Properties

Label 8.0.133...856.1
Degree $8$
Signature $[0, 4]$
Discriminant $1.331\times 10^{35}$
Root discriminant $24{,}576.64$
Ramified primes $2, 7, 11, 191$
Class number $4$ (GRH)
Class group $[2, 2]$ (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 + 823585)
 
gp: K = bnfinit(x^8 - 28*x^6 - 112*x^5 - 210*x^4 - 224*x^3 - 140*x^2 - 48*x + 823585, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![823585, -48, -140, -224, -210, -112, -28, 0, 1]);
 

\(x^{8} - 28 x^{6} - 112 x^{5} - 210 x^{4} - 224 x^{3} - 140 x^{2} - 48 x + 823585\)  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:  \(133100753213221593424899389161209856\)\(\medspace = 2^{28}\cdot 7^{8}\cdot 11^{6}\cdot 191^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $24{,}576.64$
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:  $2, 7, 11, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{14} a^{5} - \frac{3}{14} a^{4} - \frac{1}{14} a^{3} - \frac{3}{14} a^{2} - \frac{3}{7} a$, $\frac{1}{28} a^{6} - \frac{3}{28} a^{4} - \frac{3}{14} a^{3} + \frac{13}{28} a^{2} + \frac{5}{14} a - \frac{1}{4}$, $\frac{1}{28} a^{7} - \frac{1}{28} a^{5} + \frac{1}{14} a^{4} - \frac{3}{28} a^{3} - \frac{5}{14} a^{2} - \frac{5}{28} a$  Toggle raw display

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 102952955516000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 102952955516000 \cdot 4}{2\sqrt{133100753213221593424899389161209856}}\approx 0.879625946084417$ (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 ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R R ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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$2.8.28.43$x^{8} + 8 x^{5} + 26$$8$$1$$28$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4]^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.7.8.3$x^{7} + 28 x^{2} + 7$$7$$1$$8$$C_7:C_3$$[4/3]_{3}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$191$$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
191.7.6.1$x^{7} - 191$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 7.133...856.8t49.a.a$7$ $ 2^{28} \cdot 7^{8} \cdot 11^{6} \cdot 191^{6}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $1$ $-1$
14.270...576.15t72.a.a$14$ $ 2^{40} \cdot 7^{16} \cdot 11^{12} \cdot 191^{12}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $1$ $6$
20.176...744.28t433.a.a$20$ $ 2^{68} \cdot 7^{26} \cdot 11^{18} \cdot 191^{18}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $1$ $4$
21.288...696.56.a.a$21$ $ 2^{82} \cdot 7^{26} \cdot 11^{18} \cdot 191^{18}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $1$ $-3$
21.288...696.336.a.a$21$ $ 2^{82} \cdot 7^{26} \cdot 11^{18} \cdot 191^{18}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $0$ $-3$
21.288...696.336.a.b$21$ $ 2^{82} \cdot 7^{26} \cdot 11^{18} \cdot 191^{18}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $0$ $-3$
28.384...776.56.a.a$28$ $ 2^{110} \cdot 7^{34} \cdot 11^{24} \cdot 191^{24}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $1$ $-4$
35.319...016.70.a.a$35$ $ 2^{134} \cdot 7^{42} \cdot 11^{30} \cdot 191^{30}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $1$ $3$
45.761...136.336.a.a$45$ $ 2^{176} \cdot 7^{56} \cdot 11^{39} \cdot 191^{39}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $0$ $-3$
45.761...136.336.a.b$45$ $ 2^{176} \cdot 7^{56} \cdot 11^{39} \cdot 191^{39}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $0$ $-3$
56.276...296.105.a.a$56$ $ 2^{202} \cdot 7^{70} \cdot 11^{48} \cdot 191^{48}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $1$ $8$
64.295...816.168.a.a$64$ $ 2^{244} \cdot 7^{80} \cdot 11^{54} \cdot 191^{54}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (as 8T49) $1$ $0$
70.801...704.120.a.a$70$ $ 2^{272} \cdot 7^{86} \cdot 11^{60} \cdot 191^{60}$ 8.0.133100753213221593424899389161209856.1 $A_8$ (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 *.