# Properties

 Label 8.4.273318794910736.1 Degree $8$ Signature $[4, 2]$ Discriminant $2.733\times 10^{14}$ Root discriminant $$63.77$$ Ramified primes $2,19,107$ Class number $1$ (GRH) Class group trivial (GRH) Galois group $S_4\times C_2$ (as 8T24)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - x^6 + 94*x^5 + 89*x^4 - 272*x^3 - 97*x^2 + 148*x - 32)

gp: K = bnfinit(y^8 - 2*y^7 - y^6 + 94*y^5 + 89*y^4 - 272*y^3 - 97*y^2 + 148*y - 32, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - x^6 + 94*x^5 + 89*x^4 - 272*x^3 - 97*x^2 + 148*x - 32);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 2*x^7 - x^6 + 94*x^5 + 89*x^4 - 272*x^3 - 97*x^2 + 148*x - 32)

$$x^{8} - 2x^{7} - x^{6} + 94x^{5} + 89x^{4} - 272x^{3} - 97x^{2} + 148x - 32$$

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: $$273318794910736$$ 273318794910736 $$\medspace = 2^{4}\cdot 19^{4}\cdot 107^{4}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$63.77$$ 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: $$2$$, $$19$$, $$107$$ 2, 19, 107 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{9252}a^{7}-\frac{1129}{9252}a^{6}+\frac{58}{2313}a^{5}+\frac{578}{2313}a^{4}+\frac{3529}{9252}a^{3}+\frac{905}{9252}a^{2}+\frac{1157}{4626}a-\frac{823}{2313}$

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

Trivial group, which has order $1$ (assuming GRH)

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: $$-1$$ -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: $\frac{946}{2313}a^{7}-\frac{5795}{4626}a^{6}+\frac{2050}{2313}a^{5}+\frac{86948}{2313}a^{4}-\frac{8477}{2313}a^{3}-\frac{515159}{4626}a^{2}+\frac{172108}{2313}a-\frac{31003}{2313}$, $\frac{3031}{2313}a^{7}-\frac{14950}{2313}a^{6}+\frac{39361}{2313}a^{5}+\frac{177383}{2313}a^{4}-\frac{271847}{2313}a^{3}-\frac{106561}{2313}a^{2}+\frac{201949}{2313}a-\frac{89977}{2313}$, $\frac{347}{514}a^{7}-\frac{609}{514}a^{6}-\frac{451}{514}a^{5}+\frac{16403}{257}a^{4}+\frac{39279}{514}a^{3}-\frac{85857}{514}a^{2}-\frac{56707}{514}a+\frac{19167}{257}$, $\frac{183485}{9252}a^{7}-\frac{284471}{9252}a^{6}-\frac{78625}{2313}a^{5}+\frac{4277704}{2313}a^{4}+\frac{24026285}{9252}a^{3}-\frac{39391361}{9252}a^{2}-\frac{17971631}{4626}a+\frac{2880361}{2313}$, $\frac{438839}{1542}a^{7}-\frac{354277}{771}a^{6}-\frac{711535}{1542}a^{5}+\frac{20485400}{771}a^{4}+\frac{54858041}{1542}a^{3}-\frac{49094839}{771}a^{2}-\frac{80846149}{1542}a+\frac{16699949}{771}$ 946/2313*a^7 - 5795/4626*a^6 + 2050/2313*a^5 + 86948/2313*a^4 - 8477/2313*a^3 - 515159/4626*a^2 + 172108/2313*a - 31003/2313, 3031/2313*a^7 - 14950/2313*a^6 + 39361/2313*a^5 + 177383/2313*a^4 - 271847/2313*a^3 - 106561/2313*a^2 + 201949/2313*a - 89977/2313, 347/514*a^7 - 609/514*a^6 - 451/514*a^5 + 16403/257*a^4 + 39279/514*a^3 - 85857/514*a^2 - 56707/514*a + 19167/257, 183485/9252*a^7 - 284471/9252*a^6 - 78625/2313*a^5 + 4277704/2313*a^4 + 24026285/9252*a^3 - 39391361/9252*a^2 - 17971631/4626*a + 2880361/2313, 438839/1542*a^7 - 354277/771*a^6 - 711535/1542*a^5 + 20485400/771*a^4 + 54858041/1542*a^3 - 49094839/771*a^2 - 80846149/1542*a + 16699949/771 (assuming GRH) 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: $$262631.841706$$ (assuming GRH) 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 262631.841706 \cdot 1}{2\cdot\sqrt{273318794910736}}\cr\approx \mathstrut & 5.01721086726 \end{aligned} (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - x^6 + 94*x^5 + 89*x^4 - 272*x^3 - 97*x^2 + 148*x - 32)

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 - x^6 + 94*x^5 + 89*x^4 - 272*x^3 - 97*x^2 + 148*x - 32, 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 - x^6 + 94*x^5 + 89*x^4 - 272*x^3 - 97*x^2 + 148*x - 32);

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 - x^6 + 94*x^5 + 89*x^4 - 272*x^3 - 97*x^2 + 148*x - 32);

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

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: 8.0.757115775376.1 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 ${\href{/padicField/3.2.0.1}{2} }^{4}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ R ${\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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]])