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Properties

Label 8.0.4543718560000.10
Degree $8$
Signature $[0, 4]$
Discriminant $4.544\times 10^{12}$
Root discriminant \(38.21\)
Ramified primes $2,5,73$
Class number $4$
Class group [4]
Galois group $C_2^3:(C_7: C_3)$ (as 8T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^6 - 8*x^5 - 16*x^4 - 2*x^3 + 58*x^2 + 88*x + 171)
 
gp: K = bnfinit(y^8 - 2*y^6 - 8*y^5 - 16*y^4 - 2*y^3 + 58*y^2 + 88*y + 171, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^6 - 8*x^5 - 16*x^4 - 2*x^3 + 58*x^2 + 88*x + 171);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^6 - 8*x^5 - 16*x^4 - 2*x^3 + 58*x^2 + 88*x + 171)
 

\( x^{8} - 2x^{6} - 8x^{5} - 16x^{4} - 2x^{3} + 58x^{2} + 88x + 171 \) 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:   \(4543718560000\) \(\medspace = 2^{8}\cdot 5^{4}\cdot 73^{4}\) 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:  \(38.21\)
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:  $2^{31/28}5^{1/2}73^{2/3}\approx 84.13630819106092$
Ramified primes:   \(2\), \(5\), \(73\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{31273}a^{7}-\frac{5174}{31273}a^{6}+\frac{586}{31273}a^{5}+\frac{1509}{31273}a^{4}+\frac{10668}{31273}a^{3}+\frac{611}{31273}a^{2}-\frac{2683}{31273}a-\frac{3282}{31273}$ 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

$C_{4}$, which has order $4$

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:   $\frac{3517}{31273}a^{7}+\frac{3928}{31273}a^{6}-\frac{3056}{31273}a^{5}-\frac{9257}{31273}a^{4}-\frac{39517}{31273}a^{3}-\frac{71496}{31273}a^{2}+\frac{39608}{31273}a+\frac{28216}{31273}$, $\frac{938}{31273}a^{7}-\frac{5897}{31273}a^{6}-\frac{13246}{31273}a^{5}+\frac{8157}{31273}a^{4}+\frac{61770}{31273}a^{3}+\frac{104023}{31273}a^{2}+\frac{79005}{31273}a-\frac{107581}{31273}$, $\frac{77}{2843}a^{7}-\frac{378}{2843}a^{6}-\frac{366}{2843}a^{5}-\frac{370}{2843}a^{4}-\frac{191}{2843}a^{3}+\frac{10088}{2843}a^{2}+\frac{6634}{2843}a+\frac{23057}{2843}$ 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:  \( 751.513061741 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 751.513061741 \cdot 4}{2\cdot\sqrt{4543718560000}}\cr\approx \mathstrut & 1.09895596539 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^6 - 8*x^5 - 16*x^4 - 2*x^3 + 58*x^2 + 88*x + 171)
 
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^6 - 8*x^5 - 16*x^4 - 2*x^3 + 58*x^2 + 88*x + 171, 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^6 - 8*x^5 - 16*x^4 - 2*x^3 + 58*x^2 + 88*x + 171);
 
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^6 - 8*x^5 - 16*x^4 - 2*x^3 + 58*x^2 + 88*x + 171);
 
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

$F_8:C_3$ (as 8T36):

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 168
The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$
Character table for $C_2^3:(C_7: C_3)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 14 sibling: deg 14
Degree 24 sibling: deg 24
Degree 28 sibling: deg 28
Degree 42 sibling: deg 42
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ R ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(2\) Copy content Toggle raw display 2.8.8.13$x^{8} + 2 x + 2$$8$$1$$8$$C_2^3:(C_7: C_3)$$[8/7, 8/7, 8/7]_{7}^{3}$
\(5\) Copy content Toggle raw display 5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.2$x^{6} + 75 x^{2} - 375$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(73\) Copy content Toggle raw display $\Q_{73}$$x + 68$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 68$$1$$1$$0$Trivial$[\ ]$
73.3.2.1$x^{3} + 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} + 73$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.73.3t1.a.a$1$ $ 73 $ 3.3.5329.1 $C_3$ (as 3T1) $0$ $1$
1.73.3t1.a.b$1$ $ 73 $ 3.3.5329.1 $C_3$ (as 3T1) $0$ $1$
3.42632.7t3.a.a$3$ $ 2^{3} \cdot 73^{2}$ 7.7.1817487424.1 $C_7:C_3$ (as 7T3) $0$ $3$
3.42632.7t3.a.b$3$ $ 2^{3} \cdot 73^{2}$ 7.7.1817487424.1 $C_7:C_3$ (as 7T3) $0$ $3$
* 7.454...000.8t36.b.a$7$ $ 2^{8} \cdot 5^{4} \cdot 73^{4}$ 8.0.4543718560000.10 $C_2^3:(C_7: C_3)$ (as 8T36) $1$ $-1$
7.331...000.24t283.b.a$7$ $ 2^{8} \cdot 5^{4} \cdot 73^{5}$ 8.0.4543718560000.10 $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$
7.331...000.24t283.b.b$7$ $ 2^{8} \cdot 5^{4} \cdot 73^{5}$ 8.0.4543718560000.10 $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$

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 *.