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Properties

Label 21.1.265...721.1
Degree $21$
Signature $[1, 10]$
Discriminant $2.653\times 10^{29}$
Root discriminant \(25.18\)
Ramified primes $3,151$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^6:D_7$ (as 21T51)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 8*x^20 + 36*x^19 - 114*x^18 + 285*x^17 - 562*x^16 + 781*x^15 - 627*x^14 + 169*x^13 - 27*x^12 + 434*x^11 - 868*x^10 + 760*x^9 - 92*x^8 - 397*x^7 + 225*x^6 + 212*x^5 - 441*x^4 + 355*x^3 - 154*x^2 + 34*x - 3)
 
Copy content gp:K = bnfinit(y^21 - 8*y^20 + 36*y^19 - 114*y^18 + 285*y^17 - 562*y^16 + 781*y^15 - 627*y^14 + 169*y^13 - 27*y^12 + 434*y^11 - 868*y^10 + 760*y^9 - 92*y^8 - 397*y^7 + 225*y^6 + 212*y^5 - 441*y^4 + 355*y^3 - 154*y^2 + 34*y - 3, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 8*x^20 + 36*x^19 - 114*x^18 + 285*x^17 - 562*x^16 + 781*x^15 - 627*x^14 + 169*x^13 - 27*x^12 + 434*x^11 - 868*x^10 + 760*x^9 - 92*x^8 - 397*x^7 + 225*x^6 + 212*x^5 - 441*x^4 + 355*x^3 - 154*x^2 + 34*x - 3);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 8*x^20 + 36*x^19 - 114*x^18 + 285*x^17 - 562*x^16 + 781*x^15 - 627*x^14 + 169*x^13 - 27*x^12 + 434*x^11 - 868*x^10 + 760*x^9 - 92*x^8 - 397*x^7 + 225*x^6 + 212*x^5 - 441*x^4 + 355*x^3 - 154*x^2 + 34*x - 3)
 

\( x^{21} - 8 x^{20} + 36 x^{19} - 114 x^{18} + 285 x^{17} - 562 x^{16} + 781 x^{15} - 627 x^{14} + \cdots - 3 \) 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:  $[1, 10]$
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:   \(265283078340996577927735640721\) \(\medspace = 3^{16}\cdot 151^{10}\) 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:  \(25.18\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(3\), \(151\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{159}a^{19}+\frac{19}{159}a^{18}-\frac{73}{159}a^{17}-\frac{70}{159}a^{16}+\frac{76}{159}a^{15}+\frac{11}{53}a^{14}+\frac{11}{53}a^{13}-\frac{23}{53}a^{12}+\frac{40}{159}a^{11}-\frac{24}{53}a^{10}+\frac{4}{159}a^{9}-\frac{19}{159}a^{8}-\frac{5}{53}a^{7}+\frac{32}{159}a^{6}-\frac{61}{159}a^{5}-\frac{20}{159}a^{4}-\frac{23}{53}a^{3}-\frac{40}{159}a^{2}+\frac{58}{159}a+\frac{19}{53}$, $\frac{1}{20\cdots 63}a^{20}-\frac{32\cdots 96}{20\cdots 63}a^{19}+\frac{35\cdots 40}{20\cdots 63}a^{18}-\frac{68\cdots 49}{20\cdots 63}a^{17}-\frac{85\cdots 68}{20\cdots 63}a^{16}-\frac{30\cdots 08}{66\cdots 21}a^{15}+\frac{21\cdots 83}{66\cdots 21}a^{14}-\frac{90\cdots 65}{66\cdots 21}a^{13}-\frac{64\cdots 74}{28\cdots 09}a^{12}-\frac{11\cdots 03}{66\cdots 21}a^{11}+\frac{72\cdots 80}{20\cdots 63}a^{10}-\frac{87\cdots 06}{20\cdots 63}a^{9}+\frac{20\cdots 25}{66\cdots 21}a^{8}+\frac{61\cdots 28}{20\cdots 63}a^{7}-\frac{67\cdots 15}{20\cdots 63}a^{6}-\frac{86\cdots 08}{20\cdots 63}a^{5}-\frac{47\cdots 92}{95\cdots 03}a^{4}-\frac{50\cdots 16}{28\cdots 09}a^{3}+\frac{51\cdots 71}{20\cdots 63}a^{2}+\frac{12\cdots 65}{66\cdots 21}a-\frac{30\cdots 11}{66\cdots 21}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

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:  $10$
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:   $\frac{18\cdots 39}{66\cdots 21}a^{20}-\frac{42\cdots 56}{20\cdots 63}a^{19}+\frac{18\cdots 31}{20\cdots 63}a^{18}-\frac{57\cdots 14}{20\cdots 63}a^{17}+\frac{14\cdots 74}{20\cdots 63}a^{16}-\frac{27\cdots 81}{20\cdots 63}a^{15}+\frac{11\cdots 19}{66\cdots 21}a^{14}-\frac{83\cdots 80}{66\cdots 21}a^{13}+\frac{12\cdots 22}{95\cdots 03}a^{12}-\frac{74\cdots 83}{20\cdots 63}a^{11}+\frac{79\cdots 83}{66\cdots 21}a^{10}-\frac{41\cdots 86}{20\cdots 63}a^{9}+\frac{58\cdots 27}{37\cdots 71}a^{8}+\frac{10\cdots 03}{66\cdots 21}a^{7}-\frac{21\cdots 18}{20\cdots 63}a^{6}+\frac{68\cdots 37}{20\cdots 63}a^{5}+\frac{19\cdots 22}{28\cdots 09}a^{4}-\frac{98\cdots 94}{95\cdots 03}a^{3}+\frac{14\cdots 62}{20\cdots 63}a^{2}-\frac{47\cdots 96}{20\cdots 63}a+\frac{19\cdots 82}{66\cdots 21}$, $\frac{13\cdots 40}{20\cdots 63}a^{20}-\frac{10\cdots 96}{20\cdots 63}a^{19}+\frac{85\cdots 43}{37\cdots 71}a^{18}-\frac{13\cdots 30}{20\cdots 63}a^{17}+\frac{33\cdots 88}{20\cdots 63}a^{16}-\frac{21\cdots 87}{66\cdots 21}a^{15}+\frac{26\cdots 85}{66\cdots 21}a^{14}-\frac{17\cdots 25}{66\cdots 21}a^{13}-\frac{16\cdots 88}{28\cdots 09}a^{12}-\frac{28\cdots 85}{66\cdots 21}a^{11}+\frac{59\cdots 92}{20\cdots 63}a^{10}-\frac{96\cdots 47}{20\cdots 63}a^{9}+\frac{21\cdots 50}{66\cdots 21}a^{8}+\frac{17\cdots 11}{20\cdots 63}a^{7}-\frac{51\cdots 31}{20\cdots 63}a^{6}+\frac{99\cdots 73}{20\cdots 63}a^{5}+\frac{17\cdots 53}{95\cdots 03}a^{4}-\frac{67\cdots 17}{28\cdots 09}a^{3}+\frac{28\cdots 20}{20\cdots 63}a^{2}-\frac{24\cdots 11}{66\cdots 21}a+\frac{95\cdots 41}{66\cdots 21}$, $\frac{50\cdots 88}{20\cdots 63}a^{20}-\frac{12\cdots 26}{66\cdots 21}a^{19}+\frac{16\cdots 16}{20\cdots 63}a^{18}-\frac{17\cdots 05}{66\cdots 21}a^{17}+\frac{12\cdots 58}{20\cdots 63}a^{16}-\frac{23\cdots 98}{20\cdots 63}a^{15}+\frac{10\cdots 90}{66\cdots 21}a^{14}-\frac{66\cdots 01}{66\cdots 21}a^{13}+\frac{97\cdots 92}{28\cdots 09}a^{12}-\frac{10\cdots 31}{20\cdots 63}a^{11}+\frac{21\cdots 29}{20\cdots 63}a^{10}-\frac{35\cdots 97}{20\cdots 63}a^{9}+\frac{24\cdots 02}{20\cdots 63}a^{8}+\frac{51\cdots 31}{20\cdots 63}a^{7}-\frac{60\cdots 46}{66\cdots 21}a^{6}+\frac{41\cdots 74}{20\cdots 63}a^{5}+\frac{17\cdots 41}{28\cdots 09}a^{4}-\frac{24\cdots 88}{28\cdots 09}a^{3}+\frac{11\cdots 57}{20\cdots 63}a^{2}-\frac{34\cdots 02}{20\cdots 63}a+\frac{12\cdots 81}{66\cdots 21}$, $\frac{20\cdots 59}{95\cdots 03}a^{20}-\frac{35\cdots 64}{28\cdots 09}a^{19}+\frac{12\cdots 75}{28\cdots 09}a^{18}-\frac{28\cdots 82}{28\cdots 09}a^{17}+\frac{53\cdots 73}{28\cdots 09}a^{16}-\frac{60\cdots 94}{28\cdots 09}a^{15}-\frac{11\cdots 97}{95\cdots 03}a^{14}+\frac{68\cdots 27}{95\cdots 03}a^{13}-\frac{55\cdots 32}{95\cdots 03}a^{12}-\frac{11\cdots 66}{28\cdots 09}a^{11}+\frac{59\cdots 48}{95\cdots 03}a^{10}+\frac{16\cdots 10}{28\cdots 09}a^{9}-\frac{24\cdots 90}{28\cdots 09}a^{8}+\frac{95\cdots 25}{95\cdots 03}a^{7}+\frac{44\cdots 65}{28\cdots 09}a^{6}-\frac{20\cdots 12}{28\cdots 09}a^{5}+\frac{82\cdots 82}{28\cdots 09}a^{4}+\frac{12\cdots 69}{95\cdots 03}a^{3}-\frac{11\cdots 28}{28\cdots 09}a^{2}+\frac{57\cdots 88}{28\cdots 09}a-\frac{26\cdots 94}{95\cdots 03}$, $\frac{60\cdots 69}{20\cdots 63}a^{20}-\frac{47\cdots 41}{66\cdots 21}a^{19}+\frac{99\cdots 18}{20\cdots 63}a^{18}-\frac{13\cdots 79}{66\cdots 21}a^{17}+\frac{11\cdots 00}{20\cdots 63}a^{16}-\frac{27\cdots 71}{20\cdots 63}a^{15}+\frac{16\cdots 28}{66\cdots 21}a^{14}-\frac{18\cdots 08}{66\cdots 21}a^{13}+\frac{32\cdots 50}{28\cdots 09}a^{12}+\frac{11\cdots 26}{20\cdots 63}a^{11}+\frac{91\cdots 23}{20\cdots 63}a^{10}-\frac{54\cdots 66}{20\cdots 63}a^{9}+\frac{67\cdots 50}{20\cdots 63}a^{8}-\frac{28\cdots 76}{20\cdots 63}a^{7}-\frac{10\cdots 43}{66\cdots 21}a^{6}+\frac{32\cdots 53}{20\cdots 63}a^{5}+\frac{10\cdots 37}{28\cdots 09}a^{4}-\frac{44\cdots 75}{28\cdots 09}a^{3}+\frac{30\cdots 54}{20\cdots 63}a^{2}-\frac{12\cdots 61}{20\cdots 63}a+\frac{60\cdots 57}{66\cdots 21}$, $\frac{22\cdots 98}{20\cdots 63}a^{20}-\frac{56\cdots 37}{66\cdots 21}a^{19}+\frac{72\cdots 76}{20\cdots 63}a^{18}-\frac{73\cdots 28}{66\cdots 21}a^{17}+\frac{53\cdots 49}{20\cdots 63}a^{16}-\frac{19\cdots 42}{37\cdots 71}a^{15}+\frac{42\cdots 59}{66\cdots 21}a^{14}-\frac{26\cdots 31}{66\cdots 21}a^{13}-\frac{10\cdots 30}{28\cdots 09}a^{12}-\frac{48\cdots 93}{20\cdots 63}a^{11}+\frac{94\cdots 33}{20\cdots 63}a^{10}-\frac{15\cdots 42}{20\cdots 63}a^{9}+\frac{10\cdots 34}{20\cdots 63}a^{8}+\frac{27\cdots 46}{20\cdots 63}a^{7}-\frac{26\cdots 39}{66\cdots 21}a^{6}+\frac{14\cdots 63}{20\cdots 63}a^{5}+\frac{78\cdots 63}{28\cdots 09}a^{4}-\frac{10\cdots 33}{28\cdots 09}a^{3}+\frac{44\cdots 76}{20\cdots 63}a^{2}-\frac{12\cdots 43}{20\cdots 63}a+\frac{34\cdots 15}{66\cdots 21}$, $\frac{31\cdots 91}{66\cdots 21}a^{20}-\frac{25\cdots 88}{66\cdots 21}a^{19}+\frac{11\cdots 40}{66\cdots 21}a^{18}-\frac{34\cdots 30}{66\cdots 21}a^{17}+\frac{84\cdots 91}{66\cdots 21}a^{16}-\frac{16\cdots 01}{66\cdots 21}a^{15}+\frac{21\cdots 17}{66\cdots 21}a^{14}-\frac{15\cdots 26}{66\cdots 21}a^{13}+\frac{29\cdots 04}{95\cdots 03}a^{12}-\frac{37\cdots 19}{66\cdots 21}a^{11}+\frac{14\cdots 95}{66\cdots 21}a^{10}-\frac{25\cdots 86}{66\cdots 21}a^{9}+\frac{19\cdots 84}{66\cdots 21}a^{8}+\frac{11\cdots 46}{66\cdots 21}a^{7}-\frac{12\cdots 28}{66\cdots 21}a^{6}+\frac{44\cdots 34}{66\cdots 21}a^{5}+\frac{11\cdots 42}{95\cdots 03}a^{4}-\frac{18\cdots 83}{95\cdots 03}a^{3}+\frac{88\cdots 86}{66\cdots 21}a^{2}-\frac{29\cdots 92}{66\cdots 21}a+\frac{39\cdots 13}{66\cdots 21}$, $\frac{30\cdots 22}{20\cdots 63}a^{20}-\frac{22\cdots 41}{20\cdots 63}a^{19}+\frac{97\cdots 54}{20\cdots 63}a^{18}-\frac{29\cdots 52}{20\cdots 63}a^{17}+\frac{71\cdots 15}{20\cdots 63}a^{16}-\frac{44\cdots 27}{66\cdots 21}a^{15}+\frac{56\cdots 11}{66\cdots 21}a^{14}-\frac{34\cdots 13}{66\cdots 21}a^{13}-\frac{67\cdots 99}{28\cdots 09}a^{12}-\frac{22\cdots 98}{66\cdots 21}a^{11}+\frac{12\cdots 64}{20\cdots 63}a^{10}-\frac{20\cdots 77}{20\cdots 63}a^{9}+\frac{42\cdots 84}{66\cdots 21}a^{8}+\frac{41\cdots 13}{20\cdots 63}a^{7}-\frac{10\cdots 55}{20\cdots 63}a^{6}+\frac{15\cdots 66}{20\cdots 63}a^{5}+\frac{36\cdots 85}{95\cdots 03}a^{4}-\frac{13\cdots 97}{28\cdots 09}a^{3}+\frac{57\cdots 05}{20\cdots 63}a^{2}-\frac{51\cdots 39}{66\cdots 21}a+\frac{32\cdots 17}{66\cdots 21}$, $\frac{11\cdots 96}{28\cdots 09}a^{20}-\frac{30\cdots 16}{95\cdots 03}a^{19}+\frac{39\cdots 45}{28\cdots 09}a^{18}-\frac{40\cdots 68}{95\cdots 03}a^{17}+\frac{29\cdots 54}{28\cdots 09}a^{16}-\frac{55\cdots 24}{28\cdots 09}a^{15}+\frac{23\cdots 57}{95\cdots 03}a^{14}-\frac{15\cdots 89}{95\cdots 03}a^{13}+\frac{18\cdots 56}{28\cdots 09}a^{12}-\frac{41\cdots 93}{28\cdots 09}a^{11}+\frac{50\cdots 53}{28\cdots 09}a^{10}-\frac{81\cdots 18}{28\cdots 09}a^{9}+\frac{58\cdots 98}{28\cdots 09}a^{8}+\frac{10\cdots 05}{28\cdots 09}a^{7}-\frac{14\cdots 58}{95\cdots 03}a^{6}+\frac{95\cdots 85}{28\cdots 09}a^{5}+\frac{29\cdots 46}{28\cdots 09}a^{4}-\frac{41\cdots 35}{28\cdots 09}a^{3}+\frac{26\cdots 32}{28\cdots 09}a^{2}-\frac{76\cdots 98}{28\cdots 09}a+\frac{26\cdots 53}{95\cdots 03}$, $\frac{32\cdots 03}{54\cdots 53}a^{20}-\frac{13\cdots 87}{28\cdots 09}a^{19}+\frac{56\cdots 77}{28\cdots 09}a^{18}-\frac{17\cdots 58}{28\cdots 09}a^{17}+\frac{41\cdots 13}{28\cdots 09}a^{16}-\frac{26\cdots 93}{95\cdots 03}a^{15}+\frac{33\cdots 64}{95\cdots 03}a^{14}-\frac{21\cdots 69}{95\cdots 03}a^{13}+\frac{11\cdots 48}{28\cdots 09}a^{12}-\frac{12\cdots 16}{95\cdots 03}a^{11}+\frac{72\cdots 15}{28\cdots 09}a^{10}-\frac{11\cdots 98}{28\cdots 09}a^{9}+\frac{26\cdots 44}{95\cdots 03}a^{8}+\frac{19\cdots 57}{28\cdots 09}a^{7}-\frac{60\cdots 50}{28\cdots 09}a^{6}+\frac{12\cdots 18}{28\cdots 09}a^{5}+\frac{14\cdots 16}{95\cdots 03}a^{4}-\frac{57\cdots 98}{28\cdots 09}a^{3}+\frac{35\cdots 53}{28\cdots 09}a^{2}-\frac{35\cdots 16}{95\cdots 03}a+\frac{33\cdots 85}{95\cdots 03}$ 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:  \( 2440129.53307 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 2440129.53307 \cdot 1}{2\cdot\sqrt{265283078340996577927735640721}}\cr\approx \mathstrut & 0.454314701420 \end{aligned}\] (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 - 8*x^20 + 36*x^19 - 114*x^18 + 285*x^17 - 562*x^16 + 781*x^15 - 627*x^14 + 169*x^13 - 27*x^12 + 434*x^11 - 868*x^10 + 760*x^9 - 92*x^8 - 397*x^7 + 225*x^6 + 212*x^5 - 441*x^4 + 355*x^3 - 154*x^2 + 34*x - 3) 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 - 8*x^20 + 36*x^19 - 114*x^18 + 285*x^17 - 562*x^16 + 781*x^15 - 627*x^14 + 169*x^13 - 27*x^12 + 434*x^11 - 868*x^10 + 760*x^9 - 92*x^8 - 397*x^7 + 225*x^6 + 212*x^5 - 441*x^4 + 355*x^3 - 154*x^2 + 34*x - 3, 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 - 8*x^20 + 36*x^19 - 114*x^18 + 285*x^17 - 562*x^16 + 781*x^15 - 627*x^14 + 169*x^13 - 27*x^12 + 434*x^11 - 868*x^10 + 760*x^9 - 92*x^8 - 397*x^7 + 225*x^6 + 212*x^5 - 441*x^4 + 355*x^3 - 154*x^2 + 34*x - 3); 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 = PolynomialRing(QQ); K, a = NumberField(x^21 - 8*x^20 + 36*x^19 - 114*x^18 + 285*x^17 - 562*x^16 + 781*x^15 - 627*x^14 + 169*x^13 - 27*x^12 + 434*x^11 - 868*x^10 + 760*x^9 - 92*x^8 - 397*x^7 + 225*x^6 + 212*x^5 - 441*x^4 + 355*x^3 - 154*x^2 + 34*x - 3); 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

