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: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-45867792227272149237980784744990797413723399509150616187\)
\(\medspace = -\,26784394391927\cdot 906908513152058455369\cdot 1888263188591623325749\)
| 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: | $26784394391927^{1/2}906908513152058455369^{1/2}1888263188591623325749^{1/2}\approx 6.772576483678287e+27$ | ||
| Ramified primes: |
\(26784394391927\), \(906908513152058455369\), \(1888263188591623325749\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-45867\!\cdots\!16187}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{26}-a^{25}+3a^{24}-a^{23}+4a^{21}-2a^{20}-a^{19}+2a^{18}+a^{17}-6a^{16}+7a^{15}+a^{14}-5a^{13}+6a^{12}+a^{11}-4a^{10}-a^{9}+8a^{8}-10a^{7}+5a^{6}+3a^{5}-5a^{4}+6a^{3}-a^{2}+2a+4$, $139a^{26}+83a^{25}+53a^{24}+34a^{23}+19a^{22}+9a^{21}+4a^{20}+3a^{19}+6a^{18}+8a^{17}+3a^{16}-3a^{15}-a^{14}+5a^{13}+8a^{12}+8a^{11}+4a^{10}-3a^{9}-2a^{8}+9a^{7}+13a^{6}+4a^{5}-4a^{4}-5a^{3}-2a^{2}+7a+709$, $53a^{26}-11a^{25}+56a^{24}-32a^{23}+47a^{22}-60a^{21}+34a^{20}-84a^{19}+27a^{18}-91a^{17}+33a^{16}-75a^{15}+50a^{14}-41a^{13}+67a^{12}-a^{11}+66a^{10}+32a^{9}+33a^{8}+51a^{7}-33a^{6}+61a^{5}-120a^{4}+80a^{3}-207a^{2}+128a-10$, $2a^{26}+4a^{25}+7a^{24}+a^{23}-5a^{22}-6a^{21}-2a^{20}-2a^{19}-10a^{18}-11a^{17}-6a^{16}+8a^{15}+4a^{14}-2a^{12}+10a^{11}+20a^{10}+6a^{9}+a^{8}-6a^{7}+13a^{6}+3a^{5}-15a^{4}-25a^{3}-16a^{2}+12a+5$, $a^{26}+4a^{25}+a^{24}-6a^{23}-7a^{22}+3a^{21}+12a^{20}+6a^{19}-9a^{18}-14a^{17}-a^{16}+14a^{15}+9a^{14}-7a^{13}-10a^{12}+9a^{10}+2a^{9}-9a^{8}-3a^{7}+7a^{6}+7a^{5}-4a^{4}-9a^{3}+5a^{2}+9a-5$, $13a^{26}+8a^{25}-18a^{24}-13a^{23}+20a^{22}+15a^{21}-26a^{20}-22a^{19}+29a^{18}+31a^{17}-26a^{16}-34a^{15}+26a^{14}+37a^{13}-27a^{12}-42a^{11}+22a^{10}+38a^{9}-21a^{8}-28a^{7}+30a^{6}+23a^{5}-39a^{4}-16a^{3}+51a^{2}+2a-16$, $9a^{26}+a^{25}-6a^{24}-6a^{23}+5a^{21}+6a^{20}+a^{19}-10a^{18}-18a^{17}-15a^{16}-4a^{15}+8a^{14}+19a^{13}+19a^{12}+4a^{11}-10a^{10}-9a^{9}+9a^{7}+21a^{6}+20a^{5}-5a^{4}-32a^{3}-35a^{2}-23a+40$, $11a^{26}+10a^{25}+a^{24}-6a^{23}-5a^{22}+3a^{20}-3a^{19}-15a^{18}-18a^{17}-7a^{16}+6a^{15}+11a^{14}+4a^{13}-7a^{12}-4a^{11}+13a^{10}+25a^{9}+19a^{8}-a^{7}-21a^{6}-18a^{5}+4a^{4}+10a^{3}-9a^{2}-33a+14$, $89a^{26}+132a^{25}-37a^{24}-163a^{23}-29a^{22}+171a^{21}+115a^{20}-139a^{19}-199a^{18}+68a^{17}+256a^{16}+38a^{15}-267a^{14}-173a^{13}+222a^{12}+304a^{11}-123a^{10}-399a^{9}-52a^{8}+413a^{7}+256a^{6}-368a^{5}-471a^{4}+200a^{3}+603a^{2}+54a-221$, $188a^{26}+116a^{25}+67a^{24}+35a^{23}+20a^{22}+18a^{21}+18a^{20}+14a^{19}+a^{18}-6a^{17}-6a^{16}+5a^{15}+9a^{14}+a^{13}-9a^{12}-18a^{11}-3a^{10}+12a^{9}+13a^{8}+4a^{7}-15a^{6}-14a^{5}+11a^{4}+19a^{3}+23a^{2}-11a+919$, $14a^{26}+40a^{25}+24a^{24}-21a^{23}-33a^{22}-22a^{21}-17a^{20}+15a^{19}+59a^{18}+37a^{17}-25a^{16}-41a^{15}-36a^{14}-39a^{13}+16a^{12}+92a^{11}+60a^{10}-24a^{9}-41a^{8}-59a^{7}-88a^{6}+9a^{5}+134a^{4}+84a^{3}-11a^{2}-17a-16$, $2a^{26}+28a^{25}+23a^{24}-8a^{23}-34a^{22}-23a^{21}+18a^{20}+41a^{19}+19a^{18}-27a^{17}-49a^{16}-15a^{15}+41a^{14}+53a^{13}+7a^{12}-55a^{11}-62a^{10}+8a^{9}+72a^{8}+61a^{7}-21a^{6}-96a^{5}-59a^{4}+50a^{3}+109a^{2}+55a-70$, $2a^{26}+4a^{25}-3a^{24}+9a^{22}-3a^{21}-3a^{20}+2a^{19}+5a^{18}-3a^{17}-11a^{16}+10a^{15}-a^{14}-9a^{13}+a^{12}+6a^{10}-10a^{9}+3a^{8}+6a^{7}-7a^{6}+11a^{5}-12a^{4}-a^{3}+12a^{2}-10a+4$
| 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: | \( 385692004384049400 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 385692004384049400 \cdot 1}{2\cdot\sqrt{45867792227272149237980784744990797413723399509150616187}}\cr\approx \mathstrut & 1.35464346475419 \end{aligned}\] (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 $S_{27}$ |
| Character table for $S_{27}$ |
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 | $23{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.3.0.1}{3} }^{8}{,}\,{\href{/padicField/3.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.13.0.1}{13} }$ | $17{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(26784394391927\)
| $\Q_{26784394391927}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(906\!\cdots\!369\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
|
\(188\!\cdots\!749\)
| $\Q_{18\!\cdots\!49}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |