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Properties

Label 21.13.168...529.1
Degree $21$
Signature $[13, 4]$
Discriminant $1.687\times 10^{37}$
Root discriminant \(59.25\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_7\wr C_3$ (as 21T159)

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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 - 21*x^19 - 3*x^18 + 186*x^17 + 54*x^16 - 898*x^15 - 396*x^14 + 2541*x^13 + 1511*x^12 - 4209*x^11 - 3210*x^10 + 3829*x^9 + 3846*x^8 - 1479*x^7 - 2499*x^6 - 303*x^5 + 537*x^4 + 190*x^3 - 15*x^2 - 12*x - 1)
 
Copy content gp:K = bnfinit(y^21 - 21*y^19 - 3*y^18 + 186*y^17 + 54*y^16 - 898*y^15 - 396*y^14 + 2541*y^13 + 1511*y^12 - 4209*y^11 - 3210*y^10 + 3829*y^9 + 3846*y^8 - 1479*y^7 - 2499*y^6 - 303*y^5 + 537*y^4 + 190*y^3 - 15*y^2 - 12*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 54*x^16 - 898*x^15 - 396*x^14 + 2541*x^13 + 1511*x^12 - 4209*x^11 - 3210*x^10 + 3829*x^9 + 3846*x^8 - 1479*x^7 - 2499*x^6 - 303*x^5 + 537*x^4 + 190*x^3 - 15*x^2 - 12*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 54*x^16 - 898*x^15 - 396*x^14 + 2541*x^13 + 1511*x^12 - 4209*x^11 - 3210*x^10 + 3829*x^9 + 3846*x^8 - 1479*x^7 - 2499*x^6 - 303*x^5 + 537*x^4 + 190*x^3 - 15*x^2 - 12*x - 1)
 

\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 54 x^{16} - 898 x^{15} - 396 x^{14} + 2541 x^{13} + \cdots - 1 \) 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:  $[13, 4]$
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:   \(16865213361316368677229831448817302529\) \(\medspace = 7^{14}\cdot 41\cdot 167\cdot 5406197\cdot 671780022444019\) 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:  \(59.25\)
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:  $7^{2/3}41^{1/2}167^{1/2}5406197^{1/2}671780022444019^{1/2}\approx 18247708437255.99$
Ramified primes:   \(7\), \(41\), \(167\), \(5406197\), \(671780022444019\) 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:  $\Q(\sqrt{24866\!\cdots\!82321}$)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{2}{9}a^{4}+\frac{4}{9}a^{3}-\frac{2}{9}a^{2}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{2}{9}a^{3}+\frac{1}{9}a^{2}-\frac{2}{9}a$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{4}{9}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{9}a^{17}-\frac{1}{9}a^{13}-\frac{1}{9}a^{9}-\frac{1}{9}a^{6}+\frac{1}{3}a^{5}-\frac{4}{9}a^{4}+\frac{2}{9}a^{3}-\frac{2}{9}a^{2}-\frac{2}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}+\frac{2}{9}a^{6}+\frac{2}{9}a^{5}-\frac{1}{3}a^{4}-\frac{4}{9}a^{3}-\frac{4}{9}a^{2}+\frac{4}{9}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{2}{9}a^{6}-\frac{1}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{2}{9}a^{6}-\frac{4}{9}a^{5}+\frac{1}{9}a^{4}-\frac{4}{9}a^{3}-\frac{1}{3}a^{2}-\frac{4}{9}a+\frac{2}{9}$ 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:  $16$
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:   $\frac{1}{9}a^{20}+\frac{1}{9}a^{19}-\frac{22}{9}a^{18}-\frac{8}{3}a^{17}+\frac{205}{9}a^{16}+\frac{241}{9}a^{15}-\frac{1048}{9}a^{14}-\frac{1313}{9}a^{13}+353a^{12}+\frac{4199}{9}a^{11}-\frac{5774}{9}a^{10}-\frac{2669}{3}a^{9}+\frac{2024}{3}a^{8}+\frac{8987}{9}a^{7}-\frac{3194}{9}a^{6}-\frac{5692}{9}a^{5}+\frac{55}{3}a^{4}+\frac{1541}{9}a^{3}+\frac{214}{9}a^{2}-\frac{139}{9}a-\frac{22}{9}$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{19}-\frac{22}{9}a^{18}-\frac{8}{3}a^{17}+\frac{205}{9}a^{16}+\frac{241}{9}a^{15}-\frac{1048}{9}a^{14}-\frac{1313}{9}a^{13}+353a^{12}+\frac{4199}{9}a^{11}-\frac{5774}{9}a^{10}-\frac{2669}{3}a^{9}+\frac{2024}{3}a^{8}+\frac{8987}{9}a^{7}-\frac{3194}{9}a^{6}-\frac{5692}{9}a^{5}+\frac{55}{3}a^{4}+\frac{1541}{9}a^{3}+\frac{214}{9}a^{2}-\frac{139}{9}a-\frac{13}{9}$, $a$, $a^{20}-a^{19}-20a^{18}+17a^{17}+169a^{16}-115a^{15}-783a^{14}+387a^{13}+2154a^{12}-643a^{11}-3566a^{10}+356a^{9}+3473a^{8}+373a^{7}-1852a^{6}-647a^{5}+344a^{4}+193a^{3}-3a^{2}-12a$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{19}-\frac{22}{9}a^{18}-\frac{8}{3}a^{17}+\frac{205}{9}a^{16}+\frac{241}{9}a^{15}-\frac{1048}{9}a^{14}-\frac{1313}{9}a^{13}+353a^{12}+\frac{4199}{9}a^{11}-\frac{5774}{9}a^{10}-\frac{2669}{3}a^{9}+\frac{2024}{3}a^{8}+\frac{8987}{9}a^{7}-\frac{3194}{9}a^{6}-\frac{5692}{9}a^{5}+\frac{55}{3}a^{4}+\frac{1541}{9}a^{3}+\frac{214}{9}a^{2}-\frac{148}{9}a-\frac{13}{9}$, $\frac{28}{9}a^{20}-\frac{2}{3}a^{19}-\frac{196}{3}a^{18}+\frac{44}{9}a^{17}+\frac{1741}{3}a^{16}+\frac{358}{9}a^{15}-2824a^{14}-\frac{1799}{3}a^{13}+\frac{73160}{9}a^{12}+\frac{25744}{9}a^{11}-\frac{41869}{3}a^{10}-\frac{61229}{9}a^{9}+\frac{41306}{3}a^{8}+\frac{79427}{9}a^{7}-\frac{61939}{9}a^{6}-6242a^{5}+\frac{1855}{3}a^{4}+\frac{14245}{9}a^{3}+\frac{596}{3}a^{2}-\frac{1036}{9}a-\frac{41}{3}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{18}-\frac{20}{3}a^{17}+\frac{17}{3}a^{16}+\frac{169}{3}a^{15}-\frac{115}{3}a^{14}-261a^{13}+129a^{12}+718a^{11}-\frac{643}{3}a^{10}-\frac{3566}{3}a^{9}+\frac{356}{3}a^{8}+\frac{3473}{3}a^{7}+\frac{373}{3}a^{6}-\frac{1853}{3}a^{5}-\frac{646}{3}a^{4}+\frac{350}{3}a^{3}+\frac{188}{3}a^{2}-\frac{8}{3}a-\frac{10}{3}$, $\frac{25}{9}a^{20}-\frac{7}{9}a^{19}-\frac{524}{9}a^{18}+8a^{17}+\frac{4649}{9}a^{16}+\frac{47}{9}a^{15}-\frac{22624}{9}a^{14}-\frac{3593}{9}a^{13}+\frac{21754}{3}a^{12}+\frac{19810}{9}a^{11}-\frac{112685}{9}a^{10}-5556a^{9}+12503a^{8}+7489a^{7}-\frac{19268}{3}a^{6}-\frac{49291}{9}a^{5}+705a^{4}+1466a^{3}+192a^{2}-\frac{290}{3}a-\frac{37}{3}$, $\frac{20}{3}a^{20}-\frac{19}{3}a^{19}-\frac{403}{3}a^{18}+\frac{971}{9}a^{17}+\frac{10300}{9}a^{16}-\frac{6586}{9}a^{15}-\frac{16048}{3}a^{14}+\frac{22249}{9}a^{13}+\frac{133690}{9}a^{12}-\frac{37241}{9}a^{11}-\frac{223501}{9}a^{10}+\frac{7058}{3}a^{9}+\frac{219761}{9}a^{8}+\frac{21239}{9}a^{7}-13150a^{6}-\frac{12817}{3}a^{5}+\frac{7678}{3}a^{4}+1315a^{3}-\frac{593}{9}a^{2}-77a-7$, $\frac{22}{3}a^{20}-\frac{22}{9}a^{19}-\frac{1376}{9}a^{18}+\frac{257}{9}a^{17}+\frac{4046}{3}a^{16}-\frac{139}{3}a^{15}-\frac{19565}{3}a^{14}-\frac{2347}{3}a^{13}+18675a^{12}+\frac{45476}{9}a^{11}-31952a^{10}-\frac{119719}{9}a^{9}+\frac{94565}{3}a^{8}+\frac{163439}{9}a^{7}-\frac{143204}{9}a^{6}-\frac{119233}{9}a^{5}+\frac{14666}{9}a^{4}+\frac{30415}{9}a^{3}+\frac{3721}{9}a^{2}-\frac{2015}{9}a-\frac{244}{9}$, $\frac{43}{9}a^{20}-\frac{11}{3}a^{19}-\frac{881}{9}a^{18}+\frac{551}{9}a^{17}+849a^{16}-\frac{3608}{9}a^{15}-\frac{36397}{9}a^{14}+\frac{3784}{3}a^{13}+\frac{103094}{9}a^{12}-\frac{15668}{9}a^{11}-\frac{175948}{9}a^{10}-\frac{371}{9}a^{9}+\frac{58847}{3}a^{8}+\frac{28075}{9}a^{7}-\frac{96571}{9}a^{6}-\frac{34024}{9}a^{5}+\frac{19286}{9}a^{4}+\frac{10139}{9}a^{3}-\frac{788}{9}a^{2}-\frac{590}{9}a-\frac{8}{3}$, $\frac{67}{9}a^{20}-\frac{11}{9}a^{19}-\frac{1403}{9}a^{18}+\frac{26}{9}a^{17}+\frac{12415}{9}a^{16}+182a^{15}-\frac{60083}{9}a^{14}-\frac{17078}{9}a^{13}+\frac{57152}{3}a^{12}+\frac{24857}{3}a^{11}-\frac{96641}{3}a^{10}-\frac{170347}{9}a^{9}+\frac{277370}{9}a^{8}+23880a^{7}-\frac{127280}{9}a^{6}-\frac{147349}{9}a^{5}-\frac{163}{9}a^{4}+\frac{35342}{9}a^{3}+\frac{7850}{9}a^{2}-\frac{1999}{9}a-\frac{530}{9}$, $\frac{13}{9}a^{20}-\frac{2}{9}a^{19}-31a^{18}+\frac{13}{9}a^{17}+\frac{845}{3}a^{16}+\frac{53}{3}a^{15}-\frac{12643}{9}a^{14}-\frac{2171}{9}a^{13}+\frac{37471}{9}a^{12}+\frac{10361}{9}a^{11}-\frac{22307}{3}a^{10}-\frac{8404}{3}a^{9}+\frac{70282}{9}a^{8}+\frac{11416}{3}a^{7}-\frac{13345}{3}a^{6}-\frac{26269}{9}a^{5}+\frac{8107}{9}a^{4}+\frac{8141}{9}a^{3}-\frac{2}{3}a^{2}-\frac{289}{3}a-\frac{25}{3}$, $\frac{17}{9}a^{20}-\frac{1}{3}a^{19}-\frac{359}{9}a^{18}+\frac{5}{3}a^{17}+\frac{3209}{9}a^{16}+\frac{307}{9}a^{15}-\frac{15731}{9}a^{14}-\frac{3655}{9}a^{13}+5078a^{12}+\frac{5536}{3}a^{11}-\frac{79471}{9}a^{10}-\frac{12941}{3}a^{9}+\frac{79961}{9}a^{8}+\frac{50152}{9}a^{7}-\frac{41525}{9}a^{6}-3972a^{5}+\frac{4960}{9}a^{4}+\frac{9533}{9}a^{3}+\frac{967}{9}a^{2}-\frac{752}{9}a-\frac{89}{9}$, $\frac{32}{9}a^{20}-\frac{23}{9}a^{19}-\frac{655}{9}a^{18}+\frac{374}{9}a^{17}+\frac{5674}{9}a^{16}-\frac{779}{3}a^{15}-\frac{26980}{9}a^{14}+\frac{6632}{9}a^{13}+\frac{25399}{3}a^{12}-681a^{11}-14371a^{10}-\frac{10186}{9}a^{9}+\frac{128327}{9}a^{8}+\frac{10316}{3}a^{7}-\frac{22723}{3}a^{6}-3390a^{5}+\frac{11560}{9}a^{4}+\frac{8305}{9}a^{3}+\frac{161}{9}a^{2}-\frac{454}{9}a-\frac{52}{9}$, $\frac{1}{9}a^{20}-\frac{1}{3}a^{19}-2a^{18}+\frac{19}{3}a^{17}+15a^{16}-\frac{151}{3}a^{15}-\frac{553}{9}a^{14}+\frac{1954}{9}a^{13}+153a^{12}-\frac{1654}{3}a^{11}-\frac{2243}{9}a^{10}+\frac{2510}{3}a^{9}+288a^{8}-\frac{6638}{9}a^{7}-\frac{2248}{9}a^{6}+\frac{3098}{9}a^{5}+\frac{439}{3}a^{4}-\frac{539}{9}a^{3}-\frac{274}{9}a^{2}+\frac{58}{9}a+\frac{19}{9}$ 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:  \( 169796731730 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{4}\cdot 169796731730 \cdot 1}{2\cdot\sqrt{16865213361316368677229831448817302529}}\cr\approx \mathstrut & 0.263944628479659 \end{aligned}\] (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 - 21*x^19 - 3*x^18 + 186*x^17 + 54*x^16 - 898*x^15 - 396*x^14 + 2541*x^13 + 1511*x^12 - 4209*x^11 - 3210*x^10 + 3829*x^9 + 3846*x^8 - 1479*x^7 - 2499*x^6 - 303*x^5 + 537*x^4 + 190*x^3 - 15*x^2 - 12*x - 1) 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 - 21*x^19 - 3*x^18 + 186*x^17 + 54*x^16 - 898*x^15 - 396*x^14 + 2541*x^13 + 1511*x^12 - 4209*x^11 - 3210*x^10 + 3829*x^9 + 3846*x^8 - 1479*x^7 - 2499*x^6 - 303*x^5 + 537*x^4 + 190*x^3 - 15*x^2 - 12*x - 1, 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 - 21*x^19 - 3*x^18 + 186*x^17 + 54*x^16 - 898*x^15 - 396*x^14 + 2541*x^13 + 1511*x^12 - 4209*x^11 - 3210*x^10 + 3829*x^9 + 3846*x^8 - 1479*x^7 - 2499*x^6 - 303*x^5 + 537*x^4 + 190*x^3 - 15*x^2 - 12*x - 1); 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 - 21*x^19 - 3*x^18 + 186*x^17 + 54*x^16 - 898*x^15 - 396*x^14 + 2541*x^13 + 1511*x^12 - 4209*x^11 - 3210*x^10 + 3829*x^9 + 3846*x^8 - 1479*x^7 - 2499*x^6 - 303*x^5 + 537*x^4 + 190*x^3 - 15*x^2 - 12*x - 1); 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

$S_7\wr C_3$ (as 21T159):

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 non-solvable group of order 384072192000
The 1165 conjugacy class representatives for $S_7\wr C_3$
Character table for $S_7\wr C_3$

Intermediate fields

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

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 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 ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ $21$ R $15{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ $15{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ $15{,}\,{\href{/padicField/19.6.0.1}{6} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ $15{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ $15{,}\,{\href{/padicField/37.6.0.1}{6} }$ R ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ $18{,}\,{\href{/padicField/47.3.0.1}{3} }$ $21$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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

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
\(7\) Copy content Toggle raw display 7.7.3.14a1.3$x^{21} + 18 x^{15} + 12 x^{14} + 108 x^{9} + 144 x^{8} + 48 x^{7} + 216 x^{3} + 432 x^{2} + 288 x + 71$$3$$7$$14$$C_{21}$not computed
\(41\) Copy content Toggle raw display $\Q_{41}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.1.2.1a1.2$x^{2} + 246$$2$$1$$1$$C_2$$$[\ ]_{2}$$
41.3.1.0a1.1$x^{3} + x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
41.3.1.0a1.1$x^{3} + x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
41.3.1.0a1.1$x^{3} + x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
41.6.1.0a1.1$x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(167\) Copy content Toggle raw display $\Q_{167}$$x + 162$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{167}$$x + 162$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{167}$$x + 162$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{167}$$x + 162$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{167}$$x + 162$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{167}$$x + 162$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{167}$$x + 162$$1$$1$$0$Trivial$$[\ ]$$
167.1.2.1a1.2$x^{2} + 835$$2$$1$$1$$C_2$$$[\ ]_{2}$$
167.3.1.0a1.1$x^{3} + 7 x + 162$$1$$3$$0$$C_3$$$[\ ]^{3}$$
167.4.1.0a1.1$x^{4} + 3 x^{2} + 120 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
167.5.1.0a1.1$x^{5} + 3 x + 162$$1$$5$$0$$C_5$$$[\ ]^{5}$$
\(5406197\) Copy content Toggle raw display $\Q_{5406197}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{5406197}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{5406197}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $7$$1$$7$$0$$C_7$$$[\ ]^{7}$$
\(671780022444019\) Copy content Toggle raw display $\Q_{671780022444019}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{671780022444019}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$

Spectrum of ring of integers

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