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Properties

Label 7.7.5567914722008521.1
Degree $7$
Signature $[7, 0]$
Discriminant $421^{6}$
Root discriminant $177.58$
Ramified prime $421$
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![49213, -25209, -11596, 6180, 103, -180, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 180*x^5 + 103*x^4 + 6180*x^3 - 11596*x^2 - 25209*x + 49213)
 
gp: K = bnfinit(x^7 - x^6 - 180*x^5 + 103*x^4 + 6180*x^3 - 11596*x^2 - 25209*x + 49213, 1)
 

Normalized defining polynomial

\( x^{7} - x^{6} - 180 x^{5} + 103 x^{4} + 6180 x^{3} - 11596 x^{2} - 25209 x + 49213 \)

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:  \(5567914722008521=421^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.58$
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:  $421$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(421\)
Dirichlet character group:    $\lbrace$$\chi_{421}(385,·)$, $\chi_{421}(370,·)$, $\chi_{421}(33,·)$, $\chi_{421}(1,·)$, $\chi_{421}(152,·)$, $\chi_{421}(75,·)$, $\chi_{421}(247,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{13} a^{5} + \frac{3}{13} a^{4} - \frac{3}{13} a^{3} + \frac{1}{13} a^{2} - \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{1690863421} a^{6} + \frac{32924542}{1690863421} a^{5} - \frac{3279054}{9773777} a^{4} + \frac{232299659}{1690863421} a^{3} - \frac{180289649}{1690863421} a^{2} - \frac{720887298}{1690863421} a + \frac{29412496}{1690863421}$

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:  \( 428479.510505 \)
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 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

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
421Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.421.7t1.1c1$1$ $ 421 $ $x^{7} - x^{6} - 180 x^{5} + 103 x^{4} + 6180 x^{3} - 11596 x^{2} - 25209 x + 49213$ $C_7$ (as 7T1) $0$ $1$
* 1.421.7t1.1c2$1$ $ 421 $ $x^{7} - x^{6} - 180 x^{5} + 103 x^{4} + 6180 x^{3} - 11596 x^{2} - 25209 x + 49213$ $C_7$ (as 7T1) $0$ $1$
* 1.421.7t1.1c3$1$ $ 421 $ $x^{7} - x^{6} - 180 x^{5} + 103 x^{4} + 6180 x^{3} - 11596 x^{2} - 25209 x + 49213$ $C_7$ (as 7T1) $0$ $1$
* 1.421.7t1.1c4$1$ $ 421 $ $x^{7} - x^{6} - 180 x^{5} + 103 x^{4} + 6180 x^{3} - 11596 x^{2} - 25209 x + 49213$ $C_7$ (as 7T1) $0$ $1$
* 1.421.7t1.1c5$1$ $ 421 $ $x^{7} - x^{6} - 180 x^{5} + 103 x^{4} + 6180 x^{3} - 11596 x^{2} - 25209 x + 49213$ $C_7$ (as 7T1) $0$ $1$
* 1.421.7t1.1c6$1$ $ 421 $ $x^{7} - x^{6} - 180 x^{5} + 103 x^{4} + 6180 x^{3} - 11596 x^{2} - 25209 x + 49213$ $C_7$ (as 7T1) $0$ $1$

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 *.