# Properties

 Label 8.8.7644119040000.1 Degree $8$ Signature $[8, 0]$ Discriminant $7.644\times 10^{12}$ Root discriminant $$40.78$$ Ramified primes $2,3,5$ Class number $2$ Class group [2] 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 - 60*x^6 + 900*x^4 - 4500*x^2 + 5625)

gp: K = bnfinit(y^8 - 60*y^6 + 900*y^4 - 4500*y^2 + 5625, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 60*x^6 + 900*x^4 - 4500*x^2 + 5625);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 60*x^6 + 900*x^4 - 4500*x^2 + 5625)

$$x^{8} - 60x^{6} + 900x^{4} - 4500x^{2} + 5625$$

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: $[8, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$7644119040000$$ 7644119040000 $$\medspace = 2^{24}\cdot 3^{6}\cdot 5^{4}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$40.78$$ 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: $$2$$, $$3$$, $$5$$ 2, 3, 5 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 not a CM field.

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

$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{75}a^{4}$, $\frac{1}{75}a^{5}$, $\frac{1}{375}a^{6}$, $\frac{1}{375}a^{7}$

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

$C_{2}$, 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: $7$ 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: $\frac{1}{375}a^{6}-\frac{11}{75}a^{4}+\frac{8}{5}a^{2}-3$, $\frac{1}{375}a^{6}-\frac{2}{15}a^{4}+a^{2}+1$, $\frac{1}{375}a^{6}-\frac{11}{75}a^{4}+\frac{9}{5}a^{2}-6$, $\frac{4}{375}a^{6}+\frac{1}{75}a^{5}-\frac{41}{75}a^{4}-\frac{3}{5}a^{3}+5a^{2}+3a-10$, $\frac{1}{375}a^{7}-\frac{1}{75}a^{6}-\frac{2}{15}a^{5}+\frac{53}{75}a^{4}+a^{3}-\frac{36}{5}a^{2}+a+14$, $\frac{1}{125}a^{6}-\frac{31}{75}a^{4}+4a^{2}-a-8$, $\frac{2}{375}a^{7}+\frac{4}{375}a^{6}-\frac{22}{75}a^{5}-\frac{43}{75}a^{4}+\frac{17}{5}a^{3}+\frac{31}{5}a^{2}-10a-16$ 1/375*a^6 - 11/75*a^4 + 8/5*a^2 - 3, 1/375*a^6 - 2/15*a^4 + a^2 + 1, 1/375*a^6 - 11/75*a^4 + 9/5*a^2 - 6, 4/375*a^6 + 1/75*a^5 - 41/75*a^4 - 3/5*a^3 + 5*a^2 + 3*a - 10, 1/375*a^7 - 1/75*a^6 - 2/15*a^5 + 53/75*a^4 + a^3 - 36/5*a^2 + a + 14, 1/125*a^6 - 31/75*a^4 + 4*a^2 - a - 8, 2/375*a^7 + 4/375*a^6 - 22/75*a^5 - 43/75*a^4 + 17/5*a^3 + 31/5*a^2 - 10*a - 16 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: $$4900.6997022$$ 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^{8}\cdot(2\pi)^{0}\cdot 4900.6997022 \cdot 2}{2\cdot\sqrt{7644119040000}}\cr\approx \mathstrut & 0.45376849094 \end{aligned}

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

x = polygen(QQ); K.<a> = NumberField(x^8 - 60*x^6 + 900*x^4 - 4500*x^2 + 5625)

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 - 60*x^6 + 900*x^4 - 4500*x^2 + 5625, 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 - 60*x^6 + 900*x^4 - 4500*x^2 + 5625);

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 - 60*x^6 + 900*x^4 - 4500*x^2 + 5625);

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 R ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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 $$2$$ 2.8.24.12$x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 14$$8$$1$$24$$Q_8$$[2, 3, 4] $$3$$ 3.8.6.1x^{8} + 9$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$$$5$$ 5.4.2.2$x^{4} - 20 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2} 5.4.2.2x^{4} - 20 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$* 1.12.2t1.a.a$1 2^{2} \cdot 3 $$$\Q(\sqrt{3})$$$C_2$(as 2T1)$11$* 1.24.2t1.a.a$1 2^{3} \cdot 3 $$$\Q(\sqrt{6})$$$C_2$(as 2T1)$11$* 1.8.2t1.a.a$1 2^{3}$$$\Q(\sqrt{2})$$$C_2$(as 2T1)$11$*2 2.57600.8t5.e.a$2 2^{8} \cdot 3^{2} \cdot 5^{2}$8.8.7644119040000.1$Q_8$(as 8T5)$-12\$

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 *.