↑

Properties

Label 7.1.80565593759.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,7^{3}\cdot 617^{3}$
Root discriminant $36.14$
Ramified primes $7, 617$
Class number $1$
Class group Trivial
Galois group $D_{7}$ (as 7T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-189, -189, -108, -62, -15, -2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 3*x^6 - 2*x^5 - 15*x^4 - 62*x^3 - 108*x^2 - 189*x - 189)
 
gp: K = bnfinit(x^7 - 3*x^6 - 2*x^5 - 15*x^4 - 62*x^3 - 108*x^2 - 189*x - 189, 1)
 

Normalized defining polynomial

\( x^{7} - 3 x^{6} - 2 x^{5} - 15 x^{4} - 62 x^{3} - 108 x^{2} - 189 x - 189 \)

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

Invariants

Degree:  $7$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-80565593759=-\,7^{3}\cdot 617^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.14$
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:  $7, 617$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{5} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{243} a^{6} - \frac{1}{27} a^{5} - \frac{2}{243} a^{4} - \frac{10}{81} a^{3} + \frac{10}{243} a^{2} - \frac{20}{81} a + \frac{1}{27}$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{668}{81} a^{6} - \frac{383}{9} a^{5} + \frac{5684}{81} a^{4} - \frac{6599}{27} a^{3} - \frac{4228}{81} a^{2} - \frac{15133}{27} a - \frac{4408}{9} \),  \( \frac{1030}{243} a^{6} - \frac{373}{27} a^{5} - \frac{2762}{243} a^{4} - \frac{1777}{81} a^{3} - \frac{83768}{243} a^{2} - \frac{8630}{81} a - \frac{24872}{27} \),  \( \frac{1}{81} a^{6} + \frac{1}{9} a^{5} - \frac{137}{81} a^{4} + \frac{68}{27} a^{3} + \frac{361}{81} a^{2} + \frac{355}{27} a + \frac{193}{9} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5494.77424771 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 7T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

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

Sibling fields

Galois closure: 14.0.28033829543340009100599839.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
617Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7_617.2t1.1c1$1$ $ 7 \cdot 617 $ $x^{2} - x + 1080$ $C_2$ (as 2T1) $1$ $-1$
* 2.7_617.7t2.1c1$2$ $ 7 \cdot 617 $ $x^{7} - 3 x^{6} - 2 x^{5} - 15 x^{4} - 62 x^{3} - 108 x^{2} - 189 x - 189$ $D_{7}$ (as 7T2) $1$ $0$
* 2.7_617.7t2.1c2$2$ $ 7 \cdot 617 $ $x^{7} - 3 x^{6} - 2 x^{5} - 15 x^{4} - 62 x^{3} - 108 x^{2} - 189 x - 189$ $D_{7}$ (as 7T2) $1$ $0$
* 2.7_617.7t2.1c3$2$ $ 7 \cdot 617 $ $x^{7} - 3 x^{6} - 2 x^{5} - 15 x^{4} - 62 x^{3} - 108 x^{2} - 189 x - 189$ $D_{7}$ (as 7T2) $1$ $0$

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 *.