↑

Properties

Label 7.1.419904000.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,2^{9}\cdot 3^{8}\cdot 5^{3}$
Root discriminant $17.06$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $F_7$ (as 7T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 17, -24, 22, -14, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 2*x^6 + 9*x^5 - 14*x^4 + 22*x^3 - 24*x^2 + 17*x - 4)
 
gp: K = bnfinit(x^7 - 2*x^6 + 9*x^5 - 14*x^4 + 22*x^3 - 24*x^2 + 17*x - 4, 1)
 

Normalized defining polynomial

\( x^{7} - 2 x^{6} + 9 x^{5} - 14 x^{4} + 22 x^{3} - 24 x^{2} + 17 x - 4 \)

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:  \(-419904000=-\,2^{9}\cdot 3^{8}\cdot 5^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.06$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$

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:  \( a^{6} - \frac{3}{2} a^{5} + \frac{17}{2} a^{4} - 10 a^{3} + \frac{37}{2} a^{2} - 16 a + 11 \),  \( a^{6} - \frac{3}{2} a^{5} + \frac{17}{2} a^{4} - 11 a^{3} + \frac{37}{2} a^{2} - 18 a + 11 \),  \( \frac{5}{2} a^{6} - 4 a^{5} + \frac{41}{2} a^{4} - \frac{51}{2} a^{3} + \frac{83}{2} a^{2} - 37 a + 23 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 93.6182829609 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 7T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

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

Sibling fields

Galois closure: data not computed
Degree 14 sibling: Deg 14
Degree 21 sibling: Deg 21

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_5.2t1.2c1$1$ $ 2^{3} \cdot 5 $ $x^{2} + 10$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_3e2_5.6t1.4c1$1$ $ 2^{3} \cdot 3^{2} \cdot 5 $ $x^{6} + 24 x^{4} - 2 x^{3} + 309 x^{2} + 66 x + 1691$ $C_6$ (as 6T1) $0$ $-1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.2e3_3e2_5.6t1.4c2$1$ $ 2^{3} \cdot 3^{2} \cdot 5 $ $x^{6} + 24 x^{4} - 2 x^{3} + 309 x^{2} + 66 x + 1691$ $C_6$ (as 6T1) $0$ $-1$
* 6.2e9_3e8_5e3.7t4.1c1$6$ $ 2^{9} \cdot 3^{8} \cdot 5^{3}$ $x^{7} - 2 x^{6} + 9 x^{5} - 14 x^{4} + 22 x^{3} - 24 x^{2} + 17 x - 4$ $F_7$ (as 7T4) $1$ $0$

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 *.