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Properties

Label 21.3.70488525434...7463.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{29}\cdot 11^{18}\cdot 13^{14}$
Root discriminant $634.30$
Ramified primes $7, 11, 13$
Class number $12$ (GRH)
Class group $[2, 6]$ (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![-93206534790699, 0, 0, 0, 0, 0, 0, 65329261371, 0, 0, 0, 0, 0, 0, -560857, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 560857*x^14 + 65329261371*x^7 - 93206534790699)
 
gp: K = bnfinit(x^21 - 560857*x^14 + 65329261371*x^7 - 93206534790699, 1)
 

Normalized defining polynomial

\( x^{21} - 560857 x^{14} + 65329261371 x^{7} - 93206534790699 \)

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:  \(-70488525434400042968409628831807900651401450217386256557463=-\,7^{29}\cdot 11^{18}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $634.30$
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:  $7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{88} a^{7} - \frac{1}{8}$, $\frac{1}{264} a^{8} + \frac{7}{24} a$, $\frac{1}{792} a^{9} + \frac{7}{72} a^{2}$, $\frac{1}{2376} a^{10} - \frac{65}{216} a^{3}$, $\frac{1}{548856} a^{11} + \frac{1}{8316} a^{10} - \frac{1}{5544} a^{9} + \frac{1}{1848} a^{8} - \frac{3}{616} a^{7} - \frac{24257}{49896} a^{4} - \frac{65}{756} a^{3} + \frac{65}{504} a^{2} - \frac{65}{168} a + \frac{27}{56}$, $\frac{1}{1646568} a^{12} - \frac{1}{5544} a^{10} + \frac{1}{2772} a^{9} + \frac{1}{616} a^{8} + \frac{3}{616} a^{7} - \frac{24257}{149688} a^{5} + \frac{65}{504} a^{3} - \frac{65}{252} a^{2} - \frac{9}{56} a - \frac{27}{56}$, $\frac{1}{4939704} a^{13} - \frac{1}{8316} a^{10} - \frac{1}{2772} a^{9} + \frac{1}{1848} a^{8} - \frac{1}{616} a^{7} + \frac{125431}{449064} a^{6} + \frac{65}{756} a^{3} + \frac{65}{252} a^{2} - \frac{65}{168} a + \frac{9}{56}$, $\frac{1}{1863532972224} a^{14} + \frac{378355031}{84706044192} a^{7} + \frac{155993}{640192}$, $\frac{1}{5590598916672} a^{15} + \frac{378355031}{254118132576} a^{8} + \frac{155993}{1920576} a$, $\frac{1}{184489764250176} a^{16} - \frac{2509351021}{8385898375008} a^{9} - \frac{3445087}{63379008} a^{2}$, $\frac{1}{553469292750528} a^{17} - \frac{2509351021}{25157695125024} a^{10} - \frac{66824095}{190137024} a^{3}$, $\frac{1}{18264486660767424} a^{18} + \frac{515864843}{830203939125792} a^{11} - \frac{1}{5544} a^{10} - \frac{1}{2772} a^{9} + \frac{1}{924} a^{8} + \frac{1}{616} a^{7} - \frac{2554953167}{6274521792} a^{4} + \frac{65}{504} a^{3} + \frac{65}{252} a^{2} + \frac{19}{84} a - \frac{9}{56}$, $\frac{1}{54793459982302272} a^{19} + \frac{515864843}{2490611817377376} a^{12} + \frac{1}{16632} a^{10} - \frac{1}{1848} a^{9} - \frac{1}{1848} a^{8} - \frac{1}{616} a^{7} + \frac{3719568625}{18823565376} a^{5} - \frac{65}{1512} a^{3} + \frac{65}{168} a^{2} + \frac{65}{168} a + \frac{9}{56}$, $\frac{1}{1808184179415974976} a^{20} + \frac{3541080707}{82190189973453408} a^{13} - \frac{1}{8316} a^{10} - \frac{1}{2772} a^{9} + \frac{1}{1848} a^{8} - \frac{1}{616} a^{7} - \frac{29545675463}{69019739712} a^{6} + \frac{65}{756} a^{3} + \frac{65}{252} a^{2} - \frac{65}{168} a + \frac{9}{56}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 864787995831793200000 \) (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

3.3.8281.2, Deg 7

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$13$13.3.2.2$x^{3} - 13$$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}$