↑

Properties

Label 8.0.1781328125.1
Degree $8$
Signature $[0, 4]$
Discriminant $5^{7}\cdot 151^{2}$
Root discriminant $14.33$
Ramified primes $5, 151$
Class number $1$
Class group Trivial
Galois group $(C_8:C_2):C_2$ (as 8T16)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![451, -382, 453, -219, 140, -39, 18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 18*x^6 - 39*x^5 + 140*x^4 - 219*x^3 + 453*x^2 - 382*x + 451)
 
gp: K = bnfinit(x^8 - 2*x^7 + 18*x^6 - 39*x^5 + 140*x^4 - 219*x^3 + 453*x^2 - 382*x + 451, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 18 x^{6} - 39 x^{5} + 140 x^{4} - 219 x^{3} + 453 x^{2} - 382 x + 451 \)

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:  \(1781328125=5^{7}\cdot 151^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.33$
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:  $5, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5} - \frac{4}{11} a^{4} - \frac{1}{11} a^{3} - \frac{1}{11} a^{2} - \frac{5}{11} a$, $\frac{1}{121} a^{6} + \frac{4}{121} a^{5} + \frac{3}{11} a^{4} + \frac{2}{121} a^{3} - \frac{24}{121} a^{2} - \frac{18}{121} a - \frac{4}{11}$, $\frac{1}{1331} a^{7} + \frac{1}{1331} a^{6} + \frac{21}{1331} a^{5} + \frac{24}{1331} a^{4} + \frac{212}{1331} a^{3} + \frac{417}{1331} a^{2} + \frac{373}{1331} a - \frac{54}{121}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{13}{1331} a^{7} + \frac{42}{1331} a^{6} - \frac{174}{1331} a^{5} + \frac{656}{1331} a^{4} - \frac{1194}{1331} a^{3} + \frac{2697}{1331} a^{2} - \frac{2572}{1331} a + \frac{361}{121} \) (order $10$)
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:  \( 122.543627005 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 8T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 sibling: data not computed
Degree 16 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
151Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.151.2t1.1c1$1$ $ 151 $ $x^{2} - x + 38$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.5_151.2t1.1c1$1$ $ 5 \cdot 151 $ $x^{2} - x + 189$ $C_2$ (as 2T1) $1$ $-1$
1.5_151.4t1.1c1$1$ $ 5 \cdot 151 $ $x^{4} - x^{3} - 189 x^{2} + 189 x + 7031$ $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.1c1$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
1.5_151.4t1.1c2$1$ $ 5 \cdot 151 $ $x^{4} - x^{3} - 189 x^{2} + 189 x + 7031$ $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.1c2$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
2.5e2_151.4t3.2c1$2$ $ 5^{2} \cdot 151 $ $x^{4} - 2 x^{3} + 14 x^{2} - 13 x + 231$ $D_{4}$ (as 4T3) $1$ $0$
2.5_151.4t3.1c1$2$ $ 5 \cdot 151 $ $x^{4} - x^{3} + 7 x - 11$ $D_{4}$ (as 4T3) $1$ $0$
* 4.5e4_151e2.8t16.1c1$4$ $ 5^{4} \cdot 151^{2}$ $x^{8} - 2 x^{7} + 18 x^{6} - 39 x^{5} + 140 x^{4} - 219 x^{3} + 453 x^{2} - 382 x + 451$ $(C_8:C_2):C_2$ (as 8T16) $1$ $0$

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 *.