↑

Properties

Label 7.7.8233120419813614521.1
Degree $7$
Signature $[7, 0]$
Discriminant $7^{12}\cdot 29^{6}$
Root discriminant $503.76$
Ramified primes $7, 29$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_7$ (as 7T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2403694, -1467487, 133980, 70847, -2030, -609, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 609*x^5 - 2030*x^4 + 70847*x^3 + 133980*x^2 - 1467487*x - 2403694)
 
gp: K = bnfinit(x^7 - 609*x^5 - 2030*x^4 + 70847*x^3 + 133980*x^2 - 1467487*x - 2403694, 1)
 

Normalized defining polynomial

\( x^{7} - 609 x^{5} - 2030 x^{4} + 70847 x^{3} + 133980 x^{2} - 1467487 x - 2403694 \)

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:  \(8233120419813614521=7^{12}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $503.76$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(848,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(400,·)$, $\chi_{1421}(778,·)$, $\chi_{1421}(1002,·)$, $\chi_{1421}(78,·)$, $\chi_{1421}(1359,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{4} - \frac{1}{64} a^{3} + \frac{1}{64} a^{2} - \frac{15}{64} a - \frac{9}{32}$, $\frac{1}{256} a^{5} - \frac{7}{128} a^{2} - \frac{33}{256} a - \frac{9}{128}$, $\frac{1}{750221312} a^{6} - \frac{199565}{750221312} a^{5} + \frac{731005}{93777664} a^{4} + \frac{22906341}{375110656} a^{3} - \frac{176460291}{750221312} a^{2} - \frac{254388797}{750221312} a - \frac{89049843}{375110656}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 140334638.823 \) (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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

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
$7$7.7.12.4$x^{7} - 7 x^{6} + 154$$7$$1$$12$$C_7$$[2]$
$29$29.7.6.7$x^{7} - 116$$7$$1$$6$$C_7$$[\ ]_{7}$