Properties

Label 21.3.310...319.1
Degree $21$
Signature $[3, 9]$
Discriminant $-3.110\times 10^{28}$
Root discriminant $22.74$
Ramified primes $7, 71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3\times D_7$ (as 21T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*x - 1)
 
gp: K = bnfinit(x^21 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 10, 21, -47, 32, 177, -491, -233, 878, 190, -375, -195, -49, 134, 72, -30, 3, 6, -7, -1, 1]);
 

\( x^{21} - x^{20} - 7 x^{19} + 6 x^{18} + 3 x^{17} - 30 x^{16} + 72 x^{15} + 134 x^{14} - 49 x^{13} - 195 x^{12} - 375 x^{11} + 190 x^{10} + 878 x^{9} - 233 x^{8} - 491 x^{7} + 177 x^{6} + 32 x^{5} - 47 x^{4} + 21 x^{3} + 10 x^{2} - 4 x - 1 \)

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:  \(-31095511042786085990206459319\)\(\medspace = -\,7^{14}\cdot 71^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.74$
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:  $7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13} a^{19} + \frac{3}{13} a^{18} - \frac{4}{13} a^{17} + \frac{2}{13} a^{16} - \frac{5}{13} a^{15} - \frac{3}{13} a^{14} + \frac{1}{13} a^{13} - \frac{4}{13} a^{12} + \frac{4}{13} a^{11} + \frac{5}{13} a^{9} + \frac{2}{13} a^{8} - \frac{4}{13} a^{7} + \frac{6}{13} a^{6} - \frac{2}{13} a^{5} - \frac{2}{13} a^{4} + \frac{3}{13} a^{3} - \frac{4}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{32402253998138293517443} a^{20} - \frac{409394036027280199763}{32402253998138293517443} a^{19} - \frac{6335500981088739436942}{32402253998138293517443} a^{18} - \frac{1487822831922994434463}{32402253998138293517443} a^{17} - \frac{12626263934786052832741}{32402253998138293517443} a^{16} - \frac{11281663411431746104931}{32402253998138293517443} a^{15} - \frac{3352769673100889686345}{32402253998138293517443} a^{14} + \frac{5883177238115412361694}{32402253998138293517443} a^{13} + \frac{9757358992526806586124}{32402253998138293517443} a^{12} + \frac{37765152385707078011}{2492481076779868732111} a^{11} - \frac{5665347600254495165657}{32402253998138293517443} a^{10} - \frac{13389071084107555258667}{32402253998138293517443} a^{9} - \frac{12118015732057415913217}{32402253998138293517443} a^{8} + \frac{15920913657334894813858}{32402253998138293517443} a^{7} + \frac{13689202813494842405354}{32402253998138293517443} a^{6} + \frac{1364982082292010701114}{32402253998138293517443} a^{5} - \frac{13258815750152579239331}{32402253998138293517443} a^{4} + \frac{13209891103199296995134}{32402253998138293517443} a^{3} + \frac{7449802403980526072890}{32402253998138293517443} a^{2} + \frac{3877557779852202036133}{32402253998138293517443} a - \frac{199851948565594499817}{2492481076779868732111}$

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:  \( 725840.889095 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 725840.889095 \cdot 1}{2\sqrt{31095511042786085990206459319}}\approx 0.251287790113$ (assuming GRH)

Galois group

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

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

\(\Q(\zeta_{7})^+\), 7.1.357911.1

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

Sibling fields

Galois closure: Deg 42

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $21$ $21$ $21$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$