Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 + 5*x - 4)
gp: K = bnfinit(y^27 + 5*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 + 5*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 + 5*x - 4)
\( x^{27} + 5x - 4 \)
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: |
\(-47863680463851860855473749371874037931430553343164940288\)
\(\medspace = -\,2^{26}\cdot 23\cdot 4591\cdot 1054203587\cdot 40409162693929\cdot 158557479410628355703\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(115.40\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(23\), \(4591\), \(1054203587\), \(40409162693929\), \(158557479410628355703\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-71322\!\cdots\!58917}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{24}-a^{23}-a^{20}+a^{19}+a^{18}-a^{17}+2a^{16}-a^{15}-a^{14}+2a^{13}-3a^{12}+a^{11}-2a^{9}+3a^{8}+a^{7}-a^{6}+3a^{5}-3a^{4}-2a^{3}+4a^{2}-7a+5$, $4a^{26}+6a^{25}+2a^{24}+5a^{23}-a^{22}+4a^{21}-2a^{20}+5a^{19}-3a^{18}+5a^{17}-4a^{16}+5a^{15}-5a^{14}+5a^{13}-5a^{12}+6a^{11}-3a^{10}+5a^{9}-5a^{8}+2a^{7}-2a^{6}+a^{5}+a^{4}-2a^{3}+5a^{2}-7a+29$, $20a^{26}+14a^{25}+10a^{24}+8a^{23}+7a^{22}+6a^{21}+5a^{20}+5a^{19}+6a^{18}+7a^{17}+7a^{16}+5a^{15}+2a^{14}-a^{12}-2a^{11}-4a^{10}-7a^{9}-9a^{8}-8a^{7}-4a^{6}+a^{4}+a^{3}+3a^{2}+6a+109$, $a^{26}-a^{25}+2a^{23}-4a^{21}+5a^{19}-3a^{17}-a^{16}+2a^{14}+a^{13}-a^{12}-a^{10}-2a^{9}+5a^{7}+5a^{6}-9a^{5}-10a^{4}+13a^{3}+9a^{2}-13a+3$, $6a^{26}-2a^{25}-6a^{24}-6a^{23}+a^{22}+9a^{21}+4a^{20}+a^{19}+9a^{18}+10a^{17}-2a^{16}-11a^{15}-17a^{14}-10a^{13}+6a^{12}-a^{11}-9a^{10}+7a^{9}+18a^{8}+4a^{7}-10a^{6}-27a^{5}-24a^{4}+7a^{3}-a^{2}-22a+33$, $48a^{26}+75a^{25}+73a^{24}+46a^{23}-7a^{22}-63a^{21}-92a^{20}-89a^{19}-53a^{18}+15a^{17}+81a^{16}+113a^{15}+110a^{14}+60a^{13}-29a^{12}-100a^{11}-139a^{10}-140a^{9}-61a^{8}+45a^{7}+120a^{6}+181a^{5}+167a^{4}+57a^{3}-51a^{2}-159a+5$, $14a^{26}+18a^{25}+13a^{24}-a^{23}-12a^{22}-12a^{21}-8a^{20}-14a^{19}-23a^{18}-19a^{17}+15a^{15}+12a^{14}+9a^{13}+21a^{12}+40a^{11}+38a^{10}+11a^{9}-4a^{8}+2a^{7}+8a^{6}-14a^{5}-50a^{4}-47a^{3}-16a^{2}+a+53$, $a^{26}-3a^{24}+5a^{22}+7a^{21}-8a^{20}+7a^{19}-3a^{18}-17a^{17}+18a^{16}-3a^{15}-4a^{14}+19a^{13}-10a^{12}-13a^{11}+5a^{10}+3a^{9}-16a^{8}+27a^{7}+12a^{6}-41a^{5}+36a^{4}-21a^{3}-33a^{2}+56a-15$, $59a^{26}-34a^{25}-39a^{24}+74a^{23}+13a^{22}-92a^{21}+27a^{20}+76a^{19}-91a^{18}-50a^{17}+133a^{16}-6a^{15}-132a^{14}+101a^{13}+111a^{12}-180a^{11}-42a^{10}+209a^{9}-95a^{8}-204a^{7}+225a^{6}+130a^{5}-304a^{4}+51a^{3}+340a^{2}-252a+19$, $43a^{26}+35a^{25}-93a^{24}+57a^{23}+57a^{22}-124a^{21}+55a^{20}+77a^{19}-146a^{18}+70a^{17}+108a^{16}-194a^{15}+61a^{14}+153a^{13}-226a^{12}+73a^{11}+196a^{10}-297a^{9}+61a^{8}+275a^{7}-342a^{6}+54a^{5}+350a^{4}-449a^{3}+36a^{2}+486a-305$, $3664a^{26}+2967a^{25}+2386a^{24}+1871a^{23}+1464a^{22}+1197a^{21}+1007a^{20}+796a^{19}+577a^{18}+443a^{17}+415a^{16}+379a^{15}+256a^{14}+129a^{13}+122a^{12}+194a^{11}+176a^{10}+41a^{9}-36a^{8}+52a^{7}+159a^{6}+87a^{5}-85a^{4}-86a^{3}+90a^{2}+175a+18317$, $40a^{26}-6a^{25}-47a^{24}-45a^{23}+8a^{22}+62a^{21}+56a^{20}-19a^{19}-92a^{18}-62a^{17}+50a^{16}+127a^{15}+54a^{14}-107a^{13}-153a^{12}-12a^{11}+169a^{10}+154a^{9}-64a^{8}-206a^{7}-117a^{6}+125a^{5}+226a^{4}+59a^{3}-164a^{2}-218a+173$, $21a^{26}+39a^{25}+50a^{24}+64a^{23}+89a^{22}+90a^{21}+69a^{20}+80a^{19}+108a^{18}+96a^{17}+75a^{16}+86a^{15}+81a^{14}+50a^{13}+54a^{12}+64a^{11}+27a^{10}+7a^{9}+21a^{8}-17a^{7}-60a^{6}-13a^{5}+2a^{4}-98a^{3}-136a^{2}-61a+47$
| 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: | \( 346835430333748200 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 346835430333748200 \cdot 1}{2\cdot\sqrt{47863680463851860855473749371874037931430553343164940288}}\cr\approx \mathstrut & 1.19250085480711 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 + 5*x - 4)
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 - 4, 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 - 4);
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 - 4);
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 | R | ${\href{/padicField/3.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $22{,}\,{\href{/padicField/19.5.0.1}{5} }$ | R | $18{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| Deg $24$ | $2$ | $12$ | $24$ | ||||
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.21.0.1 | $x^{21} - 5 x + 18$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | |
|
\(4591\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(1054203587\)
| $\Q_{1054203587}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(40409162693929\)
| $\Q_{40409162693929}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{40409162693929}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(158\!\cdots\!703\)
| $\Q_{15\!\cdots\!03}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{15\!\cdots\!03}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |