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Properties

Label 7.7.157356790566...0369.1
Degree $7$
Signature $[7, 0]$
Discriminant $1583^{6}$
Root discriminant $552.60$
Ramified prime $1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![1084293, -1841157, -44802, 79450, -517, -678, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 678*x^5 - 517*x^4 + 79450*x^3 - 44802*x^2 - 1841157*x + 1084293)
 
gp: K = bnfinit(x^7 - x^6 - 678*x^5 - 517*x^4 + 79450*x^3 - 44802*x^2 - 1841157*x + 1084293, 1)
 

Normalized defining polynomial

\( x^{7} - x^{6} - 678 x^{5} - 517 x^{4} + 79450 x^{3} - 44802 x^{2} - 1841157 x + 1084293 \)

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:  \(15735679056639910369=1583^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $552.60$
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:  $1583$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1583\)
Dirichlet character group:    $\lbrace$$\chi_{1583}(1,·)$, $\chi_{1583}(274,·)$, $\chi_{1583}(675,·)$, $\chi_{1583}(52,·)$, $\chi_{1583}(1121,·)$, $\chi_{1583}(1304,·)$, $\chi_{1583}(1322,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{3} + \frac{1}{9} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{63} a^{4} + \frac{1}{21} a^{3} + \frac{1}{21} a^{2} + \frac{23}{63} a - \frac{1}{3}$, $\frac{1}{567} a^{5} + \frac{1}{189} a^{4} + \frac{1}{189} a^{3} + \frac{44}{567} a^{2} + \frac{7}{27} a + \frac{2}{9}$, $\frac{1}{1054591083} a^{6} + \frac{70088}{1054591083} a^{5} + \frac{4315}{16739541} a^{4} + \frac{31645265}{1054591083} a^{3} - \frac{110141558}{1054591083} a^{2} + \frac{112563986}{351530361} a + \frac{190084}{16739541}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 819365058.399 \) (assuming GRH)
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 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

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
1583Data not computed