Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 + x - 3)
gp: K = bnfinit(y^27 + y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 + x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 + x - 3)
\( x^{27} + x - 3 \)
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: |
\(-1127130637840914937096320697954724520073797550819963\)
\(\medspace = -\,991\cdot 43499487821953\cdot 240698508804661\cdot 108628318596533067121\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(77.77\) | 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: | $991^{1/2}43499487821953^{1/2}240698508804661^{1/2}108628318596533067121^{1/2}\approx 3.357276631201121e+25$ | ||
| Ramified primes: |
\(991\), \(43499487821953\), \(240698508804661\), \(108628318596533067121\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-11271\!\cdots\!19963}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a-1$, $a^{25}+a^{23}-a^{22}+a^{21}+a^{19}+a^{16}-a^{15}+2a^{14}-a^{13}+2a^{12}-a^{11}+a^{10}+a^{9}+2a^{7}-2a^{6}+3a^{5}-a^{4}+3a^{3}-a^{2}+a+1$, $3a^{26}+a^{25}-a^{24}-3a^{23}-3a^{22}+a^{21}+3a^{20}+3a^{19}+a^{18}-3a^{17}-4a^{16}-2a^{15}+2a^{14}+5a^{13}+3a^{12}-a^{11}-3a^{10}-4a^{9}-2a^{8}+3a^{7}+4a^{6}+3a^{5}-5a^{3}-4a^{2}-2a+5$, $a^{25}+a^{22}-a^{21}-2a^{20}-a^{19}-2a^{18}-a^{17}+2a^{16}+a^{15}+a^{14}+3a^{13}-a^{11}+2a^{10}-a^{8}+a^{7}-3a^{6}-5a^{5}+5a+2$, $4a^{26}-a^{25}+4a^{24}-3a^{23}+2a^{22}-3a^{21}+3a^{20}+3a^{18}-a^{14}+2a^{13}+4a^{11}-a^{10}+3a^{9}-4a^{8}+3a^{7}-4a^{6}+8a^{5}-3a^{4}+9a^{3}-9a^{2}+7a-7$, $a^{26}-6a^{25}-3a^{24}+4a^{23}-a^{21}+7a^{20}+4a^{19}-6a^{18}-2a^{17}+a^{16}-7a^{15}-3a^{14}+9a^{13}+4a^{12}-3a^{11}+6a^{10}+3a^{9}-12a^{8}-5a^{7}+7a^{6}-5a^{5}-6a^{4}+14a^{3}+7a^{2}-10a+4$, $a^{24}-a^{22}-a^{20}+a^{19}+a^{18}+a^{13}-2a^{11}-a^{9}+2a^{8}+a^{7}-a^{6}-2a^{4}+a^{2}-2$, $a^{26}+a^{25}+a^{24}+2a^{23}+a^{22}-2a^{21}-a^{20}+2a^{19}+2a^{18}-2a^{16}-4a^{15}-4a^{14}+a^{13}+4a^{12}-a^{11}-5a^{10}-4a^{9}-2a^{8}+a^{7}+4a^{6}+2a^{5}-4a^{4}-2a^{3}+4a^{2}+3a+2$, $13a^{26}+11a^{25}+10a^{24}+9a^{23}+9a^{22}+8a^{21}+5a^{20}+a^{19}-5a^{18}-9a^{17}-12a^{16}-14a^{15}-15a^{14}-18a^{13}-20a^{12}-22a^{11}-21a^{10}-16a^{9}-11a^{8}-4a^{7}+3a^{5}+7a^{4}+11a^{3}+19a^{2}+24a+41$, $3a^{25}-a^{24}-2a^{23}+3a^{22}+a^{21}-3a^{20}+4a^{18}-3a^{17}-2a^{16}+5a^{15}-4a^{13}+3a^{12}+4a^{11}-6a^{10}-a^{9}+7a^{8}-3a^{7}-5a^{6}+8a^{5}+a^{4}-6a^{3}+a^{2}+8a-8$, $6a^{26}-a^{25}-6a^{24}+a^{23}+7a^{22}-2a^{21}-8a^{20}+3a^{19}+7a^{18}-3a^{17}-8a^{16}+5a^{15}+8a^{14}-7a^{13}-8a^{12}+8a^{11}+8a^{10}-9a^{9}-9a^{8}+12a^{7}+8a^{6}-12a^{5}-7a^{4}+13a^{3}+7a^{2}-16a+1$, $5a^{26}+5a^{25}-6a^{24}-3a^{23}+7a^{22}+3a^{21}-6a^{20}-a^{19}+6a^{18}+2a^{17}-5a^{16}-2a^{15}+5a^{14}+4a^{13}-6a^{12}-5a^{11}+8a^{10}+6a^{9}-12a^{8}-6a^{7}+15a^{6}+3a^{5}-20a^{4}-a^{3}+21a^{2}-6a-17$, $2a^{26}+a^{25}+3a^{22}-3a^{21}-a^{18}-3a^{17}+2a^{15}-3a^{14}+2a^{13}+2a^{12}+2a^{11}-3a^{10}+5a^{9}-4a^{7}+a^{6}-3a^{4}-4a^{3}+4a^{2}-a-2$
| 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: | \( 813749693821532.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 813749693821532.9 \cdot 1}{2\cdot\sqrt{1127130637840914937096320697954724520073797550819963}}\cr\approx \mathstrut & 0.576556668149325 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 + 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 + 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 + 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 + 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 | $23{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.13.0.1}{13} }$ | $18{,}\,{\href{/padicField/23.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/41.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $15{,}\,{\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(991\)
| $\Q_{991}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(43499487821953\)
| $\Q_{43499487821953}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{43499487821953}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(240698508804661\)
| $\Q_{240698508804661}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{240698508804661}$ | $x$ | $1$ | $1$ | $0$ | 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(108\!\cdots\!121\)
| $\Q_{10\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{10\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |