↑

Properties

Label 8.0.80062677450625.1
Degree $8$
Signature $[0, 4]$
Discriminant $5^{4}\cdot 71^{6}$
Root discriminant $54.69$
Ramified primes $5, 71$
Class number $8$
Class group $[2, 4]$
Galois group $C_2^3:C_7$ (as 8T25)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225, 755, 751, 143, 13, 11, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 3*x^6 + 11*x^5 + 13*x^4 + 143*x^3 + 751*x^2 + 755*x + 225)
 
gp: K = bnfinit(x^8 - 2*x^7 + 3*x^6 + 11*x^5 + 13*x^4 + 143*x^3 + 751*x^2 + 755*x + 225, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 3 x^{6} + 11 x^{5} + 13 x^{4} + 143 x^{3} + 751 x^{2} + 755 x + 225 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(80062677450625=5^{4}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.69$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{4124105} a^{7} - \frac{387701}{4124105} a^{6} + \frac{35067}{4124105} a^{5} + \frac{1733363}{4124105} a^{4} - \frac{191974}{4124105} a^{3} + \frac{405034}{4124105} a^{2} - \frac{370807}{824821} a + \frac{151870}{824821}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $3$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2792.07075957 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_8$ (as 8T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 56
The 8 conjugacy class representatives for $C_2^3:C_7$
Character table for $C_2^3:C_7$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 14 sibling: data not computed
Degree 28 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.71.7t1.1c1$1$ $ 71 $ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
1.71.7t1.1c2$1$ $ 71 $ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
1.71.7t1.1c3$1$ $ 71 $ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
1.71.7t1.1c4$1$ $ 71 $ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
1.71.7t1.1c5$1$ $ 71 $ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
1.71.7t1.1c6$1$ $ 71 $ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
* 7.5e4_71e6.8t25.1c1$7$ $ 5^{4} \cdot 71^{6}$ $x^{8} - 2 x^{7} + 3 x^{6} + 11 x^{5} + 13 x^{4} + 143 x^{3} + 751 x^{2} + 755 x + 225$ $C_2^3:C_7$ (as 8T25) $1$ $-1$

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 *.