LMFDB - Number field 21.13.40453417591156198845459427294648145741.1

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*x - 1)

gp:K = bnfinit(y^21 - 21*y^19 - 3*y^18 + 186*y^17 + 45*y^16 - 907*y^15 - 270*y^14 + 2685*y^13 + 854*y^12 - 5055*y^11 - 1653*y^10 + 6124*y^9 + 2244*y^8 - 4458*y^7 - 2040*y^6 + 1560*y^5 + 960*y^4 - 80*y^3 - 123*y^2 - 21*y - 1, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*x - 1);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*x - 1)

$ x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 45 x^{16} - 907 x^{15} - 270 x^{14} + 2685 x^{13} + \cdots - 1 $ x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*x - 1

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$21$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[13, 4]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$40453417591156198845459427294648145741$ 40453417591156198845459427294648145741 $\medspace = 7^{14}\cdot 59646183107903161080974909$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$61.77$	sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:(1.0 * dK)^(1/degree(K))
Galois root discriminant:	$7^{2/3}59646183107903161080974909^{1/2}\approx 28261162488532.168$
Ramified primes:	$7$, $59646183107903161080974909$ 7, 59646183107903161080974909	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{59646\!\cdots\!74909}$)
$\Aut(K/\Q)$:	$C_1$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); 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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{2}{9}a^{4}+\frac{4}{9}a^{3}-\frac{2}{9}a^{2}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{2}{9}a^{3}+\frac{1}{9}a^{2}-\frac{2}{9}a$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{4}{9}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{9}a^{17}-\frac{1}{9}a^{13}-\frac{1}{9}a^{9}-\frac{1}{9}a^{6}+\frac{1}{3}a^{5}-\frac{4}{9}a^{4}+\frac{2}{9}a^{3}-\frac{2}{9}a^{2}-\frac{2}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}+\frac{2}{9}a^{6}+\frac{2}{9}a^{5}-\frac{1}{3}a^{4}-\frac{4}{9}a^{3}-\frac{4}{9}a^{2}+\frac{4}{9}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{2}{9}a^{6}-\frac{1}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{2}{9}a^{6}-\frac{4}{9}a^{5}+\frac{1}{9}a^{4}-\frac{4}{9}a^{3}-\frac{1}{3}a^{2}-\frac{4}{9}a+\frac{2}{9}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/3*a^7 - 1/3*a^5 - 1/3*a^4 + 1/3*a^3 + 1/3*a - 1/3, 1/3*a^8 - 1/3*a^6 - 1/3*a^5 + 1/3*a^4 + 1/3*a^2 - 1/3*a, 1/3*a^9 - 1/3*a^6 - 1/3*a^4 - 1/3*a^3 - 1/3*a^2 + 1/3*a - 1/3, 1/3*a^10 + 1/3*a^5 + 1/3*a^4 + 1/3*a^2 - 1/3, 1/3*a^11 + 1/3*a^6 + 1/3*a^5 + 1/3*a^3 - 1/3*a, 1/3*a^12 + 1/3*a^6 + 1/3*a^5 - 1/3*a^4 - 1/3*a^3 - 1/3*a^2 - 1/3*a + 1/3, 1/3*a^13 + 1/3*a^6 + 1/3*a^3 - 1/3*a^2 + 1/3, 1/9*a^14 + 1/9*a^12 + 1/9*a^11 - 1/9*a^9 + 1/9*a^8 - 1/9*a^7 - 1/9*a^6 - 2/9*a^4 + 4/9*a^3 - 2/9*a^2 + 1/9*a - 2/9, 1/9*a^15 + 1/9*a^13 + 1/9*a^12 - 1/9*a^10 + 1/9*a^9 - 1/9*a^8 - 1/9*a^7 - 2/9*a^5 + 4/9*a^4 - 2/9*a^3 + 1/9*a^2 - 2/9*a, 1/9*a^16 + 1/9*a^13 - 1/9*a^12 + 1/9*a^11 + 1/9*a^10 + 1/9*a^8 + 1/9*a^7 - 1/9*a^6 + 4/9*a^5 + 1/3*a^4 + 1/3*a^2 + 2/9*a + 2/9, 1/9*a^17 - 1/9*a^13 - 1/9*a^9 - 1/9*a^6 + 1/3*a^5 - 4/9*a^4 + 2/9*a^3 - 2/9*a^2 - 2/9*a - 4/9, 1/9*a^18 + 1/9*a^12 + 1/9*a^11 - 1/9*a^10 - 1/9*a^9 + 1/9*a^8 + 1/9*a^7 + 2/9*a^6 + 2/9*a^5 - 1/3*a^4 - 4/9*a^3 - 4/9*a^2 + 4/9, 1/9*a^19 + 1/9*a^13 + 1/9*a^12 - 1/9*a^11 - 1/9*a^10 + 1/9*a^9 + 1/9*a^8 - 1/9*a^7 + 2/9*a^6 - 1/9*a^4 + 2/9*a^3 + 1/9*a + 1/3, 1/9*a^20 + 1/9*a^13 + 1/9*a^12 + 1/9*a^11 + 1/9*a^10 - 1/9*a^9 + 1/9*a^8 - 2/9*a^6 - 4/9*a^5 + 1/9*a^4 - 4/9*a^3 - 1/3*a^2 - 4/9*a + 2/9

