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Properties

Label 21.1.29776705790...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{26}\cdot 5^{19}\cdot 7^{17}$
Root discriminant $48.89$
Ramified primes $2, 5, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 576, 7616, 32992, 63024, 72068, 59776, 28897, 13424, -1817, -5360, -107, -2990, 2345, -450, 1081, -26, 209, -2, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 21*x^19 - 2*x^18 + 209*x^17 - 26*x^16 + 1081*x^15 - 450*x^14 + 2345*x^13 - 2990*x^12 - 107*x^11 - 5360*x^10 - 1817*x^9 + 13424*x^8 + 28897*x^7 + 59776*x^6 + 72068*x^5 + 63024*x^4 + 32992*x^3 + 7616*x^2 + 576*x + 64)
 
gp: K = bnfinit(x^21 + 21*x^19 - 2*x^18 + 209*x^17 - 26*x^16 + 1081*x^15 - 450*x^14 + 2345*x^13 - 2990*x^12 - 107*x^11 - 5360*x^10 - 1817*x^9 + 13424*x^8 + 28897*x^7 + 59776*x^6 + 72068*x^5 + 63024*x^4 + 32992*x^3 + 7616*x^2 + 576*x + 64, 1)
 

Normalized defining polynomial

\( x^{21} + 21 x^{19} - 2 x^{18} + 209 x^{17} - 26 x^{16} + 1081 x^{15} - 450 x^{14} + 2345 x^{13} - 2990 x^{12} - 107 x^{11} - 5360 x^{10} - 1817 x^{9} + 13424 x^{8} + 28897 x^{7} + 59776 x^{6} + 72068 x^{5} + 63024 x^{4} + 32992 x^{3} + 7616 x^{2} + 576 x + 64 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(297767057903624960000000000000000000=2^{26}\cdot 5^{19}\cdot 7^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.89$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} + \frac{1}{4} a^{15} - \frac{1}{2} a^{14} + \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{24} a^{18} + \frac{1}{24} a^{16} - \frac{5}{12} a^{15} - \frac{1}{8} a^{14} - \frac{1}{12} a^{13} - \frac{1}{8} a^{12} - \frac{1}{12} a^{11} - \frac{1}{8} a^{10} + \frac{1}{12} a^{9} - \frac{7}{24} a^{8} + \frac{11}{24} a^{6} - \frac{1}{3} a^{5} - \frac{1}{8} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{48} a^{19} + \frac{1}{48} a^{17} - \frac{5}{24} a^{16} - \frac{1}{16} a^{15} + \frac{11}{24} a^{14} + \frac{7}{16} a^{13} - \frac{1}{24} a^{12} + \frac{7}{16} a^{11} + \frac{1}{24} a^{10} + \frac{17}{48} a^{9} - \frac{13}{48} a^{7} - \frac{1}{6} a^{6} + \frac{7}{16} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{7440802033925902251331213732125384096} a^{20} - \frac{13423536120860434815804630378162871}{1860200508481475562832803433031346024} a^{19} - \frac{69175043431590557353184973480373303}{7440802033925902251331213732125384096} a^{18} - \frac{458618434644703813612905608983058623}{3720401016962951125665606866062692048} a^{17} - \frac{777151036444819041343401231998917123}{7440802033925902251331213732125384096} a^{16} + \frac{14958960582323196342549964670728179}{1240133672320983708555202288687564016} a^{15} + \frac{3482918404487676217358458286131130549}{7440802033925902251331213732125384096} a^{14} + \frac{40924482707939025233456965421513353}{286184693612534701974277451235591696} a^{13} - \frac{558589074608128078586141369024207035}{7440802033925902251331213732125384096} a^{12} + \frac{395413220006626284755057496971335941}{1240133672320983708555202288687564016} a^{11} - \frac{1218612720420036442869787582303587573}{2480267344641967417110404577375128032} a^{10} - \frac{18958374279058514669661696141723959}{47697448935422450329046241872598616} a^{9} + \frac{2019742579311695925523986288926938843}{7440802033925902251331213732125384096} a^{8} + \frac{274862911217291360283174591439498847}{1860200508481475562832803433031346024} a^{7} + \frac{461034391872830264028668377474736317}{7440802033925902251331213732125384096} a^{6} - \frac{854119896202595236297343439102695467}{1860200508481475562832803433031346024} a^{5} + \frac{346756577912525464323433303904117029}{930100254240737781416401716515673012} a^{4} + \frac{5201726128875489544718790254764457}{465050127120368890708200858257836506} a^{3} - \frac{61259511561073803418673215576884337}{232525063560184445354100429128918253} a^{2} + \frac{97634435332045682729832776879621143}{232525063560184445354100429128918253} a + \frac{21345694356883203065026487432433916}{232525063560184445354100429128918253}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $10$
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:  \( 2812507698.960895 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.980.1, 7.1.16807000000.2

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

Sibling fields

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{7}^{3}$
2.14.20.10$x^{14} + 2 x^{12} + 2 x^{11} + 4 x^{10} + 2 x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2$$14$$1$$20$$(C_7:C_3) \times C_2$$[2]_{7}^{3}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.1$x^{14} - 5$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed