Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 3*x^6 + 15*x^5 - 6*x^4 + 12*x^3 - 33*x^2 - 82*x + 79)
gp: K = bnfinit(y^8 - 2*y^7 - 3*y^6 + 15*y^5 - 6*y^4 + 12*y^3 - 33*y^2 - 82*y + 79, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 3*x^6 + 15*x^5 - 6*x^4 + 12*x^3 - 33*x^2 - 82*x + 79);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^7 - 3*x^6 + 15*x^5 - 6*x^4 + 12*x^3 - 33*x^2 - 82*x + 79)
\( x^{8} - 2x^{7} - 3x^{6} + 15x^{5} - 6x^{4} + 12x^{3} - 33x^{2} - 82x + 79 \)
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: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(44486028125\)
\(\medspace = 5^{5}\cdot 7^{6}\cdot 11^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.43\) | 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: | $5^{3/4}7^{3/4}11^{1/2}\approx 47.725158108626026$ | ||
| Ramified primes: |
\(5\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{212133}a^{7}-\frac{24620}{212133}a^{6}+\frac{101887}{212133}a^{5}+\frac{2147}{70711}a^{4}-\frac{30482}{212133}a^{3}+\frac{20756}{212133}a^{2}-\frac{13555}{212133}a+\frac{11699}{212133}$
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: | $5$ | 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: |
$\frac{517}{70711}a^{7}-\frac{560}{70711}a^{6}-\frac{4116}{70711}a^{5}+\frac{6580}{70711}a^{4}+\frac{9359}{70711}a^{3}-\frac{17220}{70711}a^{2}-\frac{7546}{70711}a-\frac{32763}{70711}$, $\frac{4064}{212133}a^{7}+\frac{385}{212133}a^{6}-\frac{14848}{212133}a^{5}+\frac{13154}{212133}a^{4}+\frac{25845}{70711}a^{3}+\frac{206294}{212133}a^{2}-\frac{145073}{212133}a-\frac{326611}{212133}$, $\frac{2207}{70711}a^{7}-\frac{20165}{212133}a^{6}+\frac{3629}{70711}a^{5}+\frac{77839}{212133}a^{4}-\frac{153550}{212133}a^{3}+\frac{316847}{212133}a^{2}-\frac{146554}{70711}a+\frac{171956}{212133}$, $\frac{2293}{70711}a^{7}-\frac{8135}{212133}a^{6}-\frac{2253}{70711}a^{5}+\frac{42553}{212133}a^{4}+\frac{43148}{212133}a^{3}+\frac{156437}{212133}a^{2}+\frac{31225}{70711}a-\frac{203830}{212133}$, $\frac{12146}{212133}a^{7}+\frac{2299}{212133}a^{6}-\frac{64420}{212133}a^{5}+\frac{96731}{212133}a^{4}+\frac{73519}{70711}a^{3}+\frac{371216}{212133}a^{2}+\frac{612577}{212133}a-\frac{810877}{212133}$
| 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: | \( 245.026718105 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 245.026718105 \cdot 1}{2\cdot\sqrt{44486028125}}\cr\approx \mathstrut & 0.366903074326 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 3*x^6 + 15*x^5 - 6*x^4 + 12*x^3 - 33*x^2 - 82*x + 79)
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 - 3*x^6 + 15*x^5 - 6*x^4 + 12*x^3 - 33*x^2 - 82*x + 79, 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 - 3*x^6 + 15*x^5 - 6*x^4 + 12*x^3 - 33*x^2 - 82*x + 79);
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 - 3*x^6 + 15*x^5 - 6*x^4 + 12*x^3 - 33*x^2 - 82*x + 79);
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
$C_2\wr C_4$ (as 8T28):
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 64 |
| The 13 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.13475.1 |
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
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.6.20220921875.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | R | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(7\)
| 7.8.6.3 | $x^{8} - 154 x^{4} - 1421$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
|
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |