Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 2*x^6 + 16*x^5 - 28*x^4 + 28*x^3 - 20*x^2 + 8*x - 2)
gp: K = bnfinit(y^8 - 4*y^7 - 2*y^6 + 16*y^5 - 28*y^4 + 28*y^3 - 20*y^2 + 8*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 4*x^7 - 2*x^6 + 16*x^5 - 28*x^4 + 28*x^3 - 20*x^2 + 8*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 4*x^7 - 2*x^6 + 16*x^5 - 28*x^4 + 28*x^3 - 20*x^2 + 8*x - 2)
\( x^{8} - 4x^{7} - 2x^{6} + 16x^{5} - 28x^{4} + 28x^{3} - 20x^{2} + 8x - 2 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-46438023168\)
\(\medspace = -\,2^{18}\cdot 3^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.55\) | 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: | $2^{9/4}3^{25/12}\approx 46.91590565166763$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{9}a^{7}-\frac{2}{9}a^{6}+\frac{1}{3}a^{5}+\frac{4}{9}a^{4}-\frac{2}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{9}a+\frac{1}{9}$
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 $1$
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: | $4$ | 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: |
$\frac{7}{9}a^{7}-\frac{14}{9}a^{6}-\frac{20}{3}a^{5}+\frac{46}{9}a^{4}-\frac{5}{9}a^{3}-\frac{10}{3}a^{2}+\frac{43}{9}a-\frac{29}{9}$, $\frac{10}{9}a^{7}-\frac{29}{9}a^{6}-\frac{20}{3}a^{5}+\frac{121}{9}a^{4}-\frac{110}{9}a^{3}+\frac{14}{3}a^{2}-\frac{8}{9}a+\frac{1}{9}$, $\frac{20}{9}a^{7}-\frac{76}{9}a^{6}-\frac{19}{3}a^{5}+\frac{314}{9}a^{4}-\frac{490}{9}a^{3}+\frac{148}{3}a^{2}-\frac{268}{9}a+\frac{83}{9}$, $\frac{14}{3}a^{7}-\frac{43}{3}a^{6}-25a^{5}+\frac{179}{3}a^{4}-\frac{199}{3}a^{3}+36a^{2}-\frac{31}{3}a-\frac{43}{3}$
| 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: | \( 402.187110677 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 402.187110677 \cdot 1}{2\cdot\sqrt{46438023168}}\cr\approx \mathstrut & 0.925893183900 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 2*x^6 + 16*x^5 - 28*x^4 + 28*x^3 - 20*x^2 + 8*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^8 - 4*x^7 - 2*x^6 + 16*x^5 - 28*x^4 + 28*x^3 - 20*x^2 + 8*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^8 - 4*x^7 - 2*x^6 + 16*x^5 - 28*x^4 + 28*x^3 - 20*x^2 + 8*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^8 - 4*x^7 - 2*x^6 + 16*x^5 - 28*x^4 + 28*x^3 - 20*x^2 + 8*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
$S_4\wr C_2$ (as 8T47):
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 solvable group of order 1152 |
| The 20 conjugacy class representatives for $S_4\wr C_2$ |
| Character table for $S_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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]
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.18.76 | $x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $C_4\wr C_2$ | $[2, 2, 3]^{4}$ |
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.6.10.11 | $x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24$ | $6$ | $1$ | $10$ | $C_3^2:D_4$ | $[9/4, 9/4]_{4}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 2.144.4t3.b.a | $2$ | $ 2^{4} \cdot 3^{2}$ | 4.0.432.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 4.314928.6t13.b.a | $4$ | $ 2^{4} \cdot 3^{9}$ | 6.0.944784.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.1259712.12t34.b.a | $4$ | $ 2^{6} \cdot 3^{9}$ | 6.0.944784.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
| 4.5038848.12t34.c.a | $4$ | $ 2^{8} \cdot 3^{9}$ | 6.0.944784.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.1259712.6t13.c.a | $4$ | $ 2^{6} \cdot 3^{9}$ | 6.0.944784.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
| 6.2902376448.12t201.a.a | $6$ | $ 2^{14} \cdot 3^{11}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
| 6.3869835264.12t202.d.a | $6$ | $ 2^{16} \cdot 3^{10}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
| * | 6.3869835264.8t47.d.a | $6$ | $ 2^{16} \cdot 3^{10}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
| 6.2902376448.12t200.a.a | $6$ | $ 2^{14} \cdot 3^{11}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
| 9.406...664.16t1294.a.a | $9$ | $ 2^{20} \cdot 3^{18}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
| 9.121...992.18t272.a.a | $9$ | $ 2^{20} \cdot 3^{19}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.101...416.18t273.b.a | $9$ | $ 2^{18} \cdot 3^{18}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.304...248.18t274.a.a | $9$ | $ 2^{18} \cdot 3^{19}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
| 12.818...288.36t1763.b.a | $12$ | $ 2^{30} \cdot 3^{27}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
| 12.818...288.24t2821.b.a | $12$ | $ 2^{30} \cdot 3^{27}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
| 18.148...064.36t1758.b.a | $18$ | $ 2^{40} \cdot 3^{38}$ | 8.2.46438023168.3 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.