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Properties

Label 8.0.44231334895529.1
Degree $8$
Signature $[0, 4]$
Discriminant $4.423\times 10^{13}$
Root discriminant \(50.78\)
Ramified prime $89$
Class number $113$
Class group [113]
Galois group $C_8$ (as 8T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 6*x^6 - 46*x^5 - 143*x^4 + 575*x^3 + 1160*x^2 - 16*x + 512)
 
gp: K = bnfinit(y^8 - y^7 + 6*y^6 - 46*y^5 - 143*y^4 + 575*y^3 + 1160*y^2 - 16*y + 512, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - x^7 + 6*x^6 - 46*x^5 - 143*x^4 + 575*x^3 + 1160*x^2 - 16*x + 512);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - x^7 + 6*x^6 - 46*x^5 - 143*x^4 + 575*x^3 + 1160*x^2 - 16*x + 512)
 

\( x^{8} - x^{7} + 6x^{6} - 46x^{5} - 143x^{4} + 575x^{3} + 1160x^{2} - 16x + 512 \) 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:   \(44231334895529\) \(\medspace = 89^{7}\) 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:  \(50.78\)
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))
 
Galois root discriminant:  $89^{7/8}\approx 50.78275474951909$
Ramified primes:   \(89\) 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:  \(\Q(\sqrt{89}) \)
$\card{ \Gal(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(89\)
Dirichlet character group:    $\lbrace$$\chi_{89}(1,·)$, $\chi_{89}(34,·)$, $\chi_{89}(37,·)$, $\chi_{89}(12,·)$, $\chi_{89}(77,·)$, $\chi_{89}(52,·)$, $\chi_{89}(55,·)$, $\chi_{89}(88,·)$$\rbrace$
This is a CM field.
Reflex fields:  8.0.44231334895529.1$^{8}$

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{8}a^{4}-\frac{1}{8}a^{2}$, $\frac{1}{32}a^{5}-\frac{1}{32}a^{4}+\frac{3}{32}a^{3}+\frac{1}{32}a^{2}-\frac{1}{8}a$, $\frac{1}{128}a^{6}-\frac{3}{64}a^{4}-\frac{1}{32}a^{3}+\frac{5}{128}a^{2}+\frac{1}{32}a$, $\frac{1}{17152}a^{7}+\frac{61}{17152}a^{6}+\frac{85}{8576}a^{5}-\frac{113}{8576}a^{4}+\frac{1121}{17152}a^{3}+\frac{933}{17152}a^{2}+\frac{1117}{4288}a+\frac{41}{134}$ 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:  $2$

Class group and class number

$C_{113}$, which has order $113$

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:   $\frac{5}{4288}a^{7}+\frac{37}{4288}a^{6}+\frac{23}{2144}a^{5}+\frac{105}{2144}a^{4}-\frac{2167}{4288}a^{3}-\frac{9271}{4288}a^{2}+\frac{225}{1072}a-\frac{59}{67}$, $\frac{159}{8576}a^{7}-\frac{485}{8576}a^{6}+\frac{651}{4288}a^{5}-\frac{4567}{4288}a^{4}-\frac{8289}{8576}a^{3}+\frac{140307}{8576}a^{2}-\frac{7317}{2144}a-\frac{2660}{67}$, $\frac{83}{4288}a^{7}-\frac{115}{2144}a^{6}+\frac{355}{2144}a^{5}-\frac{293}{268}a^{4}-\frac{5313}{4288}a^{3}+\frac{34331}{2144}a^{2}-\frac{1851}{536}a-\frac{1033}{67}$ 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:  \( 2970.52387144 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2970.52387144 \cdot 113}{2\cdot\sqrt{44231334895529}}\cr\approx \mathstrut & 39.3310865934 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 6*x^6 - 46*x^5 - 143*x^4 + 575*x^3 + 1160*x^2 - 16*x + 512)
 
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 + 6*x^6 - 46*x^5 - 143*x^4 + 575*x^3 + 1160*x^2 - 16*x + 512, 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 + 6*x^6 - 46*x^5 - 143*x^4 + 575*x^3 + 1160*x^2 - 16*x + 512);
 
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 + 6*x^6 - 46*x^5 - 143*x^4 + 575*x^3 + 1160*x^2 - 16*x + 512);
 
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

$C_8$ (as 8T1):

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 cyclic group of order 8
The 8 conjugacy class representatives for $C_8$
Character table for $C_8$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.1.0.1}{1} }^{8}$ ${\href{/padicField/3.8.0.1}{8} }$ ${\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }$ ${\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.8.0.1}{8} }$ ${\href{/padicField/29.8.0.1}{8} }$ ${\href{/padicField/31.8.0.1}{8} }$ ${\href{/padicField/37.8.0.1}{8} }$ ${\href{/padicField/41.8.0.1}{8} }$ ${\href{/padicField/43.8.0.1}{8} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }$

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
\(89\) Copy content Toggle raw display 89.8.7.3$x^{8} + 89$$8$$1$$7$$C_8$$[\ ]_{8}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.89.8t1.a.a$1$ $ 89 $ 8.0.44231334895529.1 $C_8$ (as 8T1) $0$ $-1$
* 1.89.4t1.a.a$1$ $ 89 $ 4.4.704969.1 $C_4$ (as 4T1) $0$ $1$
* 1.89.8t1.a.b$1$ $ 89 $ 8.0.44231334895529.1 $C_8$ (as 8T1) $0$ $-1$
* 1.89.2t1.a.a$1$ $ 89 $ \(\Q(\sqrt{89}) \) $C_2$ (as 2T1) $1$ $1$
* 1.89.8t1.a.c$1$ $ 89 $ 8.0.44231334895529.1 $C_8$ (as 8T1) $0$ $-1$
* 1.89.4t1.a.b$1$ $ 89 $ 4.4.704969.1 $C_4$ (as 4T1) $0$ $1$
* 1.89.8t1.a.d$1$ $ 89 $ 8.0.44231334895529.1 $C_8$ (as 8T1) $0$ $-1$

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 *.