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Properties

Label 7.1.2270799427.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,13^{4}\cdot 43^{3}$
Root discriminant $21.71$
Ramified primes $13, 43$
Class number $1$
Class group Trivial
Galois group $F_7$ (as 7T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 55, -21, -22, 17, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 2*x^6 + x^5 + 17*x^4 - 22*x^3 - 21*x^2 + 55*x - 4)
 
gp: K = bnfinit(x^7 - 2*x^6 + x^5 + 17*x^4 - 22*x^3 - 21*x^2 + 55*x - 4, 1)
 

Normalized defining polynomial

\( x^{7} - 2 x^{6} + x^{5} + 17 x^{4} - 22 x^{3} - 21 x^{2} + 55 x - 4 \)

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:  \(-2270799427=-\,13^{4}\cdot 43^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.71$
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:  $13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{325} a^{6} + \frac{4}{325} a^{5} + \frac{1}{13} a^{4} - \frac{158}{325} a^{3} + \frac{1}{65} a^{2} + \frac{9}{325} a + \frac{109}{325}$

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:  \( \frac{14}{325} a^{6} - \frac{9}{325} a^{5} - \frac{8}{65} a^{4} + \frac{258}{325} a^{3} + \frac{1}{65} a^{2} - \frac{654}{325} a + \frac{31}{325} \),  \( \frac{69}{325} a^{6} + \frac{16}{325} a^{5} + \frac{33}{65} a^{4} + \frac{1253}{325} a^{3} + \frac{212}{65} a^{2} + \frac{1726}{325} a + \frac{3491}{325} \),  \( \frac{33}{325} a^{6} + \frac{2}{325} a^{5} - \frac{56}{65} a^{4} + \frac{51}{325} a^{3} + \frac{137}{65} a^{2} - \frac{288}{325} a - \frac{693}{325} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 226.218467209 \)
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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.6.3.2$x^{6} - 1849 x^{2} + 795070$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.43.2t1.1c1$1$ $ 43 $ $x^{2} - x + 11$ $C_2$ (as 2T1) $1$ $-1$
1.13_43.6t1.3c1$1$ $ 13 \cdot 43 $ $x^{6} - x^{5} + 24 x^{4} - 11 x^{3} + 367 x^{2} - 298 x + 2491$ $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.13_43.6t1.3c2$1$ $ 13 \cdot 43 $ $x^{6} - x^{5} + 24 x^{4} - 11 x^{3} + 367 x^{2} - 298 x + 2491$ $C_6$ (as 6T1) $0$ $-1$
* 6.13e4_43e3.7t4.1c1$6$ $ 13^{4} \cdot 43^{3}$ $x^{7} - 2 x^{6} + x^{5} + 17 x^{4} - 22 x^{3} - 21 x^{2} + 55 x - 4$ $F_7$ (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 *.