Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 4*x^6 - 4*x^5 + 18*x^4 - 32*x^3 + 40*x^2 - 32*x + 10)
gp: K = bnfinit(y^8 - 4*y^7 + 4*y^6 - 4*y^5 + 18*y^4 - 32*y^3 + 40*y^2 - 32*y + 10, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 4*x^7 + 4*x^6 - 4*x^5 + 18*x^4 - 32*x^3 + 40*x^2 - 32*x + 10);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 4*x^7 + 4*x^6 - 4*x^5 + 18*x^4 - 32*x^3 + 40*x^2 - 32*x + 10)
\( x^{8} - 4x^{7} + 4x^{6} - 4x^{5} + 18x^{4} - 32x^{3} + 40x^{2} - 32x + 10 \)
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: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2123366400\)
\(\medspace = 2^{20}\cdot 3^{4}\cdot 5^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.65\) | 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: | $2^{5/2}3^{1/2}5^{1/2}\approx 21.908902300206645$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{97}a^{7}-\frac{14}{97}a^{6}+\frac{47}{97}a^{5}+\frac{11}{97}a^{4}+\frac{5}{97}a^{3}+\frac{15}{97}a^{2}-\frac{13}{97}a+\frac{1}{97}$
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
$C_{4}$, which has order $4$
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: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{162}{97} a^{7} - \frac{522}{97} a^{6} + \frac{242}{97} a^{5} - \frac{449}{97} a^{4} + \frac{2556}{97} a^{3} - \frac{3196}{97} a^{2} + \frac{3908}{97} a - \frac{2069}{97} \)
(order $4$)
| 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: |
$\frac{162}{97}a^{7}-\frac{522}{97}a^{6}+\frac{242}{97}a^{5}-\frac{449}{97}a^{4}+\frac{2556}{97}a^{3}-\frac{3196}{97}a^{2}+\frac{4005}{97}a-\frac{2069}{97}$, $\frac{36}{97}a^{7}-\frac{116}{97}a^{6}+\frac{43}{97}a^{5}-\frac{89}{97}a^{4}+\frac{568}{97}a^{3}-\frac{624}{97}a^{2}+\frac{793}{97}a-\frac{449}{97}$, $\frac{25}{97}a^{7}-\frac{59}{97}a^{6}+\frac{11}{97}a^{5}-\frac{113}{97}a^{4}+\frac{319}{97}a^{3}-\frac{304}{97}a^{2}+\frac{548}{97}a-\frac{363}{97}$
| 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: | \( 30.7939263927 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 30.7939263927 \cdot 4}{4\cdot\sqrt{2123366400}}\cr\approx \mathstrut & 1.04153068725 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 4*x^6 - 4*x^5 + 18*x^4 - 32*x^3 + 40*x^2 - 32*x + 10)
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 + 4*x^6 - 4*x^5 + 18*x^4 - 32*x^3 + 40*x^2 - 32*x + 10, 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 + 4*x^6 - 4*x^5 + 18*x^4 - 32*x^3 + 40*x^2 - 32*x + 10);
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 + 4*x^6 - 4*x^5 + 18*x^4 - 32*x^3 + 40*x^2 - 32*x + 10);
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
$D_4:C_2^2$ (as 8T22):
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 32 |
| The 17 conjugacy class representatives for $Q_8:C_2^2$ |
| Character table for $Q_8:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{6})\) |
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
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | 16.0.4508684868648960000.1, 16.0.34789235097600000000.1, 16.0.11007531417600000000.5, 16.0.2817928042905600000000.7, 16.0.2817928042905600000000.15, 16.0.2817928042905600000000.9, 16.0.2817928042905600000000.10, 16.8.2817928042905600000000.3, 16.0.2817928042905600000000.13 |
| Minimal sibling: | 8.0.368640000.2 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.20.56 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 10$ | $8$ | $1$ | $20$ | $Q_8:C_2$ | $[2, 3, 3]^{2}$ |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.60.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{15}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.120.2t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{-30}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.120.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{30}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.40.2t1.a.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{10}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 4.921600.8t22.f.a | $4$ | $ 2^{12} \cdot 3^{2} \cdot 5^{2}$ | 8.0.2123366400.8 | $Q_8:C_2^2$ (as 8T22) | $1$ | $0$ |
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 *.