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Properties

Label 7.7.81437164.1
Degree $7$
Signature $[7, 0]$
Discriminant $2^{2}\cdot 1787\cdot 11393$
Root discriminant $13.49$
Ramified primes $2, 1787, 11393$
Class number $1$
Class group Trivial
Galois group $S_7$ (as 7T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 2, 12, -1, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 7*x^5 - x^4 + 12*x^3 + 2*x^2 - 4*x - 1)
 
gp: K = bnfinit(x^7 - 7*x^5 - x^4 + 12*x^3 + 2*x^2 - 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1 \)

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

Invariants

Degree:  $7$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(81437164=2^{2}\cdot 1787\cdot 11393\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.49$
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:  $2, 1787, 11393$
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 \(a\))

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

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:  $6$
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:  \( a \),  \( a^{6} - 7 a^{4} + 11 a^{2} - 2 a - 2 \),  \( a^{6} - 6 a^{4} - a^{3} + 7 a^{2} + a - 1 \),  \( a^{6} - 7 a^{4} + 11 a^{2} - a - 3 \),  \( a^{6} - 7 a^{4} - a^{3} + 12 a^{2} + a - 3 \),  \( a^{6} - 7 a^{4} - a^{3} + 11 a^{2} + 2 a - 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 52.1624717732 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_7$ (as 7T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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: data not computed
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 ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
1787Data not computed
11393Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_1787_11393.2t1.1c1$1$ $ 2^{2} \cdot 1787 \cdot 11393 $ $x^{2} - 20359291$ $C_2$ (as 2T1) $1$ $1$
6.2e10_1787e5_11393e5.14t46.1c1$6$ $ 2^{10} \cdot 1787^{5} \cdot 11393^{5}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $6$
* 6.2e2_1787_11393.7t7.1c1$6$ $ 2^{2} \cdot 1787 \cdot 11393 $ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $6$
14.2e8_1787e4_11393e4.21t38.1c1$14$ $ 2^{8} \cdot 1787^{4} \cdot 11393^{4}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $14$
14.2e20_1787e10_11393e10.42t413.1c1$14$ $ 2^{20} \cdot 1787^{10} \cdot 11393^{10}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $14$
14.2e18_1787e9_11393e9.30t565.1c1$14$ $ 2^{18} \cdot 1787^{9} \cdot 11393^{9}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $14$
14.2e10_1787e5_11393e5.30t565.1c1$14$ $ 2^{10} \cdot 1787^{5} \cdot 11393^{5}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $14$
15.2e10_1787e5_11393e5.42t412.1c1$15$ $ 2^{10} \cdot 1787^{5} \cdot 11393^{5}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $15$
15.2e20_1787e10_11393e10.42t411.1c1$15$ $ 2^{20} \cdot 1787^{10} \cdot 11393^{10}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $15$
20.2e20_1787e10_11393e10.70.1c1$20$ $ 2^{20} \cdot 1787^{10} \cdot 11393^{10}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $20$
21.2e20_1787e10_11393e10.84.1c1$21$ $ 2^{20} \cdot 1787^{10} \cdot 11393^{10}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $21$
21.2e22_1787e11_11393e11.42t418.1c1$21$ $ 2^{22} \cdot 1787^{11} \cdot 11393^{11}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $21$
35.2e40_1787e20_11393e20.126.1c1$35$ $ 2^{40} \cdot 1787^{20} \cdot 11393^{20}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $35$
35.2e30_1787e15_11393e15.70.1c1$35$ $ 2^{30} \cdot 1787^{15} \cdot 11393^{15}$ $x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 x - 1$ $S_7$ (as 7T7) $1$ $35$

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 *.