Properties

Label 7.1.227287.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,167\cdot 1361$
Root discriminant $5.82$
Ramified primes $167, 1361$
Class number $1$
Class group Trivial
Galois group $S_7$ (as 7T7)

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Show commands for: SageMath / Pari/GP / Magma

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

Normalized defining polynomial

\( x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $7$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-227287=-\,167\cdot 1361\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $5.82$
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:  $167, 1361$
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:  $3$
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:  \( a^{6} - 2 a^{5} + 3 a^{4} - 3 a^{3} + 3 a^{2} - 2 a + 2 \),  \( a^{6} - a^{5} + 2 a^{4} - 2 a^{3} + 2 a^{2} - a + 1 \),  \( a^{6} - a^{5} + a^{4} - a^{3} + a^{2} + 1 \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 0.440973429986 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$S_7$ (as 7T7):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 5040
The 15 conjugacy class representatives for $S_7$
Character table for $S_7$

Intermediate fields

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

Sibling fields

Degree 14 sibling: Deg 14
Degree 21 sibling: Deg 21
Degree 30 sibling: Deg 30
Degree 35 sibling: Deg 35
Degree 42 siblings: Deg 42, Deg 42, Deg 42, some 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 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
167Data not computed
1361Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.227287.2t1.a.a$1$ $ 167 \cdot 1361 $ $x^{2} - x + 56822$ $C_2$ (as 2T1) $1$ $-1$
6.606558903168239414267905207.14t46.a.a$6$ $ 167^{5} \cdot 1361^{5}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $0$
* 6.227287.7t7.a.a$6$ $ 167 \cdot 1361 $ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $0$
14.2668691580109022576161.21t38.a.a$14$ $ 167^{4} \cdot 1361^{4}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $2$
14.367913703012657638147710761445023668138230595937712849.42t413.a.a$14$ $ 167^{10} \cdot 1361^{10}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $-2$
14.1618718637725244462497682495897361785488085970327.30t565.a.a$14$ $ 167^{9} \cdot 1361^{9}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $0$
14.606558903168239414267905207.30t565.a.a$14$ $ 167^{5} \cdot 1361^{5}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $0$
15.606558903168239414267905207.42t412.a.a$15$ $ 167^{5} \cdot 1361^{5}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $-3$
15.367913703012657638147710761445023668138230595937712849.42t411.a.a$15$ $ 167^{10} \cdot 1361^{10}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $3$
20.367913703012657638147710761445023668138230595937712849.70.a.a$20$ $ 167^{10} \cdot 1361^{10}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $0$
21.367913703012657638147710761445023668138230595937712849.84.a.a$21$ $ 167^{10} \cdot 1361^{10}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $-3$
21.83622001816637916601678735836555094460134017458894940310663.42t418.a.a$21$ $ 167^{11} \cdot 1361^{11}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $3$
35.135360492864486046044476870235854960050728470870801787505812427823273468737697618838715499992135697179696801.126.a.a$35$ $ 167^{20} \cdot 1361^{20}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $-1$
35.223161332159922998005431943811725723070059325897778433408348209634456805917904743.70.a.a$35$ $ 167^{15} \cdot 1361^{15}$ $x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} + 3 x^{3} - 2 x^{2} + 2 x - 1$ $S_7$ (as 7T7) $1$ $1$

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 *.