Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 + 4*x - 2)
gp: K = bnfinit(y^27 + 4*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 + 4*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 + 4*x - 2)
\( x^{27} + 4x - 2 \)
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: |
\(-110898821147724565210111275732773790428656805531877376\)
\(\medspace = -\,2^{26}\cdot 7\cdot 211\cdot 277\cdot 9820621\cdot 41\!\cdots\!51\)
| 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: | $2^{26/27}7^{1/2}211^{1/2}277^{1/2}9820621^{1/2}411289693845079379557406117008951^{1/2}\approx 7.924178643374103e+22$ | ||
| Ramified primes: |
\(2\), \(7\), \(211\), \(277\), \(9820621\), \(41128\!\cdots\!08951\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-16525\!\cdots\!74659}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$5a^{26}+5a^{25}+4a^{24}+2a^{23}-3a^{21}-5a^{20}-6a^{19}-7a^{18}-6a^{17}-4a^{16}-2a^{15}+3a^{13}+5a^{12}+5a^{11}+7a^{10}+7a^{9}+5a^{8}+5a^{7}+3a^{6}-3a^{4}-6a^{3}-9a^{2}-12a+9$, $a^{26}+2a^{24}+a^{23}-4a^{22}-a^{20}+4a^{19}+2a^{18}-4a^{17}-a^{16}-3a^{15}+5a^{14}+3a^{13}-3a^{12}-6a^{10}+5a^{9}+2a^{8}-a^{7}+4a^{6}-9a^{5}+4a^{4}-2a^{3}+a^{2}+10a-5$, $3a^{26}+a^{25}+6a^{24}+a^{23}-3a^{22}-3a^{21}-7a^{20}+3a^{19}+11a^{17}-a^{16}+a^{15}-7a^{14}-8a^{13}+2a^{12}-a^{11}+14a^{10}-2a^{9}+9a^{8}-14a^{7}-5a^{6}-6a^{5}-a^{4}+18a^{3}+a^{2}+15a-11$, $3a^{26}-3a^{25}-2a^{24}+4a^{23}+2a^{22}-5a^{21}-a^{20}+4a^{19}+2a^{18}-5a^{17}-4a^{16}+8a^{15}+4a^{14}-10a^{13}-a^{12}+8a^{11}+2a^{10}-6a^{9}-9a^{8}+13a^{7}+5a^{6}-14a^{5}-a^{4}+12a^{3}+a^{2}-8a+3$, $a^{26}-5a^{25}-9a^{24}-8a^{23}-4a^{22}+2a^{21}+6a^{20}+5a^{19}-a^{18}-8a^{17}-8a^{16}-2a^{15}+6a^{14}+12a^{13}+12a^{12}+7a^{11}-3a^{10}-8a^{9}-2a^{8}+9a^{7}+19a^{6}+19a^{5}+13a^{4}+2a^{3}-10a^{2}-8a+9$, $11a^{26}+9a^{25}-4a^{24}+12a^{23}-13a^{22}-7a^{20}-20a^{19}+4a^{18}-26a^{17}+2a^{16}-6a^{15}-13a^{14}+22a^{13}-18a^{12}+25a^{11}+3a^{10}-4a^{9}+31a^{8}-32a^{7}+25a^{6}-22a^{5}-23a^{4}+15a^{3}-64a^{2}+26a+3$, $61a^{26}+17a^{25}-62a^{24}-48a^{23}+46a^{22}+75a^{21}-17a^{20}-93a^{19}-25a^{18}+91a^{17}+71a^{16}-68a^{15}-112a^{14}+24a^{13}+135a^{12}+39a^{11}-131a^{10}-104a^{9}+98a^{8}+160a^{7}-33a^{6}-193a^{5}-54a^{4}+190a^{3}+144a^{2}-144a+15$, $a^{26}+a^{25}+a^{24}-a^{23}+a^{19}+a^{18}-a^{17}-a^{14}+a^{13}+a^{12}+a^{9}-2a^{8}+a^{7}-a^{4}+3a^{3}-2a^{2}+3$, $3a^{26}+2a^{25}-7a^{24}-3a^{23}+9a^{22}+a^{21}-3a^{20}-4a^{19}-4a^{18}+14a^{17}+a^{16}-13a^{15}+3a^{14}+10a^{12}+4a^{11}-21a^{10}+6a^{9}+11a^{8}+4a^{6}-20a^{5}+a^{4}+26a^{3}-6a^{2}-11a+5$, $12a^{26}-3a^{25}+3a^{24}+2a^{23}-19a^{22}+28a^{21}-14a^{20}-4a^{19}+7a^{18}-12a^{17}+30a^{16}-37a^{15}+10a^{14}+20a^{13}-26a^{12}+24a^{11}-40a^{10}+39a^{9}+3a^{8}-51a^{7}+52a^{6}-36a^{5}+38a^{4}-30a^{3}-33a^{2}+94a-37$, $11a^{26}+9a^{25}+7a^{24}-a^{21}-7a^{20}-4a^{19}-10a^{18}-12a^{17}-5a^{16}-5a^{15}+a^{14}+6a^{13}+a^{12}+10a^{11}+11a^{10}+9a^{9}+17a^{8}+4a^{7}-3a^{6}+2a^{5}-10a^{4}-7a^{3}-5a^{2}-24a+33$, $32a^{26}+15a^{25}+3a^{24}+5a^{23}+5a^{22}-3a^{21}-a^{20}+6a^{19}-5a^{17}+4a^{16}+3a^{15}-7a^{14}-3a^{13}+8a^{12}-4a^{11}-6a^{10}+8a^{9}+5a^{8}-9a^{7}+12a^{5}-8a^{4}-10a^{3}+13a^{2}+2a+113$, $3a^{26}-24a^{25}-35a^{24}-21a^{23}+11a^{22}+37a^{21}+39a^{20}+13a^{19}-26a^{18}-48a^{17}-37a^{16}+2a^{15}+44a^{14}+57a^{13}+30a^{12}-21a^{11}-62a^{10}-60a^{9}-13a^{8}+46a^{7}+79a^{6}+57a^{5}-12a^{4}-74a^{3}-89a^{2}-42a+57$
| 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: | \( 15404318055100588 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 15404318055100588 \cdot 1}{2\cdot\sqrt{110898821147724565210111275732773790428656805531877376}}\cr\approx \mathstrut & 1.10031567727311 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 + 4*x - 2)
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 - 2, 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 - 2);
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 - 2);
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 | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $27$ | R | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $27$ | $27$ | $1$ | $26$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.12.0.1 | $x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(211\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
|
\(277\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(9820621\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
|
\(411\!\cdots\!951\)
| $\Q_{41\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |