Properties

Label 7.1.207911.1
Degree $7$
Signature $[1, 3]$
Discriminant $-207911$
Root discriminant \(5.75\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $S_7$ (as 7T7)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^5 - x^4 - x^3 + x^2 + x + 1)
 
gp: K = bnfinit(x^7 - x^5 - x^4 - x^3 + x^2 + x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 1, -1, -1, -1, 0, 1]);
 

\( x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1 \) Copy content Toggle raw display

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:   \(-207911\) \(\medspace = -\,11\cdot 41\cdot 461\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(5.75\)
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:   \(11\), \(41\), \(461\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

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 $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $a$, $a^{6}-a^{4}-a^{2}$, $a^{5}-a^{3}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 0.413665504587 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{3}\cdot 0.413665504587 \cdot 1}{2\sqrt{207911}}\approx 0.2250350561757$

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 ${\href{/padicField/2.7.0.1}{7} }$ ${\href{/padicField/3.7.0.1}{7} }$ ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ R ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.7.0.1}{7} }$ ${\href{/padicField/23.7.0.1}{7} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ R ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.7.0.1}{7} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

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
\(11\) Copy content Toggle raw display 11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
\(41\) Copy content Toggle raw display $\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
\(461\) Copy content Toggle raw display $\Q_{461}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{461}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.207911.2t1.a.a$1$ $ 11 \cdot 41 \cdot 461 $ \(\Q(\sqrt{-207911}) \) $C_2$ (as 2T1) $1$ $-1$
6.388...551.14t46.a.a$6$ $ 11^{5} \cdot 41^{5} \cdot 461^{5}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $0$
* 6.207911.7t7.a.a$6$ $ 11 \cdot 41 \cdot 461 $ 7.1.207911.1 $S_7$ (as 7T7) $1$ $0$
14.186...241.21t38.a.a$14$ $ 11^{4} \cdot 41^{4} \cdot 461^{4}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $2$
14.150...601.42t413.a.a$14$ $ 11^{10} \cdot 41^{10} \cdot 461^{10}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $-2$
14.725...791.30t565.a.a$14$ $ 11^{9} \cdot 41^{9} \cdot 461^{9}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $0$
14.388...551.30t565.a.a$14$ $ 11^{5} \cdot 41^{5} \cdot 461^{5}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $0$
15.388...551.42t412.a.a$15$ $ 11^{5} \cdot 41^{5} \cdot 461^{5}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $-3$
15.150...601.42t411.a.a$15$ $ 11^{10} \cdot 41^{10} \cdot 461^{10}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $3$
20.150...601.70.a.a$20$ $ 11^{10} \cdot 41^{10} \cdot 461^{10}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $0$
21.150...601.84.a.a$21$ $ 11^{10} \cdot 41^{10} \cdot 461^{10}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $-3$
21.313...511.42t418.a.a$21$ $ 11^{11} \cdot 41^{11} \cdot 461^{11}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $3$
35.227...201.126.a.a$35$ $ 11^{20} \cdot 41^{20} \cdot 461^{20}$ 7.1.207911.1 $S_7$ (as 7T7) $1$ $-1$
35.586...151.70.a.a$35$ $ 11^{15} \cdot 41^{15} \cdot 461^{15}$ 7.1.207911.1 $S_7$ (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 *.