Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 5*x - 5)
gp: K = bnfinit(y^27 - 5*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 5*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 5*x - 5)
\( x^{27} - 5x - 5 \)
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: |
\(-614890292796495532086888341869635692124068737030029296875\)
\(\medspace = -\,5^{26}\cdot 7\cdot 769\cdot 76\!\cdots\!81\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(126.85\) | 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: | $5^{26/27}7^{1/2}769^{1/2}76657233947984763773781582607681381^{1/2}\approx 9.569097694945581e+19$ | ||
| Ramified primes: |
\(5\), \(7\), \(769\), \(76657\!\cdots\!81381\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-41264\!\cdots\!73923}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a+1$, $2a^{26}-a^{24}-4a^{23}+5a^{22}+3a^{21}-5a^{20}-3a^{19}+9a^{18}+4a^{17}-16a^{16}-2a^{15}+15a^{14}+3a^{13}-20a^{12}+5a^{11}+21a^{10}-5a^{9}-21a^{8}+4a^{7}+16a^{6}-11a^{5}-9a^{4}+11a^{3}+14a^{2}-11a-14$, $34a^{26}+77a^{25}+76a^{24}+48a^{23}+34a^{22}+95a^{21}+52a^{20}-15a^{19}+3a^{18}+14a^{17}-41a^{16}-151a^{15}-105a^{14}-87a^{13}-210a^{12}-246a^{11}-176a^{10}-116a^{9}-243a^{8}-216a^{7}-a^{6}+13a^{5}-39a^{4}+79a^{3}+386a^{2}+391a+106$, $41a^{26}+48a^{25}+45a^{24}+46a^{23}+37a^{22}+30a^{21}+16a^{20}+4a^{19}-15a^{18}-33a^{17}-57a^{16}-78a^{15}-100a^{14}-120a^{13}-138a^{12}-150a^{11}-155a^{10}-144a^{9}-124a^{8}-86a^{7}-38a^{6}+26a^{5}+102a^{4}+186a^{3}+267a^{2}+343a+196$, $110a^{26}-91a^{25}+49a^{24}+4a^{23}-71a^{22}+122a^{21}-121a^{20}+121a^{19}-54a^{18}-21a^{17}+87a^{16}-144a^{15}+173a^{14}-121a^{13}+72a^{12}+19a^{11}-143a^{10}+167a^{9}-208a^{8}+170a^{7}-62a^{6}-59a^{5}+124a^{4}-274a^{3}+224a^{2}-184a-439$, $5a^{26}-3a^{25}+25a^{24}-a^{23}-18a^{22}+6a^{21}-21a^{20}-24a^{19}+27a^{18}+3a^{17}+8a^{16}+48a^{15}-13a^{14}-27a^{13}+6a^{12}-57a^{11}-27a^{10}+48a^{9}-a^{8}+41a^{7}+79a^{6}-39a^{5}-34a^{4}-13a^{3}-115a^{2}-18a+56$, $35a^{26}-8a^{25}-51a^{24}+4a^{23}+45a^{22}+28a^{21}-63a^{20}-35a^{19}+52a^{18}+59a^{17}-32a^{16}-100a^{15}+34a^{14}+93a^{13}+22a^{12}-127a^{11}-50a^{10}+132a^{9}+90a^{8}-106a^{7}-163a^{6}+92a^{5}+201a^{4}-34a^{3}-248a^{2}-35a+104$, $19a^{26}-47a^{25}+57a^{24}-65a^{23}+75a^{22}-83a^{21}+64a^{20}-50a^{19}+42a^{18}-22a^{17}-18a^{16}+35a^{15}-45a^{14}+68a^{13}-100a^{12}+82a^{11}-76a^{10}+79a^{9}-77a^{8}+18a^{7}-3a^{5}+36a^{4}-100a^{3}+87a^{2}-76a+9$, $304a^{26}-13a^{25}-345a^{24}+396a^{23}-39a^{22}-398a^{21}+438a^{20}-102a^{19}-399a^{18}+577a^{17}-281a^{16}-423a^{15}+745a^{14}-342a^{13}-467a^{12}+829a^{11}-455a^{10}-337a^{9}+1018a^{8}-777a^{7}-258a^{6}+1304a^{5}-921a^{4}-282a^{3}+1370a^{2}-1140a-1479$, $5a^{26}+93a^{25}-68a^{24}-48a^{23}-83a^{22}-24a^{21}-153a^{20}-73a^{19}+12a^{18}+94a^{17}+21a^{16}+137a^{15}+231a^{14}+167a^{13}-14a^{12}+5a^{11}-4a^{10}-228a^{9}-384a^{8}-224a^{7}-80a^{6}-176a^{5}-38a^{4}+381a^{3}+551a^{2}+309a+319$, $2a^{26}+3a^{25}-25a^{24}-33a^{23}-12a^{22}-5a^{21}-37a^{20}-58a^{19}-35a^{18}+6a^{17}-13a^{16}-56a^{15}-55a^{14}-17a^{13}-27a^{12}-100a^{11}-122a^{10}-44a^{9}+17a^{8}-48a^{7}-141a^{6}-110a^{5}-23a^{4}-66a^{3}-230a^{2}-259a-91$, $944a^{26}-711a^{25}+1196a^{24}-752a^{23}+622a^{22}-1223a^{21}+440a^{20}-612a^{19}+1172a^{18}-67a^{17}+704a^{16}-981a^{15}-266a^{14}-853a^{13}+661a^{12}+492a^{11}+974a^{10}-302a^{9}-640a^{8}-1061a^{7}-17a^{6}+825a^{5}+1241a^{4}+365a^{3}-1071a^{2}-1697a-5736$, $16a^{26}+9a^{25}+13a^{24}+a^{23}-a^{22}+4a^{21}-10a^{20}-11a^{19}-17a^{18}-36a^{17}-35a^{16}-35a^{15}-45a^{14}-26a^{13}-34a^{12}-43a^{11}-21a^{10}-21a^{9}+36a^{7}+47a^{6}+65a^{5}+74a^{4}+83a^{3}+109a^{2}+128a+69$
| 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: | \( 898727205894750200 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 898727205894750200 \cdot 1}{2\cdot\sqrt{614890292796495532086888341869635692124068737030029296875}}\cr\approx \mathstrut & 0.862119282604304 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 5*x - 5)
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 - 5, 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 - 5);
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 - 5);
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 | $23{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | R | $15{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | $25{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $23{,}\,{\href{/padicField/31.4.0.1}{4} }$ | $21{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $27$ | $27$ | $1$ | $26$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.12.0.1 | $x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(769\)
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
|
\(766\!\cdots\!381\)
| $\Q_{76\!\cdots\!81}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |