Properties

Label 7.1.199559.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,199559$
Root discriminant $5.72$
Ramified prime $199559$
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 - x^6 + x^3 - x + 1)
 
gp: K = bnfinit(x^7 - x^6 + x^3 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 1, 0, 0, -1, 1]);
 

Normalized defining polynomial

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

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:  \(-199559\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $5.72$
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:  $199559$
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:  \( a^{6} - a^{5} + a^{2} - 1 \),  \( a^{5} - a^{2} + 1 \),  \( a^{5} + a + 1 \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 0.401633777334 \)
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 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
199559Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.199559.2t1.a.a$1$ $ 199559 $ $x^{2} - x + 49890$ $C_2$ (as 2T1) $1$ $-1$
6.316487524211357779480021799.14t46.a.a$6$ $ 199559^{5}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $0$
* 6.199559.7t7.a.a$6$ $ 199559 $ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $0$
14.1585934606864926059361.21t38.a.a$14$ $ 199559^{4}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $2$
14.100164352981434776656212324209796875846869769515196401.42t413.a.a$14$ $ 199559^{10}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $-2$
14.501928517287793467877732020153422676235448010439.30t565.a.a$14$ $ 199559^{9}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $0$
14.316487524211357779480021799.30t565.a.a$14$ $ 199559^{5}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $0$
15.316487524211357779480021799.42t412.a.a$15$ $ 199559^{5}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $-3$
15.100164352981434776656212324209796875846869769515196401.42t411.a.a$15$ $ 199559^{10}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $3$
20.100164352981434776656212324209796875846869769515196401.70.a.a$20$ $ 199559^{10}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $0$
21.100164352981434776656212324209796875846869769515196401.84.a.a$21$ $ 199559^{10}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $-3$
21.19988698116622142594737075206982854747125484334683078587159.42t418.a.a$21$ $ 199559^{11}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $3$
35.10032897608189461831282689288005799510829685533126455387273609095812007302997194400815625474430069603352801.126.a.a$35$ $ 199559^{20}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (as 7T7) $1$ $-1$
35.31700768089326825660626793248681288268800058617606277681769017345553313246345399.70.a.a$35$ $ 199559^{15}$ $x^{7} - x^{6} + x^{3} - x + 1$ $S_7$ (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 *.