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Properties

Label 8.0.225...624.1
Degree $8$
Signature $[0, 4]$
Discriminant $2.253\times 10^{36}$
Root discriminant \(35\,001.66\)
Ramified primes $2,29,3917$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $A_8$ (as 8T49)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210826816)
 
gp: K = bnfinit(y^8 - 112*y^6 - 896*y^5 - 3360*y^4 - 7168*y^3 - 8960*y^2 - 6144*y + 210826816, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210826816);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210826816)
 

\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210826816 \) 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:   \(2252730971538337484304305478905626624\) \(\medspace = 2^{20}\cdot 29^{6}\cdot 3917^{6}\) 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:  \(35\,001.66\)
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:  $2^{23/8}29^{6/7}3917^{6/7}\approx 157986.40821975734$
Ramified primes:   \(2\), \(29\), \(3917\) 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 \(a\))

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}$, $\frac{1}{16}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{928}a^{5}-\frac{3}{464}a^{4}+\frac{7}{232}a^{3}-\frac{9}{116}a^{2}-\frac{25}{116}a+\frac{15}{58}$, $\frac{1}{928}a^{6}-\frac{1}{116}a^{4}+\frac{3}{29}a^{3}-\frac{21}{116}a^{2}-\frac{1}{29}a-\frac{13}{29}$, $\frac{1}{9280}a^{7}+\frac{1}{4640}a^{6}+\frac{1}{4640}a^{5}-\frac{53}{2320}a^{4}-\frac{39}{580}a^{3}-\frac{63}{290}a^{2}+\frac{193}{580}a-\frac{9}{290}$ 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

$C_{2}$, which has order $2$ (assuming GRH)

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:   $\frac{10\!\cdots\!51}{58}a^{7}+\frac{11\!\cdots\!15}{232}a^{6}-\frac{19\!\cdots\!57}{58}a^{5}-\frac{17\!\cdots\!21}{464}a^{4}+\frac{19\!\cdots\!28}{29}a^{3}+\frac{48\!\cdots\!39}{116}a^{2}+\frac{73\!\cdots\!20}{29}a-\frac{11\!\cdots\!57}{58}$, $\frac{37\!\cdots\!69}{116}a^{7}+\frac{17\!\cdots\!45}{464}a^{6}+\frac{96\!\cdots\!73}{928}a^{5}-\frac{19\!\cdots\!93}{116}a^{4}-\frac{70\!\cdots\!17}{232}a^{3}-\frac{22\!\cdots\!29}{58}a^{2}-\frac{54\!\cdots\!13}{116}a-\frac{16\!\cdots\!96}{29}$, $\frac{71\!\cdots\!27}{1160}a^{7}+\frac{83\!\cdots\!17}{580}a^{6}-\frac{27\!\cdots\!47}{2320}a^{5}-\frac{14\!\cdots\!39}{1160}a^{4}+\frac{16\!\cdots\!01}{580}a^{3}+\frac{40\!\cdots\!27}{290}a^{2}+\frac{23\!\cdots\!79}{290}a-\frac{10\!\cdots\!82}{145}$ Copy content Toggle raw display (assuming GRH)
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:  \( 816862092682000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 816862092682000 \cdot 2}{2\cdot\sqrt{2252730971538337484304305478905626624}}\cr\approx \mathstrut & 0.848229848985582 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^8 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210826816)
 
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 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210826816, 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 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210826816);
 
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 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210826816);
 
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

$A_8$ (as 8T49):

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 non-solvable group of order 20160
The 14 conjugacy class representatives for $A_8$
Character table for $A_8$

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 15 siblings: deg 15, deg 15
Degree 28 sibling: deg 28
Degree 35 sibling: deg 35
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 ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ R ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.8.20.81$x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{4} + 8 x^{3} + 12 x^{2} + 6$$8$$1$$20$$C_2^3 : C_4 $$[2, 2, 3, 7/2]^{2}$
\(29\) Copy content Toggle raw display $\Q_{29}$$x + 27$$1$$1$$0$Trivial$[\ ]$
29.7.6.3$x^{7} + 87$$7$$1$$6$$C_7$$[\ ]_{7}$
\(3917\) Copy content Toggle raw display $\Q_{3917}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $7$$7$$1$$6$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 7.225...624.8t49.a.a$7$ $ 2^{20} \cdot 29^{6} \cdot 3917^{6}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $-1$
14.123...056.15t72.a.a$14$ $ 2^{28} \cdot 29^{12} \cdot 3917^{12}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $6$
20.279...944.28t433.a.a$20$ $ 2^{48} \cdot 29^{18} \cdot 3917^{18}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $4$
21.182...984.56.a.a$21$ $ 2^{64} \cdot 29^{18} \cdot 3917^{18}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $-3$
21.182...984.336.a.a$21$ $ 2^{64} \cdot 29^{18} \cdot 3917^{18}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $0$ $-3$
21.182...984.336.a.b$21$ $ 2^{64} \cdot 29^{18} \cdot 3917^{18}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $0$ $-3$
28.412...016.56.a.a$28$ $ 2^{84} \cdot 29^{24} \cdot 3917^{24}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $-4$
35.580...624.70.a.a$35$ $ 2^{100} \cdot 29^{30} \cdot 3917^{30}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $3$
45.125...352.336.a.a$45$ $ 2^{136} \cdot 29^{39} \cdot 3917^{39}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $0$ $-3$
45.125...352.336.a.b$45$ $ 2^{136} \cdot 29^{39} \cdot 3917^{39}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $0$ $-3$
56.161...056.105.a.a$56$ $ 2^{148} \cdot 29^{48} \cdot 3917^{48}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $8$
64.239...984.168.a.a$64$ $ 2^{184} \cdot 29^{54} \cdot 3917^{54}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $0$
70.538...016.120.a.a$70$ $ 2^{204} \cdot 29^{60} \cdot 3917^{60}$ 8.0.2252730971538337484304305478905626624.1 $A_8$ (as 8T49) $1$ $-2$

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 *.