Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 4*x - 1)
gp: K = bnfinit(y^27 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 4*x - 1)
\( x^{27} - 4x - 1 \)
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: | $[3, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(110898791389876228284002382374632342908260234650878781\)
\(\medspace = 241\cdot 3119\cdot 12228551\cdot 12\!\cdots\!89\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(92.18\) | 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: | $241^{1/2}3119^{1/2}12228551^{1/2}12064779711493270243744089382808808946789^{1/2}\approx 3.330147014620769e+26$ | ||
| Ramified primes: |
\(241\), \(3119\), \(12228551\), \(12064\!\cdots\!46789\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{11089\!\cdots\!78781}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a$, $a^{13}-2$, $a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a+2$, $a^{23}-a^{22}+2a^{20}-2a^{19}+a^{18}+a^{17}-a^{16}+a^{14}+a^{13}-2a^{12}+2a^{11}-a^{9}+2a^{8}-a^{5}+3a^{4}-2a^{2}+3a$, $a^{25}-a^{24}+a^{23}+a^{21}+a^{18}+a^{16}+a^{14}+a^{12}+a^{11}+a^{10}+2a^{7}+a^{6}+2a^{5}-a^{4}+2a^{3}-a^{2}+4a$, $3a^{26}-3a^{25}+2a^{24}-a^{22}+2a^{21}-3a^{20}+4a^{19}-2a^{18}-2a^{17}+7a^{16}-8a^{15}+4a^{14}+4a^{13}-10a^{12}+12a^{11}-5a^{10}-6a^{9}+15a^{8}-14a^{7}+4a^{6}+10a^{5}-15a^{4}+12a^{3}-2a^{2}-7a$, $2a^{26}+2a^{25}+2a^{24}+2a^{23}+2a^{22}+3a^{21}+a^{20}+4a^{19}+a^{18}+4a^{17}+2a^{16}+4a^{15}+3a^{14}+4a^{13}+5a^{12}+4a^{11}+5a^{10}+7a^{9}+4a^{8}+8a^{7}+5a^{6}+10a^{5}+4a^{4}+10a^{3}+8a^{2}+8a+1$, $a^{26}-3a^{25}-4a^{23}-2a^{22}-a^{21}-2a^{20}+2a^{19}-2a^{18}+2a^{17}-a^{16}+2a^{15}+4a^{14}+a^{13}+6a^{12}-2a^{11}+2a^{10}+6a^{7}+6a^{5}-3a^{4}+a^{3}-a^{2}-3a$, $2a^{25}-2a^{24}-2a^{22}+2a^{21}-4a^{18}+2a^{17}+a^{16}+4a^{15}-2a^{14}-2a^{12}+4a^{11}-2a^{10}-4a^{8}+a^{7}-3a^{6}+4a^{5}-2a^{4}+4a^{3}-2a^{2}+2a$, $47a^{26}-37a^{25}+17a^{24}+5a^{23}-22a^{22}+32a^{21}-39a^{20}+45a^{19}-44a^{18}+30a^{17}+a^{16}-43a^{15}+75a^{14}-83a^{13}+63a^{12}-23a^{11}-16a^{10}+42a^{9}-58a^{8}+70a^{7}-79a^{6}+78a^{5}-48a^{4}-13a^{3}+84a^{2}-140a-41$, $11a^{26}+11a^{25}+9a^{24}+8a^{23}+5a^{22}+5a^{21}+3a^{20}+6a^{19}+7a^{18}+12a^{17}+16a^{16}+20a^{15}+24a^{14}+24a^{13}+25a^{12}+20a^{11}+19a^{10}+11a^{9}+12a^{8}+7a^{7}+13a^{6}+15a^{5}+27a^{4}+33a^{3}+46a^{2}+51a+11$, $20a^{26}-78a^{25}-96a^{24}+5a^{23}+116a^{22}+98a^{21}-41a^{20}-142a^{19}-78a^{18}+80a^{17}+144a^{16}+35a^{15}-110a^{14}-112a^{13}+26a^{12}+117a^{11}+40a^{10}-95a^{9}-86a^{8}+72a^{7}+154a^{6}+5a^{5}-210a^{4}-181a^{3}+125a^{2}+350a+79$, $16a^{26}+21a^{25}-18a^{24}+31a^{23}-3a^{22}-14a^{21}+36a^{20}-43a^{19}+16a^{18}-2a^{17}-51a^{16}+36a^{15}-58a^{14}-10a^{13}+9a^{12}-67a^{11}+28a^{10}-20a^{9}-43a^{8}+69a^{7}-63a^{6}+48a^{5}+45a^{4}-46a^{3}+127a^{2}-27a-16$, $25a^{26}-42a^{25}+51a^{24}-48a^{23}+35a^{22}-11a^{21}-18a^{20}+43a^{19}-59a^{18}+61a^{17}-50a^{16}+23a^{15}+14a^{14}-51a^{13}+78a^{12}-88a^{11}+80a^{10}-49a^{9}+2a^{8}+53a^{7}-99a^{6}+124a^{5}-126a^{4}+96a^{3}-44a^{2}-28a-3$
| 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: | \( 14868364781972948 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 14868364781972948 \cdot 1}{2\cdot\sqrt{110898791389876228284002382374632342908260234650878781}}\cr\approx \mathstrut & 0.676111328331640 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 4*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 - 4*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 - 4*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 - 4*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 $S_{27}$ |
| 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 | $18{,}\,{\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.9.0.1}{9} }^{3}$ | $27$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | $23{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(241\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(3119\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
|
\(12228551\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(120\!\cdots\!789\)
| $\Q_{12\!\cdots\!89}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |