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Properties

Label 7.7.394259782912576.1
Degree $7$
Signature $[7, 0]$
Discriminant $2^{6}\cdot 7^{4}\cdot 37^{6}$
Root discriminant $121.65$
Ramified primes $2, 7, 37$
Class number $1$
Class group Trivial
Galois group $C_7:C_3$ (as 7T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12679, -7145, -1951, 1203, 89, -63, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 63*x^5 + 89*x^4 + 1203*x^3 - 1951*x^2 - 7145*x + 12679)
 
gp: K = bnfinit(x^7 - x^6 - 63*x^5 + 89*x^4 + 1203*x^3 - 1951*x^2 - 7145*x + 12679, 1)
 

Normalized defining polynomial

\( x^{7} - x^{6} - 63 x^{5} + 89 x^{4} + 1203 x^{3} - 1951 x^{2} - 7145 x + 12679 \)

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:  \(394259782912576=2^{6}\cdot 7^{4}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.65$
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:  $2, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{33698} a^{6} + \frac{3299}{16849} a^{5} + \frac{2323}{33698} a^{4} - \frac{216}{2407} a^{3} - \frac{4957}{33698} a^{2} + \frac{3782}{16849} a + \frac{953}{33698}$

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 479900.742126 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:C_3$ (as 7T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

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

Sibling fields

Galois closure: Deg 21

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$37$37.7.6.1$x^{7} - 37$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 3.2e3_7e2_37e3.7t3.1c1$3$ $ 2^{3} \cdot 7^{2} \cdot 37^{3}$ $x^{7} - x^{6} - 63 x^{5} + 89 x^{4} + 1203 x^{3} - 1951 x^{2} - 7145 x + 12679$ $C_7:C_3$ (as 7T3) $0$ $3$
* 3.2e3_7e2_37e3.7t3.1c2$3$ $ 2^{3} \cdot 7^{2} \cdot 37^{3}$ $x^{7} - x^{6} - 63 x^{5} + 89 x^{4} + 1203 x^{3} - 1951 x^{2} - 7145 x + 12679$ $C_7:C_3$ (as 7T3) $0$ $3$

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 *.