Properties

Label 8.0.757115775376.1
Degree $8$
Signature $[0, 4]$
Discriminant $757115775376$
Root discriminant \(30.54\)
Ramified primes $2,19,107$
Class number $3$
Class group [3]
Galois group $S_4\times C_2$ (as 8T24)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 5*x^6 + 8*x^5 - 20*x^4 + 24*x^3 + 45*x^2 - 27*x + 81)
 
gp: K = bnfinit(y^8 - y^7 + 5*y^6 + 8*y^5 - 20*y^4 + 24*y^3 + 45*y^2 - 27*y + 81, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - x^7 + 5*x^6 + 8*x^5 - 20*x^4 + 24*x^3 + 45*x^2 - 27*x + 81);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - x^7 + 5*x^6 + 8*x^5 - 20*x^4 + 24*x^3 + 45*x^2 - 27*x + 81)
 

\( x^{8} - x^{7} + 5x^{6} + 8x^{5} - 20x^{4} + 24x^{3} + 45x^{2} - 27x + 81 \) Copy content Toggle raw display

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:   \(757115775376\) \(\medspace = 2^{4}\cdot 19^{2}\cdot 107^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(30.54\)
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:  $2^{2/3}19^{1/2}107^{1/2}\approx 71.57401056858944$
Ramified primes:   \(2\), \(19\), \(107\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{1}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{9}a^{2}$, $\frac{1}{1782}a^{7}-\frac{47}{891}a^{6}-\frac{163}{1782}a^{5}-\frac{277}{1782}a^{4}-\frac{395}{1782}a^{3}-\frac{23}{594}a^{2}-\frac{4}{99}a-\frac{17}{66}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

$C_{3}$, which has order $3$

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:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{1}{891}a^{7}+\frac{5}{891}a^{6}+\frac{35}{891}a^{5}-\frac{79}{891}a^{4}+\frac{100}{891}a^{3}+\frac{109}{297}a^{2}-\frac{8}{99}a+\frac{16}{33}$, $\frac{67}{891}a^{7}-\frac{160}{891}a^{6}-\frac{130}{891}a^{5}-\frac{640}{891}a^{4}-\frac{5279}{891}a^{3}-\frac{2168}{297}a^{2}-\frac{437}{99}a-\frac{248}{33}$, $\frac{200}{297}a^{7}+\frac{340}{297}a^{6}+\frac{1225}{297}a^{5}+\frac{4660}{297}a^{4}+\frac{2246}{297}a^{3}+\frac{1549}{99}a^{2}+\frac{358}{33}a+\frac{21}{11}$ Copy content Toggle raw display
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:  \( 934.707000099 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 934.707000099 \cdot 3}{2\cdot\sqrt{757115775376}}\cr\approx \mathstrut & 2.51133749018 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 5*x^6 + 8*x^5 - 20*x^4 + 24*x^3 + 45*x^2 - 27*x + 81)
 
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 - x^7 + 5*x^6 + 8*x^5 - 20*x^4 + 24*x^3 + 45*x^2 - 27*x + 81, 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 - x^7 + 5*x^6 + 8*x^5 - 20*x^4 + 24*x^3 + 45*x^2 - 27*x + 81);
 
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 - x^7 + 5*x^6 + 8*x^5 - 20*x^4 + 24*x^3 + 45*x^2 - 27*x + 81);
 
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

$C_2\times S_4$ (as 8T24):

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 48
The 10 conjugacy class representatives for $S_4\times C_2$
Character table for $S_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-107}) \), 4.2.870124.1

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 6 siblings: 6.0.618032.1, 6.2.11742608.1
Degree 8 sibling: 8.4.273318794910736.1
Degree 12 siblings: deg 12, deg 12, deg 12, deg 12, deg 12, deg 12
Degree 16 sibling: deg 16
Degree 24 siblings: deg 24, deg 24, deg 24, deg 24
Minimal sibling: 6.0.618032.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ R ${\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(19\) Copy content Toggle raw display 19.2.1.1$x^{2} + 38$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} + 38$$2$$1$$1$$C_2$$[\ ]_{2}$
\(107\) Copy content Toggle raw display 107.8.4.1$x^{8} + 454 x^{6} + 158 x^{5} + 71649 x^{4} - 31758 x^{3} + 4642389 x^{2} - 5206732 x + 101871387$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$