$C_3^6:D_7$ (as 21T51):

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); G, transitive_group_identification(G)
 
A solvable group of order 10206
The 96 conjugacy class representatives for $C_3^6:D_7$
Character table for $C_3^6:D_7$

Intermediate fields

7.1.3442951.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 21 siblings: data not computed
Degree 42 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }^{3}$ R ${\href{/padicField/5.7.0.1}{7} }^{3}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.7.0.1}{7} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.7.0.1}{7} }^{3}$ ${\href{/padicField/19.7.0.1}{7} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }^{3}$ ${\href{/padicField/31.7.0.1}{7} }^{3}$ ${\href{/padicField/37.7.0.1}{7} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }^{3}$ ${\href{/padicField/47.7.0.1}{7} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.7.0.1}{7} }^{3}$

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
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.3.8a1.2$x^{6} + 6 x^{5} + 21 x^{4} + 44 x^{3} + 60 x^{2} + 57 x + 23$$3$$2$$8$$S_3\times C_3$$$[2]^{6}$$
3.2.3.8a4.3$x^{6} + 9 x^{5} + 33 x^{4} + 68 x^{3} + 84 x^{2} + 60 x + 41$$3$$2$$8$$S_3\times C_3$$$[2, 2]^{2}$$
3.6.1.0a1.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(151\) Copy content Toggle raw display $\Q_{151}$$x + 145$$1$$1$$0$Trivial$$[\ ]$$
151.1.2.1a1.1$x^{2} + 151$$2$$1$$1$$C_2$$$[\ ]_{2}$$
151.1.2.1a1.1$x^{2} + 151$$2$$1$$1$$C_2$$$[\ ]_{2}$$
151.1.2.1a1.1$x^{2} + 151$$2$$1$$1$$C_2$$$[\ ]_{2}$$
151.1.2.1a1.1$x^{2} + 151$$2$$1$$1$$C_2$$$[\ ]_{2}$$
151.1.2.1a1.1$x^{2} + 151$$2$$1$$1$$C_2$$$[\ ]_{2}$$
151.1.2.1a1.1$x^{2} + 151$$2$$1$$1$$C_2$$$[\ ]_{2}$$
151.1.2.1a1.1$x^{2} + 151$$2$$1$$1$$C_2$$$[\ ]_{2}$$
151.3.2.3a1.2$x^{6} + 2 x^{4} + 290 x^{3} + x^{2} + 290 x + 21176$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Spectrum of ring of integers

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