Properties

Label 7.3.50808384.1
Degree $7$
Signature $[3, 2]$
Discriminant $2^{6}\cdot 3^{8}\cdot 11^{2}$
Root discriminant $12.61$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $A_7$ (as 7T6)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 2*x^6 + 2*x + 2)
 
gp: K = bnfinit(x^7 - 2*x^6 + 2*x + 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 2, 0, 0, 0, 0, -2, 1]);
 

Normalized defining polynomial

\( x^{7} - 2 x^{6} + 2 x + 2 \)

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

Invariants

Degree:  $7$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(50808384=2^{6}\cdot 3^{8}\cdot 11^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $12.61$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{6} - 2 a^{5} + a^{4} - a^{3} - a + 1 \),  \( a^{3} - a - 1 \),  \( a^{3} - a^{2} - a - 1 \),  \( a^{6} - 3 a^{5} + 3 a^{4} - 3 a^{3} + 2 a^{2} - a + 3 \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 38.0037816148 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$A_7$ (as 7T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 2520
The 9 conjugacy class representatives for $A_7$
Character table for $A_7$

Intermediate fields

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

Sibling fields

Degree 15 siblings: Deg 15, Deg 15
Degree 21 sibling: Deg 21
Degree 35 sibling: data not computed
Degree 42 siblings: data not computed

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 ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ R ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$3$3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$11$11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 6.50808384.7t6.a.a$6$ $ 2^{6} \cdot 3^{8} \cdot 11^{2}$ $x^{7} - 2 x^{6} + 2 x + 2$ $A_7$ (as 7T6) $1$ $2$
10.351405582803624448.70.a.a$10$ $ 2^{9} \cdot 3^{18} \cdot 11^{6}$ $x^{7} - 2 x^{6} + 2 x + 2$ $A_7$ (as 7T6) $0$ $-2$
10.351405582803624448.70.a.b$10$ $ 2^{9} \cdot 3^{18} \cdot 11^{6}$ $x^{7} - 2 x^{6} + 2 x + 2$ $A_7$ (as 7T6) $0$ $-2$
14.2430423365276781059668217856.15t47.a.a$14$ $ 2^{12} \cdot 3^{28} \cdot 11^{10}$ $x^{7} - 2 x^{6} + 2 x + 2$ $A_7$ (as 7T6) $1$ $2$
14.247977080428199271469056.21t33.a.a$14$ $ 2^{12} \cdot 3^{24} \cdot 11^{8}$ $x^{7} - 2 x^{6} + 2 x + 2$ $A_7$ (as 7T6) $1$ $2$
15.2231793723853793443221504.42t294.a.a$15$ $ 2^{12} \cdot 3^{26} \cdot 11^{8}$ $x^{7} - 2 x^{6} + 2 x + 2$ $A_7$ (as 7T6) $1$ $-1$
21.11862004710203242703974199567822205967663104.42t299.a.a$21$ $ 2^{18} \cdot 3^{44} \cdot 11^{16}$ $x^{7} - 2 x^{6} + 2 x + 2$ $A_7$ (as 7T6) $1$ $1$
35.2941505296061748107334365673192725862524976759105923348765368909824.70.a.a$35$ $ 2^{30} \cdot 3^{68} \cdot 11^{24}$ $x^{7} - 2 x^{6} + 2 x + 2$ $A_7$ (as 7T6) $1$ $-1$

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 *.