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Properties

Label 21.7.297...000.1
Degree $21$
Signature $[7, 7]$
Discriminant $-2.973\times 10^{40}$
Root discriminant \(84.59\)
Ramified primes $2,5,7,41$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_7^2:S_3$ (as 21T18)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 9*x^19 + 124*x^18 - 631*x^17 + 802*x^16 + 1826*x^15 - 27193*x^14 + 110548*x^13 - 224650*x^12 + 303049*x^11 + 450353*x^10 - 265331*x^9 + 796296*x^8 + 5445541*x^7 - 12181122*x^6 - 15402246*x^5 + 20340053*x^4 + 8297097*x^3 - 25045665*x^2 + 26864880*x + 1121605)
 
Copy content gp:K = bnfinit(y^21 - 3*y^20 - 9*y^19 + 124*y^18 - 631*y^17 + 802*y^16 + 1826*y^15 - 27193*y^14 + 110548*y^13 - 224650*y^12 + 303049*y^11 + 450353*y^10 - 265331*y^9 + 796296*y^8 + 5445541*y^7 - 12181122*y^6 - 15402246*y^5 + 20340053*y^4 + 8297097*y^3 - 25045665*y^2 + 26864880*y + 1121605, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^20 - 9*x^19 + 124*x^18 - 631*x^17 + 802*x^16 + 1826*x^15 - 27193*x^14 + 110548*x^13 - 224650*x^12 + 303049*x^11 + 450353*x^10 - 265331*x^9 + 796296*x^8 + 5445541*x^7 - 12181122*x^6 - 15402246*x^5 + 20340053*x^4 + 8297097*x^3 - 25045665*x^2 + 26864880*x + 1121605);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 3*x^20 - 9*x^19 + 124*x^18 - 631*x^17 + 802*x^16 + 1826*x^15 - 27193*x^14 + 110548*x^13 - 224650*x^12 + 303049*x^11 + 450353*x^10 - 265331*x^9 + 796296*x^8 + 5445541*x^7 - 12181122*x^6 - 15402246*x^5 + 20340053*x^4 + 8297097*x^3 - 25045665*x^2 + 26864880*x + 1121605)
 

\( x^{21} - 3 x^{20} - 9 x^{19} + 124 x^{18} - 631 x^{17} + 802 x^{16} + 1826 x^{15} - 27193 x^{14} + \cdots + 1121605 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $21$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[7, 7]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-29731207187895151973927972800000000000000\) \(\medspace = -\,2^{18}\cdot 5^{14}\cdot 7^{7}\cdot 41^{12}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(84.59\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(5\), \(7\), \(41\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$:   $C_7$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}$, $\frac{1}{5}a^{12}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}$, $\frac{1}{5}a^{14}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}$, $\frac{1}{5}a^{15}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{9}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{3}$, $\frac{1}{25}a^{17}+\frac{1}{25}a^{15}-\frac{1}{25}a^{14}+\frac{2}{25}a^{13}+\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{9}{25}a^{7}-\frac{1}{5}a^{6}-\frac{1}{25}a^{5}-\frac{4}{25}a^{4}+\frac{3}{25}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{25}a^{18}+\frac{1}{25}a^{16}-\frac{1}{25}a^{15}+\frac{2}{25}a^{14}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{11}{25}a^{8}+\frac{2}{5}a^{7}-\frac{6}{25}a^{6}+\frac{11}{25}a^{5}-\frac{12}{25}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{25}a^{19}-\frac{1}{25}a^{16}+\frac{1}{25}a^{15}+\frac{1}{25}a^{14}-\frac{2}{25}a^{13}+\frac{2}{5}a^{10}-\frac{11}{25}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{6}{25}a^{6}-\frac{6}{25}a^{5}-\frac{6}{25}a^{4}+\frac{7}{25}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{99\cdots 75}a^{20}+\frac{99\cdots 76}{99\cdots 75}a^{19}-\frac{22\cdots 73}{19\cdots 75}a^{18}+\frac{10\cdots 09}{99\cdots 75}a^{17}-\frac{13\cdots 06}{19\cdots 75}a^{16}+\frac{76\cdots 12}{99\cdots 75}a^{15}-\frac{66\cdots 41}{99\cdots 75}a^{14}+\frac{83\cdots 33}{98\cdots 75}a^{13}-\frac{52\cdots 24}{19\cdots 75}a^{12}-\frac{77\cdots 06}{19\cdots 75}a^{11}+\frac{21\cdots 29}{99\cdots 75}a^{10}+\frac{12\cdots 69}{99\cdots 75}a^{9}-\frac{63\cdots 24}{19\cdots 75}a^{8}+\frac{18\cdots 46}{99\cdots 75}a^{7}+\frac{98\cdots 07}{19\cdots 75}a^{6}+\frac{44\cdots 38}{99\cdots 75}a^{5}+\frac{42\cdots 71}{99\cdots 75}a^{4}-\frac{12\cdots 78}{99\cdots 75}a^{3}-\frac{65\cdots 23}{19\cdots 75}a^{2}+\frac{30\cdots 66}{39\cdots 55}a-\frac{85\cdots 24}{19\cdots 75}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $5$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $13$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{41\cdots 12}{41\cdots 75}a^{20}+\frac{22\cdots 16}{41\cdots 75}a^{19}-\frac{10\cdots 44}{82\cdots 15}a^{18}-\frac{25\cdots 48}{41\cdots 75}a^{17}+\frac{28\cdots 27}{41\cdots 75}a^{16}-\frac{13\cdots 56}{41\cdots 75}a^{15}+\frac{27\cdots 36}{82\cdots 15}a^{14}+\frac{46\cdots 88}{41\cdots 75}a^{13}-\frac{11\cdots 92}{82\cdots 15}a^{12}+\frac{45\cdots 04}{82\cdots 15}a^{11}-\frac{45\cdots 28}{41\cdots 75}a^{10}+\frac{66\cdots 84}{41\cdots 75}a^{9}+\frac{11\cdots 76}{82\cdots 15}a^{8}+\frac{99\cdots 88}{41\cdots 75}a^{7}-\frac{51\cdots 52}{41\cdots 75}a^{6}+\frac{14\cdots 56}{41\cdots 75}a^{5}-\frac{45\cdots 84}{82\cdots 15}a^{4}-\frac{43\cdots 08}{41\cdots 75}a^{3}+\frac{78\cdots 96}{16\cdots 23}a^{2}+\frac{95\cdots 12}{82\cdots 15}a-\frac{18\cdots 19}{82\cdots 15}$, $\frac{18\cdots 74}{19\cdots 75}a^{20}+\frac{31\cdots 94}{19\cdots 75}a^{19}+\frac{20\cdots 31}{19\cdots 75}a^{18}-\frac{20\cdots 52}{19\cdots 75}a^{17}+\frac{91\cdots 63}{19\cdots 75}a^{16}-\frac{30\cdots 07}{19\cdots 75}a^{15}-\frac{15\cdots 70}{79\cdots 31}a^{14}+\frac{18\cdots 03}{78\cdots 31}a^{13}-\frac{29\cdots 26}{39\cdots 55}a^{12}+\frac{45\cdots 33}{39\cdots 55}a^{11}-\frac{27\cdots 91}{19\cdots 75}a^{10}-\frac{11\cdots 59}{19\cdots 75}a^{9}-\frac{10\cdots 01}{19\cdots 75}a^{8}-\frac{29\cdots 98}{19\cdots 75}a^{7}-\frac{14\cdots 03}{19\cdots 75}a^{6}+\frac{38\cdots 42}{19\cdots 75}a^{5}+\frac{13\cdots 75}{79\cdots 31}a^{4}+\frac{13\cdots 43}{39\cdots 55}a^{3}-\frac{10\cdots 69}{39\cdots 55}a^{2}+\frac{15\cdots 30}{79\cdots 31}a+\frac{61\cdots 23}{78\cdots 31}$, $\frac{19\cdots 04}{99\cdots 75}a^{20}+\frac{50\cdots 96}{99\cdots 75}a^{19}+\frac{37\cdots 92}{19\cdots 75}a^{18}-\frac{22\cdots 71}{99\cdots 75}a^{17}+\frac{22\cdots 74}{19\cdots 75}a^{16}-\frac{11\cdots 33}{99\cdots 75}a^{15}-\frac{38\cdots 51}{99\cdots 75}a^{14}+\frac{49\cdots 48}{98\cdots 75}a^{13}-\frac{38\cdots 79}{19\cdots 75}a^{12}+\frac{71\cdots 04}{19\cdots 75}a^{11}-\frac{44\cdots 16}{99\cdots 75}a^{10}-\frac{10\cdots 26}{99\cdots 75}a^{9}+\frac{22\cdots 26}{19\cdots 75}a^{8}-\frac{14\cdots 24}{99\cdots 75}a^{7}-\frac{22\cdots 73}{19\cdots 75}a^{6}+\frac{18\cdots 58}{99\cdots 75}a^{5}+\frac{34\cdots 06}{99\cdots 75}a^{4}-\frac{28\cdots 68}{99\cdots 75}a^{3}-\frac{59\cdots 68}{19\cdots 75}a^{2}+\frac{15\cdots 84}{39\cdots 55}a-\frac{50\cdots 19}{19\cdots 75}$, $\frac{56\cdots 74}{99\cdots 75}a^{20}-\frac{65\cdots 01}{99\cdots 75}a^{19}+\frac{10\cdots 33}{19\cdots 75}a^{18}+\frac{12\cdots 21}{99\cdots 75}a^{17}-\frac{18\cdots 44}{19\cdots 75}a^{16}+\frac{30\cdots 68}{99\cdots 75}a^{15}-\frac{37\cdots 64}{99\cdots 75}a^{14}-\frac{25\cdots 73}{98\cdots 75}a^{13}+\frac{37\cdots 09}{19\cdots 75}a^{12}-\frac{10\cdots 44}{19\cdots 75}a^{11}+\frac{84\cdots 21}{99\cdots 75}a^{10}-\frac{46\cdots 44}{99\cdots 75}a^{9}-\frac{65\cdots 76}{19\cdots 75}a^{8}-\frac{84\cdots 51}{99\cdots 75}a^{7}-\frac{26\cdots 27}{19\cdots 75}a^{6}-\frac{38\cdots 68}{99\cdots 75}a^{5}+\frac{20\cdots 09}{99\cdots 75}a^{4}+\frac{12\cdots 18}{99\cdots 75}a^{3}-\frac{26\cdots 47}{19\cdots 75}a^{2}-\frac{94\cdots 29}{79\cdots 31}a+\frac{13\cdots 94}{19\cdots 75}$, $\frac{55\cdots 46}{99\cdots 75}a^{20}-\frac{94\cdots 64}{99\cdots 75}a^{19}-\frac{11\cdots 09}{19\cdots 75}a^{18}+\frac{60\cdots 09}{99\cdots 75}a^{17}-\frac{10\cdots 61}{39\cdots 55}a^{16}+\frac{11\cdots 92}{99\cdots 75}a^{15}+\frac{10\cdots 24}{99\cdots 75}a^{14}-\frac{13\cdots 47}{98\cdots 75}a^{13}+\frac{88\cdots 41}{19\cdots 75}a^{12}-\frac{14\cdots 76}{19\cdots 75}a^{11}+\frac{85\cdots 34}{99\cdots 75}a^{10}+\frac{36\cdots 09}{99\cdots 75}a^{9}+\frac{44\cdots 42}{19\cdots 75}a^{8}+\frac{11\cdots 96}{99\cdots 75}a^{7}+\frac{82\cdots 16}{19\cdots 75}a^{6}-\frac{97\cdots 17}{99\cdots 75}a^{5}-\frac{72\cdots 44}{99\cdots 75}a^{4}-\frac{92\cdots 73}{99\cdots 75}a^{3}+\frac{27\cdots 02}{19\cdots 75}a^{2}-\frac{33\cdots 07}{39\cdots 55}a-\frac{68\cdots 59}{19\cdots 75}$, $\frac{90\cdots 36}{19\cdots 75}a^{20}-\frac{20\cdots 59}{19\cdots 75}a^{19}-\frac{93\cdots 07}{19\cdots 75}a^{18}+\frac{10\cdots 82}{19\cdots 75}a^{17}-\frac{49\cdots 92}{19\cdots 75}a^{16}+\frac{39\cdots 47}{19\cdots 75}a^{15}+\frac{17\cdots 27}{19\cdots 75}a^{14}-\frac{23\cdots 91}{19\cdots 75}a^{13}+\frac{16\cdots 62}{39\cdots 55}a^{12}-\frac{60\cdots 31}{79\cdots 31}a^{11}+\frac{19\cdots 54}{19\cdots 75}a^{10}+\frac{50\cdots 09}{19\cdots 75}a^{9}+\frac{20\cdots 77}{19\cdots 75}a^{8}+\frac{11\cdots 43}{19\cdots 75}a^{7}+\frac{60\cdots 17}{19\cdots 75}a^{6}-\frac{58\cdots 67}{19\cdots 75}a^{5}-\frac{15\cdots 52}{19\cdots 75}a^{4}+\frac{59\cdots 11}{19\cdots 75}a^{3}+\frac{94\cdots 58}{39\cdots 55}a^{2}-\frac{80\cdots 13}{79\cdots 31}a+\frac{20\cdots 83}{39\cdots 55}$, $\frac{19\cdots 96}{19\cdots 75}a^{20}+\frac{61\cdots 71}{19\cdots 75}a^{19}+\frac{31\cdots 93}{39\cdots 55}a^{18}-\frac{48\cdots 28}{39\cdots 55}a^{17}+\frac{12\cdots 38}{19\cdots 75}a^{16}-\frac{18\cdots 41}{19\cdots 75}a^{15}-\frac{30\cdots 71}{19\cdots 75}a^{14}+\frac{52\cdots 36}{19\cdots 75}a^{13}-\frac{44\cdots 74}{39\cdots 55}a^{12}+\frac{96\cdots 57}{39\cdots 55}a^{11}-\frac{69\cdots 44}{19\cdots 75}a^{10}-\frac{71\cdots 81}{19\cdots 75}a^{9}+\frac{12\cdots 67}{39\cdots 55}a^{8}-\frac{37\cdots 36}{39\cdots 55}a^{7}-\frac{10\cdots 33}{19\cdots 75}a^{6}+\frac{25\cdots 01}{19\cdots 75}a^{5}+\frac{23\cdots 26}{19\cdots 75}a^{4}-\frac{43\cdots 11}{19\cdots 75}a^{3}-\frac{34\cdots 87}{39\cdots 55}a^{2}+\frac{11\cdots 92}{39\cdots 55}a-\frac{13\cdots 08}{39\cdots 55}$, $\frac{91\cdots 88}{19\cdots 75}a^{20}-\frac{15\cdots 51}{19\cdots 75}a^{19}-\frac{10\cdots 24}{19\cdots 75}a^{18}+\frac{10\cdots 69}{19\cdots 75}a^{17}-\frac{89\cdots 09}{39\cdots 55}a^{16}+\frac{16\cdots 63}{19\cdots 75}a^{15}+\frac{18\cdots 73}{19\cdots 75}a^{14}-\frac{22\cdots 04}{19\cdots 75}a^{13}+\frac{14\cdots 94}{39\cdots 55}a^{12}-\frac{22\cdots 06}{39\cdots 55}a^{11}+\frac{14\cdots 82}{19\cdots 75}a^{10}+\frac{56\cdots 86}{19\cdots 75}a^{9}+\frac{56\cdots 09}{19\cdots 75}a^{8}+\frac{13\cdots 51}{19\cdots 75}a^{7}+\frac{13\cdots 06}{39\cdots 55}a^{6}-\frac{21\cdots 73}{19\cdots 75}a^{5}-\frac{16\cdots 53}{19\cdots 75}a^{4}-\frac{42\cdots 36}{19\cdots 75}a^{3}+\frac{64\cdots 97}{39\cdots 55}a^{2}-\frac{77\cdots 20}{79\cdots 31}a-\frac{17\cdots 58}{39\cdots 55}$, $\frac{52\cdots 12}{99\cdots 75}a^{20}-\frac{15\cdots 13}{99\cdots 75}a^{19}-\frac{92\cdots 86}{19\cdots 75}a^{18}+\frac{65\cdots 08}{99\cdots 75}a^{17}-\frac{66\cdots 87}{19\cdots 75}a^{16}+\frac{43\cdots 19}{99\cdots 75}a^{15}+\frac{92\cdots 58}{99\cdots 75}a^{14}-\frac{14\cdots 79}{98\cdots 75}a^{13}+\frac{11\cdots 47}{19\cdots 75}a^{12}-\frac{24\cdots 27}{19\cdots 75}a^{11}+\frac{16\cdots 48}{99\cdots 75}a^{10}+\frac{22\cdots 53}{99\cdots 75}a^{9}-\frac{26\cdots 08}{19\cdots 75}a^{8}+\frac{41\cdots 52}{99\cdots 75}a^{7}+\frac{57\cdots 24}{19\cdots 75}a^{6}-\frac{65\cdots 44}{99\cdots 75}a^{5}-\frac{76\cdots 98}{99\cdots 75}a^{4}+\frac{10\cdots 14}{99\cdots 75}a^{3}+\frac{79\cdots 24}{19\cdots 75}a^{2}-\frac{52\cdots 88}{39\cdots 55}a+\frac{28\cdots 37}{19\cdots 75}$, $\frac{45\cdots 63}{99\cdots 75}a^{20}-\frac{94\cdots 12}{99\cdots 75}a^{19}-\frac{21\cdots 58}{39\cdots 55}a^{18}+\frac{52\cdots 62}{99\cdots 75}a^{17}-\frac{46\cdots 49}{19\cdots 75}a^{16}+\frac{94\cdots 06}{99\cdots 75}a^{15}+\frac{11\cdots 12}{99\cdots 75}a^{14}-\frac{11\cdots 06}{98\cdots 75}a^{13}+\frac{77\cdots 33}{19\cdots 75}a^{12}-\frac{10\cdots 73}{19\cdots 75}a^{11}+\frac{48\cdots 02}{99\cdots 75}a^{10}+\frac{30\cdots 22}{99\cdots 75}a^{9}+\frac{22\cdots 84}{19\cdots 75}a^{8}+\frac{13\cdots 03}{99\cdots 75}a^{7}+\frac{48\cdots 42}{19\cdots 75}a^{6}-\frac{35\cdots 31}{99\cdots 75}a^{5}-\frac{13\cdots 72}{99\cdots 75}a^{4}+\frac{14\cdots 96}{99\cdots 75}a^{3}+\frac{33\cdots 91}{19\cdots 75}a^{2}+\frac{18\cdots 83}{39\cdots 55}a+\frac{13\cdots 18}{19\cdots 75}$, $\frac{81\cdots 77}{99\cdots 75}a^{20}+\frac{82\cdots 63}{99\cdots 75}a^{19}+\frac{25\cdots 89}{19\cdots 75}a^{18}-\frac{85\cdots 73}{99\cdots 75}a^{17}+\frac{60\cdots 77}{19\cdots 75}a^{16}+\frac{41\cdots 21}{99\cdots 75}a^{15}-\frac{29\cdots 68}{99\cdots 75}a^{14}+\frac{18\cdots 44}{98\cdots 75}a^{13}-\frac{90\cdots 87}{19\cdots 75}a^{12}-\frac{14\cdots 98}{19\cdots 75}a^{11}+\frac{12\cdots 67}{99\cdots 75}a^{10}-\frac{80\cdots 78}{99\cdots 75}a^{9}-\frac{16\cdots 02}{39\cdots 55}a^{8}-\frac{76\cdots 37}{99\cdots 75}a^{7}-\frac{44\cdots 74}{19\cdots 75}a^{6}+\frac{34\cdots 29}{99\cdots 75}a^{5}+\frac{42\cdots 58}{99\cdots 75}a^{4}+\frac{24\cdots 21}{99\cdots 75}a^{3}-\frac{10\cdots 24}{19\cdots 75}a^{2}-\frac{34\cdots 29}{39\cdots 55}a-\frac{47\cdots 82}{19\cdots 75}$, $\frac{60\cdots 21}{99\cdots 75}a^{20}+\frac{51\cdots 26}{99\cdots 75}a^{19}-\frac{38\cdots 54}{19\cdots 75}a^{18}-\frac{23\cdots 11}{99\cdots 75}a^{17}+\frac{12\cdots 22}{19\cdots 75}a^{16}-\frac{37\cdots 63}{99\cdots 75}a^{15}+\frac{72\cdots 84}{99\cdots 75}a^{14}-\frac{12\cdots 17}{98\cdots 75}a^{13}-\frac{26\cdots 69}{19\cdots 75}a^{12}+\frac{13\cdots 29}{19\cdots 75}a^{11}-\frac{17\cdots 41}{99\cdots 75}a^{10}+\frac{35\cdots 69}{99\cdots 75}a^{9}-\frac{33\cdots 33}{19\cdots 75}a^{8}+\frac{36\cdots 16}{99\cdots 75}a^{7}+\frac{93\cdots 59}{19\cdots 75}a^{6}+\frac{27\cdots 88}{99\cdots 75}a^{5}-\frac{62\cdots 54}{99\cdots 75}a^{4}-\frac{55\cdots 53}{99\cdots 75}a^{3}+\frac{62\cdots 27}{19\cdots 75}a^{2}-\frac{24\cdots 19}{39\cdots 55}a+\frac{11\cdots 76}{19\cdots 75}$, $\frac{12\cdots 57}{99\cdots 75}a^{20}-\frac{18\cdots 73}{99\cdots 75}a^{19}-\frac{34\cdots 58}{19\cdots 75}a^{18}+\frac{13\cdots 28}{99\cdots 75}a^{17}-\frac{10\cdots 08}{19\cdots 75}a^{16}-\frac{25\cdots 46}{99\cdots 75}a^{15}+\frac{39\cdots 73}{99\cdots 75}a^{14}-\frac{29\cdots 54}{98\cdots 75}a^{13}+\frac{16\cdots 32}{19\cdots 75}a^{12}-\frac{10\cdots 02}{19\cdots 75}a^{11}-\frac{80\cdots 97}{99\cdots 75}a^{10}+\frac{10\cdots 63}{99\cdots 75}a^{9}+\frac{20\cdots 79}{19\cdots 75}a^{8}-\frac{14\cdots 68}{99\cdots 75}a^{7}+\frac{45\cdots 21}{39\cdots 55}a^{6}-\frac{77\cdots 29}{99\cdots 75}a^{5}-\frac{45\cdots 63}{99\cdots 75}a^{4}+\frac{13\cdots 89}{99\cdots 75}a^{3}+\frac{98\cdots 54}{19\cdots 75}a^{2}-\frac{96\cdots 87}{39\cdots 55}a-\frac{88\cdots 88}{19\cdots 75}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4451604187499.273 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 4451604187499.273 \cdot 1}{2\cdot\sqrt{29731207187895151973927972800000000000000}}\cr\approx \mathstrut & 0.638777077448137 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 9*x^19 + 124*x^18 - 631*x^17 + 802*x^16 + 1826*x^15 - 27193*x^14 + 110548*x^13 - 224650*x^12 + 303049*x^11 + 450353*x^10 - 265331*x^9 + 796296*x^8 + 5445541*x^7 - 12181122*x^6 - 15402246*x^5 + 20340053*x^4 + 8297097*x^3 - 25045665*x^2 + 26864880*x + 1121605) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 - 3*x^20 - 9*x^19 + 124*x^18 - 631*x^17 + 802*x^16 + 1826*x^15 - 27193*x^14 + 110548*x^13 - 224650*x^12 + 303049*x^11 + 450353*x^10 - 265331*x^9 + 796296*x^8 + 5445541*x^7 - 12181122*x^6 - 15402246*x^5 + 20340053*x^4 + 8297097*x^3 - 25045665*x^2 + 26864880*x + 1121605, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^20 - 9*x^19 + 124*x^18 - 631*x^17 + 802*x^16 + 1826*x^15 - 27193*x^14 + 110548*x^13 - 224650*x^12 + 303049*x^11 + 450353*x^10 - 265331*x^9 + 796296*x^8 + 5445541*x^7 - 12181122*x^6 - 15402246*x^5 + 20340053*x^4 + 8297097*x^3 - 25045665*x^2 + 26864880*x + 1121605); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 3*x^20 - 9*x^19 + 124*x^18 - 631*x^17 + 802*x^16 + 1826*x^15 - 27193*x^14 + 110548*x^13 - 224650*x^12 + 303049*x^11 + 450353*x^10 - 265331*x^9 + 796296*x^8 + 5445541*x^7 - 12181122*x^6 - 15402246*x^5 + 20340053*x^4 + 8297097*x^3 - 25045665*x^2 + 26864880*x + 1121605); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_7^2:S_3$ (as 21T18):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 294
The 20 conjugacy class representatives for $C_7^2:S_3$
Character table for $C_7^2:S_3$

Intermediate fields

3.1.175.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 14 sibling: data not computed
Degree 21 sibling: data not computed
Degree 42 siblings: data not computed
Minimal sibling: $ x^{14} - 7 x^{13} + 25 x^{12} + 23 x^{11} - 191 x^{10} - 721 x^{9} + 18403 x^{8} - 110198 x^{7} + 452906 x^{6} - 1297862 x^{5} + 2876330 x^{4} - 4653736 x^{3} + 5702548 x^{2} - 4378774 x + 2373928 $

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.7.0.1}{7} }$ R R ${\href{/padicField/11.3.0.1}{3} }^{7}$ ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.7.0.1}{7} }$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ ${\href{/padicField/23.3.0.1}{3} }^{7}$ ${\href{/padicField/29.3.0.1}{3} }^{7}$ ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.7.0.1}{7} }$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ R ${\href{/padicField/43.3.0.1}{3} }^{7}$ ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.7.0.1}{7} }$ ${\href{/padicField/53.7.0.1}{7} }^{3}$ ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.7.0.1}{7} }$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.21.18.1$x^{21} + 7 x^{19} + 7 x^{18} + 21 x^{17} + 42 x^{16} + 56 x^{15} + 105 x^{14} + 140 x^{13} + 175 x^{12} + 231 x^{11} + 245 x^{10} + 252 x^{9} + 252 x^{8} + 211 x^{7} + 168 x^{6} + 126 x^{5} + 77 x^{4} + 42 x^{3} + 21 x^{2} + 7 x + 3$$7$$3$$18$21T2$$[\ ]_{7}^{3}$$
\(5\) Copy content Toggle raw display 5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
\(7\) Copy content Toggle raw display 7.7.0.1$x^{7} + 6 x + 4$$1$$7$$0$$C_7$$$[\ ]^{7}$$
7.14.7.1$x^{14} + 12 x^{8} + 8 x^{7} + 36 x^{2} + 48 x + 23$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(41\) Copy content Toggle raw display 41.7.0.1$x^{7} + 6 x + 35$$1$$7$$0$$C_7$$$[\ ]^{7}$$
41.14.12.4$x^{14} + 266 x^{13} + 30366 x^{12} + 1930096 x^{11} + 73890236 x^{10} + 1710174648 x^{9} + 22401114568 x^{8} + 134798826368 x^{7} + 134406687408 x^{6} + 61566287328 x^{5} + 15960290976 x^{4} + 2501404416 x^{3} + 236126016 x^{2} + 12410619 x + 281371$$7$$2$$12$$C_7 \wr C_2$$$[\ ]_{7}^{14}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)