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Properties

Label 21.3.71307894054...4375.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,5^{18}\cdot 7^{15}\cdot 13^{14}$
Root discriminant $88.18$
Ramified primes $5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78125, 0, 0, 0, 0, 0, 0, -18500, 0, 0, 0, 0, 0, 0, -995, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 995*x^14 - 18500*x^7 + 78125)
 
gp: K = bnfinit(x^21 - 995*x^14 - 18500*x^7 + 78125, 1)
 

Normalized defining polynomial

\( x^{21} - 995 x^{14} - 18500 x^{7} + 78125 \)

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:  \(-71307894054048265215993997676849365234375=-\,5^{18}\cdot 7^{15}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.18$
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:  $5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{5} a^{8}$, $\frac{1}{5} a^{9}$, $\frac{1}{5} a^{10}$, $\frac{1}{25} a^{11} + \frac{1}{5} a^{4}$, $\frac{1}{25} a^{12} + \frac{1}{5} a^{5}$, $\frac{1}{25} a^{13} + \frac{1}{5} a^{6}$, $\frac{1}{406375} a^{14} - \frac{1234}{81275} a^{7} - \frac{1539}{3251}$, $\frac{1}{406375} a^{15} - \frac{1234}{81275} a^{8} - \frac{1539}{3251} a$, $\frac{1}{2031875} a^{16} + \frac{31276}{406375} a^{9} + \frac{1712}{16255} a^{2}$, $\frac{1}{2031875} a^{17} + \frac{31276}{406375} a^{10} + \frac{1712}{16255} a^{3}$, $\frac{1}{71115625} a^{18} + \frac{3}{14223125} a^{17} + \frac{2}{2844625} a^{15} - \frac{1}{2844625} a^{14} - \frac{3}{175} a^{13} + \frac{2}{175} a^{12} + \frac{112551}{14223125} a^{11} + \frac{13404}{406375} a^{10} + \frac{2}{35} a^{9} - \frac{34978}{568925} a^{8} - \frac{47531}{568925} a^{7} + \frac{2}{35} a^{6} + \frac{12}{35} a^{5} + \frac{7211}{81275} a^{4} - \frac{11119}{113785} a^{3} - \frac{1}{7} a^{2} - \frac{6329}{22757} a + \frac{8041}{22757}$, $\frac{1}{71115625} a^{19} - \frac{3}{14223125} a^{17} + \frac{3}{14223125} a^{16} - \frac{3}{2844625} a^{15} - \frac{2}{2844625} a^{14} - \frac{2}{175} a^{13} - \frac{49999}{14223125} a^{12} - \frac{1}{175} a^{11} + \frac{68722}{2844625} a^{10} + \frac{12553}{2844625} a^{9} - \frac{12553}{568925} a^{8} - \frac{13787}{568925} a^{7} + \frac{3}{35} a^{6} - \frac{144583}{568925} a^{5} - \frac{1}{35} a^{4} - \frac{5136}{113785} a^{3} - \frac{27374}{113785} a^{2} - \frac{8387}{22757} a - \frac{173}{22757}$, $\frac{1}{355578125} a^{20} - \frac{3}{14223125} a^{17} - \frac{3}{14223125} a^{16} - \frac{2}{2844625} a^{14} - \frac{293824}{71115625} a^{13} + \frac{3}{175} a^{12} - \frac{2}{175} a^{11} + \frac{231272}{2844625} a^{10} - \frac{13404}{406375} a^{9} - \frac{2}{35} a^{8} + \frac{34978}{568925} a^{7} - \frac{680998}{2844625} a^{6} - \frac{2}{35} a^{5} - \frac{12}{35} a^{4} - \frac{5378}{16255} a^{3} + \frac{11119}{113785} a^{2} + \frac{1}{7} a + \frac{6329}{22757}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1669652271476.0906 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.169.1, 7.1.7500386359375.1

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

Sibling fields

Degree 21 siblings: 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 $21$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$