Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3)
gp: K = bnfinit(y^27 - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 3)
\( x^{27} - 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1127130637840908780976740490797413723399509150616187\)
\(\medspace = -\,3^{107}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(77.77\) | 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: | $3^{661/162}\approx 88.465212890785$ | ||
| Ramified primes: |
\(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$
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: | $13$ | 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: |
$a^{9}-a^{6}+1$, $a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{18}-2$, $a^{24}+a^{21}+a^{12}+2a^{9}+a^{6}+1$, $a^{26}-a^{24}+a^{23}-a^{21}+a^{20}-a^{18}+a^{17}-a^{15}+a^{14}-2a^{12}+2a^{11}-2a^{9}+2a^{8}-2a^{6}+2a^{5}-2a^{3}+2a^{2}-2$, $a^{24}+a^{9}-a^{6}-3a^{3}+1$, $2a^{26}-a^{25}-6a^{24}-4a^{23}-2a^{22}-6a^{21}-4a^{20}+3a^{19}+3a^{18}+2a^{17}+8a^{16}+8a^{15}+a^{14}+2a^{13}+3a^{12}-6a^{11}-10a^{10}-4a^{9}-7a^{8}-11a^{7}+7a^{5}+2a^{4}+7a^{3}+17a^{2}+9a+2$, $2a^{26}-3a^{25}-a^{24}+2a^{23}+a^{22}-3a^{21}-2a^{20}+4a^{19}-3a^{17}-2a^{16}+5a^{15}-4a^{13}+4a^{11}+a^{10}-6a^{9}+3a^{8}+3a^{7}-6a^{5}+3a^{4}+5a^{3}-5a^{2}-3a+2$, $2a^{26}-a^{24}-2a^{23}+2a^{22}+3a^{21}-2a^{20}-2a^{19}+2a^{18}+a^{17}-2a^{16}+a^{15}+a^{14}+a^{13}-5a^{12}-a^{11}+6a^{10}-a^{9}-3a^{8}+4a^{6}-3a^{4}+6a^{2}-3a-5$, $3a^{26}-7a^{25}+4a^{24}-2a^{23}+8a^{22}-8a^{21}+7a^{20}-5a^{19}+4a^{18}-13a^{17}+8a^{16}-4a^{15}+6a^{14}-7a^{13}+11a^{12}-4a^{11}-8a^{9}+3a^{8}-3a^{7}-2a^{6}+2a^{5}+3a^{4}+8a^{3}-8a^{2}+3a-10$, $20a^{26}+18a^{25}+23a^{24}+7a^{23}+8a^{22}-17a^{21}-10a^{20}-37a^{19}-17a^{18}-36a^{17}-15a^{15}+30a^{14}+12a^{13}+51a^{12}+19a^{11}+44a^{10}-3a^{9}+9a^{8}-39a^{7}-28a^{6}-60a^{5}-36a^{4}-40a^{3}-8a^{2}+12a+38$, $11a^{26}-30a^{25}-25a^{24}+22a^{23}+40a^{22}-15a^{21}-47a^{20}+2a^{19}+46a^{18}+20a^{17}-44a^{16}-42a^{15}+39a^{14}+55a^{13}-22a^{12}-63a^{11}-6a^{10}+71a^{9}+33a^{8}-71a^{7}-53a^{6}+52a^{5}+75a^{4}-21a^{3}-99a^{2}-6a+107$, $5a^{26}+15a^{25}-4a^{24}-8a^{23}+9a^{22}-6a^{21}-16a^{20}+13a^{19}+17a^{18}-6a^{17}-9a^{16}-5a^{15}+a^{14}+2a^{13}-11a^{12}+9a^{11}+30a^{10}-19a^{9}-39a^{8}+16a^{7}+21a^{6}-11a^{5}+5a^{4}+13a^{3}-4a^{2}-15a-22$
| 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: | \( 1439757597579506.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 1439757597579506.5 \cdot 1}{2\cdot\sqrt{1127130637840908780976740490797413723399509150616187}}\cr\approx \mathstrut & 1.02009481503433 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 3)
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^27 - 3, 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^27 - 3);
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^27 - 3);
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
$C_{27}:C_{18}$ (as 27T176):
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 solvable group of order 486 |
| The 31 conjugacy class representatives for $C_{27}:C_{18}$ |
| Character table for $C_{27}:C_{18}$ is not computed |
Intermediate fields
| 3.1.243.1, 9.1.2541865828329.2 |
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 | $18{,}\,{\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $27$ | $18{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $27$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $27$ | $18{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $27$ | $27$ | $18{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $27$ | $18{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $27$ | $27$ | $1$ | $107$ |