Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^6 - 12*x^5 + 6*x^4 + 48*x^3 - 32*x^2 + 16)
gp: K = bnfinit(y^8 - 2*y^6 - 12*y^5 + 6*y^4 + 48*y^3 - 32*y^2 + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^6 - 12*x^5 + 6*x^4 + 48*x^3 - 32*x^2 + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 2*x^6 - 12*x^5 + 6*x^4 + 48*x^3 - 32*x^2 + 16)
\( x^{8} - 2x^{6} - 12x^{5} + 6x^{4} + 48x^{3} - 32x^{2} + 16 \)
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: |
\(-119439360000\)
\(\medspace = -\,2^{18}\cdot 3^{6}\cdot 5^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.25\) | 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: | $2^{11/4}3^{3/4}5^{2/3}\approx 44.838743292597435$ | ||
| 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: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2600}a^{7}+\frac{32}{325}a^{6}+\frac{267}{1300}a^{5}+\frac{24}{325}a^{4}-\frac{121}{1300}a^{3}+\frac{62}{325}a^{2}-\frac{57}{325}a+\frac{33}{325}$
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: |
$\frac{29}{1300}a^{7}-\frac{51}{1300}a^{6}-\frac{57}{650}a^{5}-\frac{141}{650}a^{4}+\frac{391}{650}a^{3}+\frac{1017}{650}a^{2}-\frac{706}{325}a-\frac{361}{325}$, $\frac{383}{1300}a^{7}-\frac{51}{650}a^{6}-\frac{439}{650}a^{5}-\frac{1116}{325}a^{4}+\frac{1757}{650}a^{3}+\frac{4917}{325}a^{2}-\frac{4337}{325}a+\frac{253}{325}$, $\frac{8}{65}a^{7}-\frac{63}{260}a^{6}+\frac{29}{130}a^{5}-\frac{243}{130}a^{4}+\frac{274}{65}a^{3}-\frac{319}{130}a^{2}-\frac{8}{65}a+\frac{97}{65}$, $\frac{33}{260}a^{7}-\frac{1}{130}a^{6}-\frac{47}{65}a^{5}-\frac{41}{65}a^{4}+\frac{297}{130}a^{3}+\frac{127}{65}a^{2}-\frac{252}{65}a+\frac{163}{65}$
| 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: | \( 922.131306311 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 922.131306311 \cdot 1}{2\cdot\sqrt{119439360000}}\cr\approx \mathstrut & 1.32369714903 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^6 - 12*x^5 + 6*x^4 + 48*x^3 - 32*x^2 + 16)
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 - 2*x^6 - 12*x^5 + 6*x^4 + 48*x^3 - 32*x^2 + 16, 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 - 2*x^6 - 12*x^5 + 6*x^4 + 48*x^3 - 32*x^2 + 16);
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 - 2*x^6 - 12*x^5 + 6*x^4 + 48*x^3 - 32*x^2 + 16);
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
| \(\Q(\sqrt{2}) \) |
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 | R | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.10.2 | $x^{4} + 4 x^{3} + 2$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ |
| 2.4.8.3 | $x^{4} + 6 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
|
\(3\)
| 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.6.4.2 | $x^{6} + 10 x^{3} - 25$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 2.1152.4t3.c.a | $2$ | $ 2^{7} \cdot 3^{2}$ | 4.0.4608.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 4.1843200.12t34.b.a | $4$ | $ 2^{13} \cdot 3^{2} \cdot 5^{2}$ | 6.0.460800.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.23040000.12t34.b.a | $4$ | $ 2^{12} \cdot 3^{2} \cdot 5^{4}$ | 6.0.460800.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
| 4.115200.6t13.b.a | $4$ | $ 2^{9} \cdot 3^{2} \cdot 5^{2}$ | 6.0.460800.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.5760000.6t13.b.a | $4$ | $ 2^{10} \cdot 3^{2} \cdot 5^{4}$ | 6.0.460800.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
| 6.6635520000.12t201.a.a | $6$ | $ 2^{17} \cdot 3^{4} \cdot 5^{4}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
| 6.14929920000.12t202.a.a | $6$ | $ 2^{15} \cdot 3^{6} \cdot 5^{4}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
| * | 6.14929920000.8t47.a.a | $6$ | $ 2^{15} \cdot 3^{6} \cdot 5^{4}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
| 6.26542080000.12t200.a.a | $6$ | $ 2^{19} \cdot 3^{4} \cdot 5^{4}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
| 9.238...000.16t1294.a.a | $9$ | $ 2^{21} \cdot 3^{6} \cdot 5^{6}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
| 9.238...000.18t272.a.a | $9$ | $ 2^{21} \cdot 3^{6} \cdot 5^{6}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.764...000.18t273.b.a | $9$ | $ 2^{26} \cdot 3^{6} \cdot 5^{6}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.305...000.18t274.b.a | $9$ | $ 2^{28} \cdot 3^{6} \cdot 5^{6}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
| 12.396...000.36t1763.a.a | $12$ | $ 2^{34} \cdot 3^{10} \cdot 5^{8}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
| 12.990...000.24t2821.a.a | $12$ | $ 2^{32} \cdot 3^{10} \cdot 5^{8}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
| 18.262...000.36t1758.a.a | $18$ | $ 2^{51} \cdot 3^{14} \cdot 5^{12}$ | 8.2.119439360000.2 | $S_4\wr C_2$ (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 *.