864719a^{7}+305792a^{6}-545826a^{5}+116480a^{4}+104613a^{3}-70309a^{2}+16654a-1451$, $a$, $a-1$, $162a^{20}-889a^{19}-608a^{18}+11303a^{17}-10459a^{16}-53143a^{15}+91093a^{14}+105902a^{13}-296900a^{12}-41595a^{11}+469630a^{10}-149007a^{9}-388603a^{8}+265677a^{7}+125741a^{6}-176762a^{5}+29647a^{4}+35879a^{3}-22072a^{2}+5027a-424$, $436a^{20}-2454a^{19}-1326a^{18}+30789a^{17}-32291a^{16}-140889a^{15}+266479a^{14}+259722a^{13}-851518a^{12}-21262a^{11}+1322750a^{10}-557247a^{9}-1062598a^{8}+861851a^{7}+305705a^{6}-544291a^{5}+115807a^{4}+104443a^{3}-70065a^{2}+16564a-1438$, $1066a^{20}-5922a^{19}-3699a^{18}+75106a^{17}-73205a^{16}-351269a^{15}+625173a^{14}+689385a^{13}-2033216a^{12}-228072a^{11}+3230510a^{10}-1076197a^{9}-2705837a^{8}+1853968a^{7}+916570a^{6}-1228421a^{5}+171392a^{4}+257196a^{3}-144647a^{2}+30606a-2406$, $4311a^{20}-23948a^{19}-14966a^{18}+303735a^{17}-295963a^{16}-1420686a^{15}+2527889a^{14}+2788836a^{13}-8221947a^{12}-925137a^{11}+13064908a^{10}-4347931a^{9}-10945004a^{8}+7494154a^{7}+3709834a^{6}-4966842a^{5}+691233a^{4}+1040500a^{3}-584602a^{2}+123550a-9687$, $3428a^{20}-18739a^{19}-13431a^{18}+239678a^{17}-214906a^{16}-1139995a^{15}+1904857a^{14}+2341267a^{13}-6278418a^{12}-1180098a^{11}+10096637a^{10}-2690693a^{9}-8617241a^{8}+5237411a^{7}+3108127a^{6}-3611261a^{5}+373588a^{4}+787621a^{3}-412448a^{2}+83630a-6333$, $11972a^{20}-65530a^{19}-46440a^{18}+837400a^{17}-756540a^{16}-3976044a^{15}+6681345a^{14}+8129037a^{13}-21987208a^{12}-3962117a^{11}+35296031a^{10}-9659381a^{9}-30033439a^{8}+18525454a^{7}+10725936a^{6}-12708856a^{5}+1397715a^{4}+2751321a^{3}-1463540a^{2}+300529a-23073$
| 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: | \( 361691.21971 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 361691.21971 \cdot 1}{2\cdot\sqrt{10367263422675536780303271503}}\cr\approx \mathstrut & 0.21686262273 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - x^19 + 72*x^18 - 100*x^17 - 299*x^16 + 733*x^15 + 386*x^14 - 2195*x^13 + 634*x^12 + 3126*x^11 - 2357*x^10 - 2090*x^9 + 2868*x^8 + 87*x^7 - 1535*x^6 + 673*x^5 + 170*x^4 - 243*x^3 + 89*x^2 - 15*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^21 - 6*x^20 - x^19 + 72*x^18 - 100*x^17 - 299*x^16 + 733*x^15 + 386*x^14 - 2195*x^13 + 634*x^12 + 3126*x^11 - 2357*x^10 - 2090*x^9 + 2868*x^8 + 87*x^7 - 1535*x^6 + 673*x^5 + 170*x^4 - 243*x^3 + 89*x^2 - 15*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^21 - 6*x^20 - x^19 + 72*x^18 - 100*x^17 - 299*x^16 + 733*x^15 + 386*x^14 - 2195*x^13 + 634*x^12 + 3126*x^11 - 2357*x^10 - 2090*x^9 + 2868*x^8 + 87*x^7 - 1535*x^6 + 673*x^5 + 170*x^4 - 243*x^3 + 89*x^2 - 15*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^21 - 6*x^20 - x^19 + 72*x^18 - 100*x^17 - 299*x^16 + 733*x^15 + 386*x^14 - 2195*x^13 + 634*x^12 + 3126*x^11 - 2357*x^10 - 2090*x^9 + 2868*x^8 + 87*x^7 - 1535*x^6 + 673*x^5 + 170*x^4 - 243*x^3 + 89*x^2 - 15*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
$C_3^6.S_7$ (as 21T130):
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 3674160 |
| The 143 conjugacy class representatives for $C_3^6.S_7$ |
| Character table for $C_3^6.S_7$ |
Intermediate fields
| 7.1.184607.1 |
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 42 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 | ${\href{/padicField/2.7.0.1}{7} }^{3}$ | ${\href{/padicField/3.7.0.1}{7} }^{3}$ | $15{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.7.0.1}{7} }^{3}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.9.0.1}{9} }$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.7.0.1}{7} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.6.4.2 | $x^{6} - 110 x^{3} - 16819$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| 11.15.0.1 | $x^{15} + 10 x^{6} + 7 x^{5} + 5 x^{3} + 9$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(103\)
| $\Q_{103}$ | $x + 98$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{103}$ | $x + 98$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{103}$ | $x + 98$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.3.2.3 | $x^{3} + 412$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 103.3.2.3 | $x^{3} + 412$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(184607\)
| $\Q_{184607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{184607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{184607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{184607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{184607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{184607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ |