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Properties

Label 8.0.7765332671265625.1
Degree $8$
Signature $[0, 4]$
Discriminant $7.765\times 10^{15}$
Root discriminant \(96.89\)
Ramified primes $5,89$
Class number $144$
Class group [2, 6, 12]
Galois group $Q_8$ (as 8T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 78*x^6 - 100*x^5 + 2541*x^4 - 7670*x^3 + 65772*x^2 - 36712*x + 1485296)
 
gp: K = bnfinit(y^8 - y^7 + 78*y^6 - 100*y^5 + 2541*y^4 - 7670*y^3 + 65772*y^2 - 36712*y + 1485296, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - x^7 + 78*x^6 - 100*x^5 + 2541*x^4 - 7670*x^3 + 65772*x^2 - 36712*x + 1485296);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - x^7 + 78*x^6 - 100*x^5 + 2541*x^4 - 7670*x^3 + 65772*x^2 - 36712*x + 1485296)
 

\( x^{8} - x^{7} + 78x^{6} - 100x^{5} + 2541x^{4} - 7670x^{3} + 65772x^{2} - 36712x + 1485296 \) 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:   \(7765332671265625\) \(\medspace = 5^{6}\cdot 89^{6}\) 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:  \(96.89\)
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:   \(5\), \(89\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is a CM field.
Reflex fields:  8.0.7765332671265625.1$^{8}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{20}a^{6}-\frac{1}{4}a^{5}-\frac{3}{10}a^{4}+\frac{1}{5}a^{3}+\frac{9}{20}a^{2}-\frac{1}{10}a+\frac{1}{5}$, $\frac{1}{800847072035240}a^{7}+\frac{5836312763973}{800847072035240}a^{6}-\frac{7426852614307}{100105884004405}a^{5}+\frac{16753299162772}{100105884004405}a^{4}+\frac{386333588283621}{800847072035240}a^{3}-\frac{6088416141048}{20021176800881}a^{2}+\frac{10006463012541}{100105884004405}a+\frac{49581642084244}{100105884004405}$ Copy content Toggle raw display

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

$C_{2}\times C_{6}\times C_{12}$, which has order $144$

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:   $\frac{62631}{428031572440}a^{7}+\frac{14980157}{428031572440}a^{6}-\frac{21599143}{214015786220}a^{5}+\frac{137040969}{53503946555}a^{4}-\frac{2810569893}{428031572440}a^{3}+\frac{8873281973}{214015786220}a^{2}-\frac{2392402829}{10700789311}a+\frac{29794446921}{53503946555}$, $\frac{28333}{727519640}a^{7}+\frac{46351}{727519640}a^{6}+\frac{813011}{363759820}a^{5}+\frac{99752}{90939955}a^{4}+\frac{47264721}{727519640}a^{3}-\frac{46800161}{363759820}a^{2}+\frac{5364641}{18187991}a-\frac{524800172}{90939955}$, $\frac{16210691021}{200211768008810}a^{7}+\frac{8343038787}{200211768008810}a^{6}+\frac{97597478492}{100105884004405}a^{5}-\frac{776469624109}{100105884004405}a^{4}+\frac{8947680086587}{200211768008810}a^{3}+\frac{10994582817893}{100105884004405}a^{2}+\frac{29520268516964}{20021176800881}a-\frac{395662214754241}{100105884004405}$ 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:  \( 162.044654479 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 162.044654479 \cdot 144}{2\cdot\sqrt{7765332671265625}}\cr\approx \mathstrut & 0.206351032396 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 78*x^6 - 100*x^5 + 2541*x^4 - 7670*x^3 + 65772*x^2 - 36712*x + 1485296)
 
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 + 78*x^6 - 100*x^5 + 2541*x^4 - 7670*x^3 + 65772*x^2 - 36712*x + 1485296, 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 + 78*x^6 - 100*x^5 + 2541*x^4 - 7670*x^3 + 65772*x^2 - 36712*x + 1485296);
 
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 + 78*x^6 - 100*x^5 + 2541*x^4 - 7670*x^3 + 65772*x^2 - 36712*x + 1485296);
 
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

\(\Q(\sqrt{89}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{445}) \), \(\Q(\sqrt{5}, \sqrt{89})\)

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 ${\href{/padicField/2.4.0.1}{4} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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 $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
\(89\) Copy content Toggle raw display 89.4.3.1$x^{4} + 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} + 89$$4$$1$$3$$C_4$$[\ ]_{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 1.445.2t1.a.a$1$ $ 5 \cdot 89 $ \(\Q(\sqrt{445}) \) $C_2$ (as 2T1) $1$ $1$
* 1.89.2t1.a.a$1$ $ 89 $ \(\Q(\sqrt{89}) \) $C_2$ (as 2T1) $1$ $1$
*2 2.198025.8t5.a.a$2$ $ 5^{2} \cdot 89^{2}$ 8.0.7765332671265625.1 $Q_8$ (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 *.