# Properties

 Label 8.0.359729184374784.1 Degree $8$ Signature $[0, 4]$ Discriminant $3.597\times 10^{14}$ Root discriminant $$65.99$$ Ramified primes $2,3,7$ Class number $72$ Class group [2, 6, 6] Galois group $Q_8$ (as 8T5)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 84*x^6 + 1890*x^4 + 10584*x^2 + 1764)

gp: K = bnfinit(y^8 + 84*y^6 + 1890*y^4 + 10584*y^2 + 1764, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 + 84*x^6 + 1890*x^4 + 10584*x^2 + 1764);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 + 84*x^6 + 1890*x^4 + 10584*x^2 + 1764)

$$x^{8} + 84x^{6} + 1890x^{4} + 10584x^{2} + 1764$$

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: $$359729184374784$$ 359729184374784 $$\medspace = 2^{22}\cdot 3^{6}\cdot 7^{6}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$65.99$$ 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)) Ramified primes: $$2$$, $$3$$, $$7$$ 2, 3, 7 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q$$ $\card{ \Gal(K/\Q) }$: $8$ sage: K.automorphisms()  magma: Automorphisms(K);  oscar: automorphisms(K) This field is Galois over $\Q$. This is a CM field. Reflex fields: 8.0.359729184374784.1$^{8}$

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{210}a^{4}-\frac{2}{5}$, $\frac{1}{210}a^{5}-\frac{2}{5}a$, $\frac{1}{5250}a^{6}+\frac{1}{750}a^{4}+\frac{53}{125}a^{2}+\frac{46}{125}$, $\frac{1}{5250}a^{7}+\frac{1}{750}a^{5}+\frac{53}{125}a^{3}+\frac{46}{125}a$

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

$C_{2}\times C_{6}\times C_{6}$, which has order $72$

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: $$-1$$ -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: $\frac{1}{5250}a^{6}+\frac{107}{5250}a^{4}+\frac{53}{125}a^{2}+\frac{96}{125}$, $\frac{1}{875}a^{6}+\frac{467}{5250}a^{4}+\frac{193}{125}a^{2}+\frac{51}{125}$, $\frac{1}{210}a^{4}-\frac{2}{5}$ 1/5250*a^6 + 107/5250*a^4 + 53/125*a^2 + 96/125, 1/875*a^6 + 467/5250*a^4 + 193/125*a^2 + 51/125, 1/210*a^4 - 2/5 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: $$97.698846762$$ 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 97.698846762 \cdot 72}{2\cdot\sqrt{359729184374784}}\cr\approx \mathstrut & 0.28901712397 \end{aligned}

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

x = polygen(QQ); K.<a> = NumberField(x^8 + 84*x^6 + 1890*x^4 + 10584*x^2 + 1764)

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 + 84*x^6 + 1890*x^4 + 10584*x^2 + 1764, 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 + 84*x^6 + 1890*x^4 + 10584*x^2 + 1764);

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 + 84*x^6 + 1890*x^4 + 10584*x^2 + 1764);

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

$Q_8$ (as 8T5):

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 8 The 5 conjugacy class representatives for $Q_8$ Character table for $Q_8$

## 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]

## 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 ${\href{/padicField/5.2.0.1}{2} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.1.0.1}{1} }^{8}$ ${\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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$LabelPolynomial$efc$Galois group Slope content $$2$$ 2.8.22.4$x^{8} + 16 x^{7} + 68 x^{6} + 24 x^{5} + 48 x^{4} + 128 x^{3} + 568 x^{2} + 336 x + 588$$4$$2$$22$$Q_8$$[3, 4]^{2} $$3$$ 3.8.6.1x^{8} + 9$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$$$7$$ 7.8.6.1$x^{8} + 14 x^{4} - 245$$4$$2$$6$$Q_8[\ ]_{4}^{2}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$* 1.21.2t1.a.a$1 3 \cdot 7 $$$\Q(\sqrt{21})$$$C_2$(as 2T1)$11$* 1.24.2t1.a.a$1 2^{3} \cdot 3 $$$\Q(\sqrt{6})$$$C_2$(as 2T1)$11$* 1.56.2t1.a.a$1 2^{3} \cdot 7 $$$\Q(\sqrt{14})$$$C_2$(as 2T1)$11$*2 2.112896.8t5.h.a$2 2^{8} \cdot 3^{2} \cdot 7^{2}$8.0.359729184374784.1$Q_8$(as 8T5)$-1-2\$

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 *.