Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*x - 1)
gp: K = bnfinit(y^21 - y^20 - 20*y^19 + 19*y^18 + 171*y^17 - 153*y^16 - 816*y^15 + 680*y^14 + 2380*y^13 - 1820*y^12 - 4368*y^11 + 3003*y^10 + 5005*y^9 - 3003*y^8 - 3432*y^7 + 1716*y^6 + 1287*y^5 - 495*y^4 - 220*y^3 + 55*y^2 + 11*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*x - 1)
\( x^{21} - x^{20} - 20 x^{19} + 19 x^{18} + 171 x^{17} - 153 x^{16} - 816 x^{15} + 680 x^{14} + 2380 x^{13} + \cdots - 1 \)
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: | $[21, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(467056167777397914441056671494001\)
\(\medspace = 43^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.95\) | 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: | $43^{20/21}\approx 35.948787215302666$ | ||
| Ramified primes: |
\(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $21$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{43}(1,·)$, $\chi_{43}(4,·)$, $\chi_{43}(6,·)$, $\chi_{43}(9,·)$, $\chi_{43}(10,·)$, $\chi_{43}(11,·)$, $\chi_{43}(13,·)$, $\chi_{43}(14,·)$, $\chi_{43}(15,·)$, $\chi_{43}(16,·)$, $\chi_{43}(17,·)$, $\chi_{43}(21,·)$, $\chi_{43}(23,·)$, $\chi_{43}(24,·)$, $\chi_{43}(25,·)$, $\chi_{43}(31,·)$, $\chi_{43}(35,·)$, $\chi_{43}(36,·)$, $\chi_{43}(38,·)$, $\chi_{43}(40,·)$, $\chi_{43}(41,·)$$\rbrace$ | ||
| 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: | $20$ | 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: |
$a^{10}-a^{9}-9a^{8}+8a^{7}+28a^{6}-21a^{5}-35a^{4}+20a^{3}+15a^{2}-5a-1$, $a^{17}-a^{16}-16a^{15}+15a^{14}+105a^{13}-91a^{12}-364a^{11}+287a^{10}+714a^{9}-504a^{8}-784a^{7}+490a^{6}+441a^{5}-245a^{4}-99a^{3}+50a^{2}+2a-1$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{19}-a^{18}-18a^{17}+18a^{16}+135a^{15}-135a^{14}-545a^{13}+545a^{12}+1275a^{11}-1275a^{10}-1728a^{9}+1728a^{8}+1275a^{7}-1275a^{6}-441a^{5}+441a^{4}+54a^{3}-54a^{2}-a+1$, $a-1$, $a^{7}-7a^{5}+14a^{3}-7a-1$, $a^{20}-a^{19}-19a^{18}+18a^{17}+153a^{16}-136a^{15}-681a^{14}+561a^{13}+1833a^{12}-1378a^{11}-3068a^{10}+2067a^{9}+3159a^{8}-1872a^{7}-1898a^{6}+975a^{5}+585a^{4}-260a^{3}-64a^{2}+25a-1$, $a^{20}-20a^{18}-a^{17}+170a^{16}+16a^{15}-799a^{14}-104a^{13}+2262a^{12}+351a^{11}-3939a^{10}-650a^{9}+4135a^{8}+637a^{7}-2465a^{6}-287a^{5}+748a^{4}+43a^{3}-94a^{2}-2a+3$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{20}-19a^{18}+153a^{16}-680a^{14}+1820a^{12}-3003a^{10}+3003a^{8}-1716a^{6}+495a^{4}-55a^{2}+1$, $a^{3}-3a-1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{5}-5a^{3}+5a$, $a^{16}-a^{15}-15a^{14}+14a^{13}+91a^{12}-78a^{11}-286a^{10}+220a^{9}+495a^{8}-330a^{7}-462a^{6}+253a^{5}+209a^{4}-88a^{3}-33a^{2}+11a$, $a^{19}-19a^{17}+152a^{15}-a^{14}-665a^{13}+14a^{12}+1729a^{11}-77a^{10}-2717a^{9}+210a^{8}+2508a^{7}-294a^{6}-1254a^{5}+196a^{4}+285a^{3}-49a^{2}-19a+2$, $a^{12}-12a^{10}-a^{9}+54a^{8}+9a^{7}-111a^{6}-27a^{5}+99a^{4}+29a^{3}-27a^{2}-6a+1$, $a^{3}-3a$, $a^{8}-8a^{6}+21a^{4}-20a^{2}+5$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+661a^{8}-680a^{6}+356a^{4}-80a^{2}+5$, $a^{20}-a^{19}-19a^{18}+19a^{17}+152a^{16}-153a^{15}-664a^{14}+679a^{13}+1715a^{12}-1807a^{11}-2639a^{10}+2937a^{9}+2288a^{8}-2837a^{7}-924a^{6}+1499a^{5}+34a^{4}-354a^{3}+60a^{2}+17a-3$
| 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: | \( 2620717953.48 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 2620717953.48 \cdot 1}{2\cdot\sqrt{467056167777397914441056671494001}}\cr\approx \mathstrut & 0.127155643882 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*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 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*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 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*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 - x^20 - 20*x^19 + 19*x^18 + 171*x^17 - 153*x^16 - 816*x^15 + 680*x^14 + 2380*x^13 - 1820*x^12 - 4368*x^11 + 3003*x^10 + 5005*x^9 - 3003*x^8 - 3432*x^7 + 1716*x^6 + 1287*x^5 - 495*x^4 - 220*x^3 + 55*x^2 + 11*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
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 cyclic group of order 21 |
| The 21 conjugacy class representatives for $C_{21}$ |
| Character table for $C_{21}$ is not computed |
Intermediate fields
| 3.3.1849.1, 7.7.6321363049.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/padicField/7.3.0.1}{3} }^{7}$ | ${\href{/padicField/11.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ | $21$ | $21$ | $21$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }^{3}$ | R | ${\href{/padicField/47.7.0.1}{7} }^{3}$ | $21$ | ${\href{/padicField/59.7.0.1}{7} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(43\)
| 43.21.20.1 | $x^{21} + 43$ | $21$ | $1$ | $20$ | $C_{21}$ | $[\ ]_{21}$ |