Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 5*x - 3)
gp: K = bnfinit(y^27 - 5*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 5*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 5*x - 3)
\( x^{27} - 5x - 3 \)
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: | $[3, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(45865537965996467420418831264009202586276600490849383813\)
\(\medspace = 17\cdot 359\cdot 5984861568115119773\cdot 12\!\cdots\!27\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(115.22\) | 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: | $17^{1/2}359^{1/2}5984861568115119773^{1/2}1255709015602496871819754676848927^{1/2}\approx 6.772410055954709e+27$ | ||
| Ramified primes: |
\(17\), \(359\), \(5984861568115119773\), \(12557\!\cdots\!48927\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{45865\!\cdots\!83813}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $14$ | 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: |
$a+1$, $4a^{26}-3a^{25}-2a^{24}-6a^{23}+a^{21}+5a^{20}+a^{19}-3a^{18}-7a^{17}-7a^{16}+a^{15}+4a^{14}+9a^{13}-2a^{12}-7a^{11}-15a^{10}-5a^{9}+4a^{8}+14a^{7}+7a^{6}-8a^{5}-19a^{4}-20a^{3}+a^{2}+12a+1$, $6a^{26}-4a^{25}+2a^{24}-a^{23}+a^{22}-a^{18}+a^{17}+a^{15}+a^{14}-a^{13}+2a^{9}+a^{8}-a^{7}+a^{6}-3a^{5}+a^{4}+a^{3}+a^{2}+a-31$, $7a^{26}-2a^{25}+3a^{24}+6a^{23}+a^{22}-a^{21}-12a^{20}-5a^{19}-5a^{18}+3a^{17}-3a^{16}-4a^{15}+a^{14}+10a^{13}+21a^{12}+9a^{11}+3a^{10}-13a^{9}-5a^{7}-4a^{6}-27a^{5}-26a^{4}-11a^{3}+17a^{2}+31a-19$, $2a^{26}+a^{25}-6a^{24}+a^{23}+3a^{22}+4a^{21}-6a^{20}-a^{19}+2a^{18}+7a^{17}-4a^{16}-5a^{15}-3a^{14}+11a^{13}-9a^{11}-7a^{10}+14a^{9}+5a^{8}-6a^{7}-12a^{6}+9a^{5}+12a^{4}+a^{3}-23a^{2}-2a+7$, $10a^{26}+9a^{25}-2a^{24}-10a^{23}-15a^{22}-22a^{21}-23a^{20}-15a^{19}-6a^{18}+5a^{17}+25a^{16}+30a^{15}+36a^{14}+39a^{13}+18a^{12}+2a^{11}-19a^{10}-46a^{9}-58a^{8}-59a^{7}-53a^{6}-24a^{5}+20a^{4}+46a^{3}+87a^{2}+109a+41$, $7a^{26}-8a^{25}+4a^{24}+3a^{23}-7a^{22}+7a^{21}-6a^{20}+6a^{18}-13a^{17}+9a^{16}-9a^{14}+12a^{13}-9a^{12}+6a^{11}+6a^{10}-15a^{9}+19a^{8}-6a^{7}-10a^{6}+17a^{5}-15a^{4}+10a^{3}-15a-5$, $6a^{26}-a^{25}-8a^{24}+6a^{23}+3a^{22}-9a^{21}+2a^{20}+10a^{19}-10a^{18}-a^{17}+13a^{16}-4a^{15}-8a^{14}+15a^{13}+3a^{12}-14a^{11}+11a^{10}+10a^{9}-15a^{8}-2a^{7}+20a^{6}-15a^{5}-15a^{4}+20a^{3}-4a^{2}-33a-14$, $5a^{26}+a^{25}+2a^{24}+5a^{22}+3a^{21}+5a^{20}+a^{19}+5a^{18}+5a^{17}+10a^{16}+4a^{15}+3a^{14}+8a^{12}+9a^{11}+15a^{10}+6a^{9}+7a^{8}+8a^{7}+16a^{6}+11a^{5}+11a^{4}+2a^{3}+14a^{2}+23a+8$, $48a^{26}-50a^{25}+40a^{24}-21a^{23}-6a^{22}+37a^{21}-58a^{20}+65a^{19}-54a^{18}+34a^{17}-a^{16}-36a^{15}+70a^{14}-82a^{13}+73a^{12}-48a^{11}+11a^{10}+36a^{9}-83a^{8}+107a^{7}-102a^{6}+69a^{5}-29a^{4}-32a^{3}+91a^{2}-139a-103$, $52a^{26}-33a^{25}+17a^{24}-8a^{23}+6a^{22}-6a^{21}+4a^{20}+a^{19}-3a^{17}+a^{16}+4a^{15}-a^{14}-4a^{13}+5a^{12}+3a^{11}-7a^{10}-a^{9}+7a^{8}-a^{7}-6a^{6}-a^{5}+4a^{4}-a^{3}-9a^{2}+3a-248$, $3a^{26}-3a^{25}+a^{24}-8a^{23}-4a^{22}+8a^{21}-a^{19}+3a^{18}-5a^{17}+9a^{16}+15a^{15}-10a^{14}-8a^{13}-2a^{12}-13a^{11}+10a^{10}+10a^{9}-25a^{8}-3a^{7}+17a^{6}+2a^{5}+22a^{4}+6a^{3}-32a^{2}+9a+10$, $a^{26}-10a^{25}+7a^{24}-6a^{23}-3a^{22}+7a^{21}-5a^{20}+4a^{19}+7a^{18}-3a^{17}+7a^{16}-5a^{14}+4a^{13}-13a^{12}+a^{11}-6a^{10}-10a^{9}+16a^{8}-16a^{7}+18a^{6}+14a^{5}-14a^{4}+38a^{3}-18a^{2}-5a+13$, $2a^{25}-4a^{24}+2a^{23}+2a^{22}-2a^{21}+2a^{20}-2a^{19}+5a^{18}-2a^{17}-a^{16}+4a^{15}-a^{14}+a^{13}-6a^{12}+7a^{11}+3a^{10}-3a^{9}+5a^{8}+a^{7}+4a^{6}-13a^{5}+4a^{4}+12a^{3}-5a^{2}+4a+5$
| 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: | \( 276724601545482200 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 276724601545482200 \cdot 1}{2\cdot\sqrt{45865537965996467420418831264009202586276600490849383813}}\cr\approx \mathstrut & 0.618761005473397 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 5*x - 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 - 5*x - 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 - 5*x - 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 - 5*x - 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
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 10888869450418352160768000000 |
| The 3010 conjugacy class representatives for $S_{27}$ are not computed |
| Character table for $S_{27}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | $23{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $27$ | ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $27$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.21.0.1 | $x^{21} + x^{9} + 10 x^{8} + 9 x^{7} + 12 x^{6} + 16 x^{5} + 6 x^{3} + 6 x^{2} + 3 x + 14$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | |
|
\(359\)
| 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(5984861568115119773\)
| $\Q_{5984861568115119773}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(125\!\cdots\!927\)
| $\Q_{12\!\cdots\!27}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |