Properties

Label 21.3.108...223.1
Degree $21$
Signature $[3, 9]$
Discriminant $-1.086\times 10^{26}$
Root discriminant $17.37$
Ramified primes $3, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^14 - 12*x^7 + 1)
 
gp: K = bnfinit(x^21 - 9*x^14 - 12*x^7 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 1]);
 

\( x^{21} - 9 x^{14} - 12 x^{7} + 1 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-108608979330127274981177223\)\(\medspace = -\,3^{28}\cdot 7^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.37$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $3$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{3}{17} a^{7} + \frac{7}{17}$, $\frac{1}{17} a^{15} + \frac{3}{17} a^{8} + \frac{7}{17} a$, $\frac{1}{17} a^{16} + \frac{3}{17} a^{9} + \frac{7}{17} a^{2}$, $\frac{1}{17} a^{17} + \frac{3}{17} a^{10} + \frac{7}{17} a^{3}$, $\frac{1}{119} a^{18} + \frac{2}{119} a^{17} + \frac{2}{119} a^{16} - \frac{1}{119} a^{15} - \frac{1}{119} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{12} - \frac{48}{119} a^{11} + \frac{6}{119} a^{10} - \frac{45}{119} a^{9} - \frac{54}{119} a^{8} - \frac{3}{119} a^{7} - \frac{1}{7} a^{5} + \frac{41}{119} a^{4} - \frac{3}{119} a^{3} + \frac{48}{119} a^{2} - \frac{58}{119} a - \frac{41}{119}$, $\frac{1}{119} a^{19} - \frac{2}{119} a^{17} + \frac{2}{119} a^{16} + \frac{1}{119} a^{15} + \frac{1}{7} a^{13} + \frac{3}{119} a^{12} - \frac{1}{7} a^{11} - \frac{57}{119} a^{10} + \frac{57}{119} a^{9} - \frac{2}{17} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{44}{119} a^{5} + \frac{2}{7} a^{4} + \frac{54}{119} a^{3} + \frac{2}{17} a^{2} - \frac{44}{119} a - \frac{3}{7}$, $\frac{1}{119} a^{20} - \frac{1}{119} a^{17} - \frac{2}{119} a^{16} - \frac{2}{119} a^{15} + \frac{1}{119} a^{14} + \frac{20}{119} a^{13} + \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{48}{119} a^{10} - \frac{6}{119} a^{9} + \frac{45}{119} a^{8} + \frac{54}{119} a^{7} - \frac{44}{119} a^{6} + \frac{1}{7} a^{4} - \frac{41}{119} a^{3} + \frac{3}{119} a^{2} - \frac{48}{119} a + \frac{58}{119}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 51151.7616302 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 51151.7616302 \cdot 1}{2\sqrt{108608979330127274981177223}}\approx 0.299644655153$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\zeta_{9})^+\), 7.1.110270727.1

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 $21$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
7Data not computed