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Properties

Label 7.7.8515170979269874921.1
Degree $7$
Signature $[7, 0]$
Discriminant $1429^{6}$
Root discriminant $506.19$
Ramified prime $1429$
Class number $1$
Class group Trivial
Galois group $C_7$ (as 7T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43521943, -6889121, -607744, 118732, 1983, -612, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 612*x^5 + 1983*x^4 + 118732*x^3 - 607744*x^2 - 6889121*x + 43521943)
 
gp: K = bnfinit(x^7 - x^6 - 612*x^5 + 1983*x^4 + 118732*x^3 - 607744*x^2 - 6889121*x + 43521943, 1)
 

Normalized defining polynomial

\( x^{7} - x^{6} - 612 x^{5} + 1983 x^{4} + 118732 x^{3} - 607744 x^{2} - 6889121 x + 43521943 \)

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

Invariants

Degree:  $7$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8515170979269874921=1429^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $506.19$
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:  $1429$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1429\)
Dirichlet character group:    $\lbrace$$\chi_{1429}(1,·)$, $\chi_{1429}(1362,·)$, $\chi_{1429}(756,·)$, $\chi_{1429}(1365,·)$, $\chi_{1429}(1238,·)$, $\chi_{1429}(792,·)$, $\chi_{1429}(202,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{31} a^{5} + \frac{7}{31} a^{4} - \frac{14}{31} a^{3} - \frac{8}{31} a^{2} + \frac{7}{31} a + \frac{9}{31}$, $\frac{1}{55029250007} a^{6} - \frac{66998871}{55029250007} a^{5} + \frac{2666704154}{55029250007} a^{4} - \frac{2511326754}{55029250007} a^{3} - \frac{4371266313}{55029250007} a^{2} - \frac{2350045973}{55029250007} a + \frac{19131882898}{55029250007}$

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:  $6$
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:  \( 9587300.65259 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7$ (as 7T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 7
The 7 conjugacy class representatives for $C_7$
Character table for $C_7$

Intermediate fields

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{7}$

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
1429Data not computed