Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58)
gp: K = bnfinit(y^8 - 2*y^7 + 5*y^6 - 2*y^5 + 21*y^4 - 30*y^3 + 75*y^2 - 62*y + 58, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 2*x^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58)
\( x^{8} - 2x^{7} + 5x^{6} - 2x^{5} + 21x^{4} - 30x^{3} + 75x^{2} - 62x + 58 \)
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: |
\(1429145856\)
\(\medspace = 2^{8}\cdot 3^{4}\cdot 41^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.94\) | 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: | $2\cdot 3^{1/2}41^{1/2}\approx 22.181073012818835$ | ||
| Ramified primes: |
\(2\), \(3\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{1888}a^{7}+\frac{65}{1888}a^{6}+\frac{7}{118}a^{5}-\frac{25}{944}a^{4}+\frac{447}{1888}a^{3}+\frac{655}{1888}a^{2}+\frac{2}{59}a-\frac{247}{944}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
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: |
\( -\frac{51}{1888} a^{7} - \frac{11}{1888} a^{6} - \frac{3}{118} a^{5} - \frac{141}{944} a^{4} - \frac{1085}{1888} a^{3} - \frac{837}{1888} a^{2} - \frac{27}{118} a - \frac{1091}{944} \)
(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: |
$\frac{97}{1888}a^{7}-\frac{67}{1888}a^{6}+\frac{1}{236}a^{5}+\frac{171}{944}a^{4}+\frac{1823}{1888}a^{3}-\frac{893}{1888}a^{2}+\frac{9}{236}a+\frac{1293}{944}$, $\frac{5}{944}a^{7}-\frac{265}{944}a^{6}+\frac{81}{236}a^{5}-\frac{243}{472}a^{4}-\frac{597}{944}a^{3}-\frac{4631}{944}a^{2}+\frac{1319}{236}a-\frac{3005}{472}$, $\frac{327}{1888}a^{7}+\frac{15}{1888}a^{6}+\frac{47}{118}a^{5}+\frac{793}{944}a^{4}+\frac{7401}{1888}a^{3}+\frac{3201}{1888}a^{2}+\frac{777}{118}a+\frac{5607}{944}$
| 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: | \( 152.32463607 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 152.32463607 \cdot 2}{4\cdot\sqrt{1429145856}}\cr\approx \mathstrut & 3.1399400030 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58)
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^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58, 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^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58);
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^7 + 5*x^6 - 2*x^5 + 21*x^4 - 30*x^3 + 75*x^2 - 62*x + 58);
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 solvable group of order 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.0.656.1 |
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: | 16.0.3433371692450636169216.3 |
| Degree 8 sibling: | 8.2.14648745024.1 |
| 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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
|
\(3\)
| 3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
|
\(41\)
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.164.2t1.a.a | $1$ | $ 2^{2} \cdot 41 $ | \(\Q(\sqrt{-41}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.164.4t3.a.a | $2$ | $ 2^{2} \cdot 41 $ | 4.2.6724.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.1476.8t6.a.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 41 $ | 8.0.1429145856.1 | $D_{8}$ (as 8T6) | $1$ | $0$ |
| * | 2.1476.8t6.a.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 41 $ | 8.0.1429145856.1 | $D_{8}$ (as 8T6) | $1$ | $0$ |
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 *.