Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 2*x - 4)
gp: K = bnfinit(y^27 - 2*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 2*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 2*x - 4)
\( x^{27} - 2x - 4 \)
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: |
\(-499253841597823035730555735945079609621365309403299840\)
\(\medspace = -\,2^{50}\cdot 5\cdot 139\cdot 42307\cdot 361717434139\cdot 41692232937963286631\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(97.46\) | 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: |
\(2\), \(5\), \(139\), \(42307\), \(361717434139\), \(41692232937963286631\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-44342\!\cdots\!60785}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{2}a^{26}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: |
$5a^{26}+4a^{25}-2a^{24}-6a^{23}+a^{22}+8a^{21}-6a^{19}-5a^{18}+9a^{17}+5a^{16}-5a^{15}-10a^{14}+3a^{13}+12a^{12}-2a^{11}-12a^{10}-8a^{9}+15a^{8}+8a^{7}-10a^{6}-16a^{5}+8a^{4}+22a^{3}-3a^{2}-20a-17$, $3a^{26}-6a^{25}-4a^{24}+6a^{23}+4a^{22}-5a^{21}-2a^{20}+4a^{18}+3a^{17}-3a^{16}-9a^{15}+5a^{14}+12a^{13}-7a^{12}-13a^{11}+6a^{10}+13a^{9}-5a^{8}-9a^{7}+5a^{5}+10a^{4}-3a^{3}-17a^{2}-3a+23$, $4a^{26}+7a^{25}-12a^{24}-11a^{23}+15a^{22}-a^{21}-16a^{20}-5a^{19}+12a^{18}+3a^{17}-29a^{16}+5a^{15}+16a^{14}-11a^{13}-23a^{12}+a^{11}+26a^{10}-25a^{9}-29a^{8}+21a^{7}+11a^{6}-27a^{5}-35a^{4}+31a^{3}+12a^{2}-58a-21$, $5a^{26}-3a^{25}-3a^{23}+9a^{22}-2a^{21}-7a^{20}-2a^{19}+6a^{18}+2a^{17}-a^{16}-3a^{15}-a^{14}+12a^{13}-4a^{12}-10a^{11}-5a^{10}+14a^{9}-a^{8}+2a^{7}-13a^{6}+12a^{5}+9a^{4}-28a^{2}+8a-3$, $20a^{26}+a^{25}-14a^{24}+36a^{23}-54a^{22}+56a^{21}-67a^{20}+59a^{19}-47a^{18}+32a^{17}+4a^{16}-19a^{15}+45a^{14}-72a^{13}+74a^{12}-89a^{11}+71a^{10}-55a^{9}+49a^{8}-2a^{7}-15a^{6}+46a^{5}-77a^{4}+72a^{3}-112a^{2}+100a-119$, $13a^{26}+29a^{25}+19a^{24}-8a^{23}-30a^{22}-32a^{21}-6a^{20}+32a^{19}+42a^{18}+21a^{17}-19a^{16}-55a^{15}-41a^{14}+11a^{13}+54a^{12}+58a^{11}+11a^{10}-54a^{9}-74a^{8}-29a^{7}+38a^{6}+87a^{5}+69a^{4}-25a^{3}-102a^{2}-102a-43$, $15a^{26}+9a^{25}-13a^{24}-13a^{23}+12a^{22}+20a^{21}-5a^{20}-25a^{19}-3a^{18}+25a^{17}+8a^{16}-27a^{15}-22a^{14}+23a^{13}+30a^{12}-12a^{11}-39a^{10}+7a^{9}+47a^{8}+12a^{7}-48a^{6}-30a^{5}+40a^{4}+41a^{3}-37a^{2}-68a-13$, $3a^{26}-6a^{25}+2a^{24}-3a^{23}+4a^{21}-5a^{20}+6a^{19}-4a^{18}+3a^{17}+5a^{16}-a^{15}+8a^{14}-5a^{13}+7a^{12}+4a^{11}+6a^{9}-9a^{8}+5a^{7}-3a^{6}-3a^{5}+4a^{4}-15a^{3}-a^{2}-11a-9$, $7a^{26}+10a^{25}+11a^{24}+20a^{23}+16a^{22}+26a^{21}+18a^{20}+24a^{19}+16a^{18}+16a^{17}+13a^{16}+9a^{15}+15a^{14}+11a^{13}+26a^{12}+24a^{11}+42a^{10}+41a^{9}+53a^{8}+52a^{7}+51a^{6}+51a^{5}+36a^{4}+40a^{3}+20a^{2}+33a+11$, $a^{26}+16a^{25}+14a^{24}-6a^{23}-22a^{22}-13a^{21}+14a^{20}+28a^{19}+10a^{18}-22a^{17}-31a^{16}-4a^{15}+29a^{14}+29a^{13}-4a^{12}-30a^{11}-17a^{10}+18a^{9}+29a^{8}+a^{7}-29a^{6}-18a^{5}+24a^{4}+39a^{3}-3a^{2}-55a-45$, $7a^{26}-6a^{25}+3a^{24}+5a^{23}-13a^{22}+9a^{21}-12a^{20}+8a^{19}-15a^{18}+15a^{17}-6a^{16}-a^{15}+5a^{14}-9a^{13}+4a^{12}-21a^{11}+19a^{10}-21a^{9}+11a^{8}-a^{7}+6a^{6}-9a^{5}-10a^{4}+13a^{3}-38a^{2}+16a-27$, $22a^{26}+7a^{25}-31a^{24}+47a^{23}-73a^{22}+78a^{21}-75a^{20}+80a^{19}-54a^{18}+29a^{17}-10a^{16}-35a^{15}+59a^{14}-74a^{13}+109a^{12}-104a^{11}+94a^{10}-91a^{9}+46a^{8}-12a^{7}-11a^{6}+71a^{5}-104a^{4}+113a^{3}-151a^{2}+132a-143$, $51a^{26}-56a^{25}+63a^{24}-44a^{23}+42a^{22}-18a^{21}-5a^{20}+26a^{19}-42a^{18}+73a^{17}-64a^{16}+82a^{15}-71a^{14}+53a^{13}-31a^{12}+16a^{11}+40a^{10}-39a^{9}+81a^{8}-95a^{7}+97a^{6}-87a^{5}+99a^{4}-28a^{3}+35a^{2}+24a-165$
| 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: | \( 87322201666955410 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 87322201666955410 \cdot 1}{2\cdot\sqrt{499253841597823035730555735945079609621365309403299840}}\cr\approx \mathstrut & 2.93969489431734 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 2*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 - 2*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 - 2*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 - 2*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 $S_{27}$ |
| 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 | R | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | $26{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/41.5.0.1}{5} }$ | $22{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/59.7.0.1}{7} }$ |
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 | $[\ ]$ |
| Deg $26$ | $26$ | $1$ | $50$ | ||||
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.7.0.1 | $x^{7} + 3 x + 3$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 5.8.0.1 | $x^{8} + x^{4} + 3 x^{2} + 4 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 5.9.0.1 | $x^{9} + 2 x^{3} + x + 3$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(139\)
| 139.2.1.1 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 139.4.0.1 | $x^{4} + 7 x^{2} + 96 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 139.10.0.1 | $x^{10} + 110 x^{5} + 48 x^{4} + 130 x^{3} + 66 x^{2} + 106 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 139.11.0.1 | $x^{11} + 7 x + 137$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
|
\(42307\)
| $\Q_{42307}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
|
\(361717434139\)
| $\Q_{361717434139}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{361717434139}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
|
\(41692232937963286631\)
| $\Q_{41692232937963286631}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{41692232937963286631}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |