Properties

Label 8.0.195...664.1
Degree $8$
Signature $[0, 4]$
Discriminant $1.950\times 10^{34}$
Root discriminant $19{,}331.23$
Ramified primes $2, 51473$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $A_8$ (as 8T49)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 28*x^6 - 112*x^5 - 210*x^4 - 224*x^3 - 140*x^2 - 48*x + 823561)
 
gp: K = bnfinit(x^8 - 28*x^6 - 112*x^5 - 210*x^4 - 224*x^3 - 140*x^2 - 48*x + 823561, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![823561, -48, -140, -224, -210, -112, -28, 0, 1]);
 

\(x^{8} - 28 x^{6} - 112 x^{5} - 210 x^{4} - 224 x^{3} - 140 x^{2} - 48 x + 823561\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(19501894337558159417628591379185664\)\(\medspace = 2^{20}\cdot 51473^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19{,}331.23$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 51473$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{80} a^{7} + \frac{1}{80} a^{6} + \frac{3}{80} a^{5} - \frac{9}{80} a^{4} + \frac{11}{80} a^{3} - \frac{13}{80} a^{2} - \frac{23}{80} a + \frac{29}{80}$  Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

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

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 73355994708200 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 73355994708200 \cdot 2}{2\sqrt{19501894337558159417628591379185664}}\approx 0.818684719333306$ (assuming GRH)

Galois group

$A_8$ (as 8T49):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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 $\Q$.

Sibling fields

Degree 15 siblings: Deg 15, Deg 15
Degree 28 sibling: Deg 28
Degree 35 sibling: Deg 35

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.20.79$x^{8} + 8 x^{7} + 16$$8$$1$$20$$C_2^4:C_6$$[2, 2, 2, 3, 3]^{3}$
$51473$Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 7.195...664.8t49.a.a$7$ $ 2^{20} \cdot 51473^{6}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $1$ $-1$
14.232...544.15t72.a.a$14$ $ 2^{26} \cdot 51473^{12}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $1$ $6$
20.452...216.28t433.a.a$20$ $ 2^{46} \cdot 51473^{18}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $1$ $4$
21.741...944.56.a.a$21$ $ 2^{60} \cdot 51473^{18}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $1$ $-3$
21.741...944.336.a.a$21$ $ 2^{60} \cdot 51473^{18}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $0$ $-3$
21.741...944.336.a.b$21$ $ 2^{60} \cdot 51473^{18}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $0$ $-3$
28.144...816.56.a.a$28$ $ 2^{80} \cdot 51473^{24}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $1$ $-4$
35.110...304.70.a.a$35$ $ 2^{92} \cdot 51473^{30}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $1$ $3$
45.480...168.336.a.a$45$ $ 2^{126} \cdot 51473^{39}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $0$ $-3$
45.480...168.336.a.b$45$ $ 2^{126} \cdot 51473^{39}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $0$ $-3$
56.498...664.105.a.a$56$ $ 2^{138} \cdot 51473^{48}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $1$ $8$
64.159...064.168.a.a$64$ $ 2^{172} \cdot 51473^{54}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (as 8T49) $1$ $0$
70.310...496.120.a.a$70$ $ 2^{192} \cdot 51473^{60}$ 8.0.19501894337558159417628591379185664.1 $A_8$ (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 *.