Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 + 5*x - 1)
gp: K = bnfinit(y^27 + 5*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 + 5*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 + 5*x - 1)
\( x^{27} + 5x - 1 \)
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: |
\(-45866665096634308772626296247537769948249630619149892803\)
\(\medspace = -\,83\cdot 563\cdot 796853\cdot 12\!\cdots\!19\)
| 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: | $83^{1/2}563^{1/2}796853^{1/2}1231778062127328031049100178271442365420402719^{1/2}\approx 6.772493270327724e+27$ | ||
| Ramified primes: |
\(83\), \(563\), \(796853\), \(12317\!\cdots\!02719\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-45866\!\cdots\!92803}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{26}+5$, $8a^{26}+10a^{25}+9a^{24}+4a^{23}-3a^{22}-10a^{21}-14a^{20}-13a^{19}-7a^{18}+3a^{17}+13a^{16}+19a^{15}+18a^{14}+10a^{13}-3a^{12}-16a^{11}-24a^{10}-24a^{9}-14a^{8}+2a^{7}+17a^{6}+30a^{5}+31a^{4}+21a^{3}+4a^{2}-19a+4$, $15a^{26}-11a^{25}-7a^{24}+19a^{23}-10a^{22}-10a^{21}+24a^{20}-10a^{19}-17a^{18}+28a^{17}-11a^{16}-24a^{15}+33a^{14}-8a^{13}-28a^{12}+40a^{11}-4a^{10}-37a^{9}+44a^{8}-2a^{7}-52a^{6}+49a^{5}+5a^{4}-66a^{3}+61a^{2}+17a-4$, $11a^{26}-17a^{25}+14a^{24}-7a^{23}-3a^{22}+14a^{21}-18a^{20}+15a^{19}-a^{18}-16a^{17}+30a^{16}-39a^{15}+32a^{14}-13a^{13}-19a^{12}+51a^{11}-68a^{10}+73a^{9}-51a^{8}+11a^{7}+38a^{6}-83a^{5}+97a^{4}-90a^{3}+53a^{2}+6a-4$, $4a^{26}+4a^{25}+11a^{24}+13a^{23}+8a^{22}+9a^{21}+16a^{20}+15a^{19}+11a^{18}+14a^{17}+13a^{16}+11a^{15}+19a^{14}+20a^{13}+7a^{12}+4a^{11}+16a^{10}+14a^{9}-3a^{8}-2a^{7}+7a^{6}-3a^{5}-11a^{4}-10a^{3}-25a^{2}-33a+7$, $27a^{26}+68a^{25}+58a^{24}+16a^{23}-34a^{22}-69a^{21}-41a^{20}+20a^{19}+70a^{18}+87a^{17}+27a^{16}-64a^{15}-120a^{14}-122a^{13}-32a^{12}+82a^{11}+131a^{10}+112a^{9}-4a^{8}-125a^{7}-133a^{6}-62a^{5}+97a^{4}+234a^{3}+189a^{2}+48a-21$, $31a^{26}+11a^{25}-14a^{24}-14a^{23}+14a^{22}+39a^{21}+32a^{20}-a^{19}-24a^{18}-3a^{17}+45a^{16}+64a^{15}+30a^{14}-33a^{13}-55a^{12}-4a^{11}+73a^{10}+94a^{9}+21a^{8}-73a^{7}-101a^{6}-22a^{5}+85a^{4}+95a^{3}-3a^{2}-120a+24$, $45a^{26}+14a^{25}-13a^{24}-49a^{23}-66a^{22}-51a^{21}-35a^{20}+22a^{19}+49a^{18}+94a^{17}+73a^{16}+48a^{15}-10a^{14}-74a^{13}-100a^{12}-126a^{11}-56a^{10}-16a^{9}+97a^{8}+128a^{7}+155a^{6}+113a^{5}+9a^{4}-73a^{3}-205a^{2}-169a+42$, $3a^{26}+3a^{24}-a^{23}-3a^{21}-a^{20}+a^{19}-a^{18}-2a^{16}-2a^{15}-6a^{14}-6a^{13}-a^{12}-4a^{11}+2a^{10}-9a^{9}-11a^{7}-4a^{6}-11a^{5}+3a^{3}-7a^{2}+a+2$, $5a^{26}+20a^{25}-11a^{24}+22a^{23}-71a^{22}+28a^{21}+17a^{20}+35a^{19}-23a^{18}-26a^{17}+19a^{16}-63a^{15}+94a^{14}-4a^{13}+25a^{12}-117a^{11}+26a^{10}+42a^{9}-24a^{8}+94a^{7}-85a^{6}+10a^{5}-164a^{4}+205a^{3}+17a^{2}-23a+1$, $13a^{26}+10a^{25}-7a^{24}-6a^{23}+14a^{22}+18a^{21}+a^{20}-3a^{19}+11a^{18}+11a^{17}-2a^{16}+6a^{15}+24a^{14}+8a^{13}-23a^{12}-8a^{11}+34a^{10}+25a^{9}-24a^{8}-26a^{7}+16a^{6}+15a^{5}-24a^{4}-14a^{3}+26a^{2}-3a-1$, $135a^{26}-44a^{25}+83a^{24}-155a^{23}+11a^{22}-2a^{21}+67a^{20}+121a^{19}-156a^{18}+46a^{17}-225a^{16}+242a^{15}-24a^{14}+140a^{13}-127a^{12}-195a^{11}+120a^{10}-102a^{9}+444a^{8}-301a^{7}+137a^{6}-431a^{5}+164a^{4}+215a^{3}+46a^{2}+239a-49$, $58a^{26}+63a^{25}-55a^{24}-17a^{23}+5a^{22}-76a^{21}-8a^{20}+78a^{19}+87a^{18}-49a^{17}-19a^{16}+32a^{15}-157a^{14}-32a^{13}+134a^{12}+84a^{11}-18a^{10}-2a^{9}+52a^{8}-234a^{7}-115a^{6}+233a^{5}+48a^{4}+8a^{3}+112a^{2}+26a-9$
| 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: | \( 96868394873426400 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 96868394873426400 \cdot 1}{2\cdot\sqrt{45866665096634308772626296247537769948249630619149892803}}\cr\approx \mathstrut & 0.340229377031490 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 + 5*x - 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^27 + 5*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^27 + 5*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^27 + 5*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 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 | $23{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $27$ | $25{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.13.0.1}{13} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(83\)
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.5.0.1 | $x^{5} + 9 x + 81$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 83.17.0.1 | $x^{17} + 7 x + 81$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(563\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
|
\(796853\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
|
\(123\!\cdots\!719\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |