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Properties

Label 21.3.36209581211...2303.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{29}\cdot 13^{18}$
Root discriminant $132.39$
Ramified primes $7, 13$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_3\times D_7$ (as 21T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-62748517, 0, 0, 0, 0, 0, 0, 927134, 0, 0, 0, 0, 0, 0, -2899, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2899*x^14 + 927134*x^7 - 62748517)
 
gp: K = bnfinit(x^21 - 2899*x^14 + 927134*x^7 - 62748517, 1)
 

Normalized defining polynomial

\( x^{21} - 2899 x^{14} + 927134 x^{7} - 62748517 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-362095812116921075757469201554122678840602303=-\,7^{29}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.39$
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:  $7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7}$, $\frac{1}{13} a^{8}$, $\frac{1}{13} a^{9}$, $\frac{1}{13} a^{10}$, $\frac{1}{1183} a^{11} - \frac{1}{91} a^{10} - \frac{3}{91} a^{9} + \frac{3}{91} a^{8} - \frac{2}{91} a^{7} - \frac{2}{91} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{1183} a^{12} - \frac{2}{91} a^{10} - \frac{1}{91} a^{9} + \frac{2}{91} a^{8} + \frac{2}{91} a^{7} - \frac{2}{91} a^{5} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{1183} a^{13} + \frac{1}{91} a^{10} + \frac{1}{91} a^{9} + \frac{3}{91} a^{8} - \frac{3}{91} a^{7} - \frac{2}{91} a^{6} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{10134761} a^{14} + \frac{21045}{779597} a^{7} - \frac{1390}{4613}$, $\frac{1}{10134761} a^{15} + \frac{21045}{779597} a^{8} - \frac{1390}{4613} a$, $\frac{1}{131751893} a^{16} - \frac{38924}{10134761} a^{9} + \frac{17062}{59969} a^{2}$, $\frac{1}{131751893} a^{17} - \frac{38924}{10134761} a^{10} + \frac{17062}{59969} a^{3}$, $\frac{1}{1712774609} a^{18} - \frac{38924}{131751893} a^{11} - \frac{282783}{779597} a^{4}$, $\frac{1}{1712774609} a^{19} - \frac{38924}{131751893} a^{12} - \frac{282783}{779597} a^{5}$, $\frac{1}{22266069917} a^{20} - \frac{261666}{1712774609} a^{13} + \frac{2}{91} a^{10} + \frac{2}{91} a^{9} - \frac{1}{91} a^{8} + \frac{1}{91} a^{7} - \frac{3366903}{10134761} a^{6} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$

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

Class group and class number

$C_{7}$, which has order $7$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15804226575243.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_7$ (as 21T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 15 conjugacy class representatives for $C_3\times D_7$
Character table for $C_3\times D_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.194779153450063.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 $21$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $21$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
7Data not computed
$13$13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$