Properties

Label 7.3.112021056.1
Degree $7$
Signature $[3, 2]$
Discriminant $2^{6}\cdot 3^{6}\cdot 7^{4}$
Root discriminant $14.12$
Ramified primes $2, 3, 7$
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 - 3*x^6 + 3*x^5 + 3*x^4 - 9*x^3 + 3*x^2 + x - 3)
 
gp: K = bnfinit(x^7 - 3*x^6 + 3*x^5 + 3*x^4 - 9*x^3 + 3*x^2 + x - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 1, 3, -9, 3, 3, -3, 1]);
 

Normalized defining polynomial

\( x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3 \)

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:  \(112021056=2^{6}\cdot 3^{6}\cdot 7^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $14.12$
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:  $2, 3, 7$
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 \(a\))

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

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:  \( \frac{1}{2} a^{6} - \frac{3}{2} a^{5} + \frac{3}{2} a^{4} + a^{3} - \frac{7}{2} a^{2} + \frac{3}{2} a - \frac{1}{2} \),  \( \frac{1}{2} a^{5} - a^{4} + a^{3} + a^{2} - \frac{3}{2} a + 1 \),  \( \frac{1}{2} a^{5} - \frac{3}{2} a^{4} + a^{3} + 2 a^{2} - \frac{9}{2} a - \frac{1}{2} \),  \( \frac{1}{2} a^{6} - a^{5} + \frac{1}{2} a^{4} + 2 a^{3} - \frac{5}{2} a^{2} - a + \frac{1}{2} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 50.0931046712 \)
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 ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

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$$[\ ]_{7}^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 6.112021056.7t6.a.a$6$ $ 2^{6} \cdot 3^{6} \cdot 7^{4}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ $A_7$ (as 7T6) $1$ $2$
10.14117306610774528.70.a.a$10$ $ 2^{9} \cdot 3^{14} \cdot 7^{8}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ $A_7$ (as 7T6) $0$ $-2$
10.14117306610774528.70.a.b$10$ $ 2^{9} \cdot 3^{14} \cdot 7^{8}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ $A_7$ (as 7T6) $0$ $-2$
14.21964383255760327839744.15t47.a.a$14$ $ 2^{12} \cdot 3^{18} \cdot 7^{12}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ $A_7$ (as 7T6) $1$ $2$
14.21964383255760327839744.21t33.a.a$14$ $ 2^{12} \cdot 3^{18} \cdot 7^{12}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ $A_7$ (as 7T6) $1$ $2$
15.197679449301842950557696.42t294.a.a$15$ $ 2^{12} \cdot 3^{20} \cdot 7^{12}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ $A_7$ (as 7T6) $1$ $-1$
21.1793685113483563715477940482605056.42t299.a.a$21$ $ 2^{18} \cdot 3^{30} \cdot 7^{16}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ $A_7$ (as 7T6) $1$ $1$
35.354574685454344552857366707698313319755263283342109310976.70.a.a$35$ $ 2^{30} \cdot 3^{50} \cdot 7^{28}$ $x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 9 x^{3} + 3 x^{2} + x - 3$ $A_7$ (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 *.