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Properties

Label 21.3.67442474391...0368.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{18}\cdot 7^{29}\cdot 19^{14}$
Root discriminant $189.48$
Ramified primes $2, 7, 19$
Class number $21$ (GRH)
Class group $[21]$ (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![2097152, 0, 0, 0, 0, 0, 0, -181248, 0, 0, 0, 0, 0, 0, -830, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 830*x^14 - 181248*x^7 + 2097152)
 
gp: K = bnfinit(x^21 - 830*x^14 - 181248*x^7 + 2097152, 1)
 

Normalized defining polynomial

\( x^{21} - 830 x^{14} - 181248 x^{7} + 2097152 \)

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:  \(-674424743915820495717627003682966750085368250368=-\,2^{18}\cdot 7^{29}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $189.48$
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:  $2, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a$, $\frac{1}{8} a^{9} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{10} + \frac{1}{8} a^{3}$, $\frac{1}{448} a^{11} + \frac{1}{112} a^{10} - \frac{1}{56} a^{9} + \frac{1}{14} a^{8} + \frac{1}{7} a^{7} - \frac{31}{224} a^{4} + \frac{25}{56} a^{3} + \frac{3}{28} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{896} a^{12} - \frac{3}{112} a^{10} - \frac{3}{56} a^{9} - \frac{1}{14} a^{8} + \frac{3}{14} a^{7} - \frac{31}{448} a^{5} - \frac{19}{56} a^{3} + \frac{9}{28} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{1792} a^{13} + \frac{3}{112} a^{10} - \frac{1}{56} a^{9} + \frac{1}{28} a^{8} - \frac{1}{7} a^{7} - \frac{31}{896} a^{6} + \frac{19}{56} a^{3} + \frac{3}{28} a^{2} - \frac{3}{14} a - \frac{1}{7}$, $\frac{1}{896000} a^{14} + \frac{92513}{448000} a^{7} + \frac{417}{875}$, $\frac{1}{1792000} a^{15} + \frac{92513}{896000} a^{8} - \frac{229}{875} a$, $\frac{1}{7168000} a^{16} - \frac{131487}{3584000} a^{9} + \frac{2167}{7000} a^{2}$, $\frac{1}{14336000} a^{17} - \frac{131487}{7168000} a^{10} - \frac{4833}{14000} a^{3}$, $\frac{1}{57344000} a^{18} - \frac{3487}{28672000} a^{11} + \frac{1}{56} a^{10} - \frac{1}{28} a^{9} - \frac{3}{28} a^{8} - \frac{3}{14} a^{7} - \frac{6333}{56000} a^{4} - \frac{3}{28} a^{3} + \frac{3}{14} a^{2} - \frac{5}{14} a + \frac{2}{7}$, $\frac{1}{114688000} a^{19} - \frac{3487}{57344000} a^{12} + \frac{1}{112} a^{10} + \frac{1}{56} a^{9} + \frac{3}{28} a^{8} - \frac{1}{14} a^{7} - \frac{6333}{112000} a^{5} + \frac{25}{56} a^{3} - \frac{3}{28} a^{2} + \frac{5}{14} a + \frac{3}{7}$, $\frac{1}{458752000} a^{20} - \frac{3487}{229376000} a^{13} + \frac{3}{112} a^{10} - \frac{1}{56} a^{9} + \frac{1}{28} a^{8} - \frac{1}{7} a^{7} + \frac{105667}{448000} a^{6} + \frac{19}{56} a^{3} + \frac{3}{28} a^{2} - \frac{3}{14} a - \frac{1}{7}$

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

Class group and class number

$C_{21}$, which has order $21$ (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:  \( 1867922722744152.2 \) (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

3.3.17689.2, 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ R $21$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
7Data not computed
19Data not computed