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

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

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$16$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity sage:UK.torsion_generator() gp:K.tu[2] magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar:torsion_units_generator(OK)
Fundamental units:	$\frac{1}{3}a^{19}-\frac{1}{3}a^{18}-\frac{20}{3}a^{17}+\frac{17}{3}a^{16}+\frac{169}{3}a^{15}-\frac{124}{3}a^{14}-261a^{13}+171a^{12}+724a^{11}-\frac{1318}{3}a^{10}-\frac{3737}{3}a^{9}+\frac{2084}{3}a^{8}+\frac{4040}{3}a^{7}-\frac{1796}{3}a^{6}-\frac{2663}{3}a^{5}+\frac{623}{3}a^{4}+\frac{944}{3}a^{3}+\frac{17}{3}a^{2}-\frac{110}{3}a-\frac{16}{3}$, $\frac{1}{9}a^{20}+\frac{7}{9}a^{19}-\frac{28}{9}a^{18}-16a^{17}+\frac{307}{9}a^{16}+\frac{1246}{9}a^{15}-\frac{1810}{9}a^{14}-\frac{5885}{9}a^{13}+726a^{12}+\frac{16583}{9}a^{11}-\frac{15320}{9}a^{10}-3216a^{9}+2575a^{8}+\frac{31787}{9}a^{7}-\frac{20441}{9}a^{6}-\frac{21256}{9}a^{5}+\frac{2813}{3}a^{4}+\frac{7565}{9}a^{3}-\frac{746}{9}a^{2}-\frac{862}{9}a-\frac{82}{9}$, $a$, $\frac{4}{9}a^{20}-\frac{4}{9}a^{19}-9a^{18}+\frac{70}{9}a^{17}+\frac{694}{9}a^{16}-59a^{15}-\frac{3266}{9}a^{14}+\frac{770}{3}a^{13}+\frac{3071}{3}a^{12}-\frac{6311}{9}a^{11}-\frac{5360}{3}a^{10}+\frac{10799}{9}a^{9}+\frac{17423}{9}a^{8}-\frac{10595}{9}a^{7}-\frac{11227}{9}a^{6}+\frac{1670}{3}a^{5}+\frac{3773}{9}a^{4}-\frac{239}{3}a^{3}-\frac{409}{9}a^{2}-\frac{22}{9}a-\frac{8}{9}$, $a+1$, $3a^{20}+\frac{2}{9}a^{19}-\frac{571}{9}a^{18}-\frac{121}{9}a^{17}+\frac{5101}{9}a^{16}+\frac{1552}{9}a^{15}-\frac{25159}{9}a^{14}-\frac{8830}{9}a^{13}+\frac{25226}{3}a^{12}+\frac{9046}{3}a^{11}-\frac{145720}{9}a^{10}-\frac{16931}{3}a^{9}+\frac{60674}{3}a^{8}+\frac{65335}{9}a^{7}-\frac{46204}{3}a^{6}-\frac{18770}{3}a^{5}+\frac{53405}{9}a^{4}+\frac{8570}{3}a^{3}-\frac{1861}{3}a^{2}-\frac{1105}{3}a-\frac{236}{9}$, $\frac{17}{9}a^{20}-\frac{26}{9}a^{19}-37a^{18}+\frac{158}{3}a^{17}+306a^{16}-413a^{15}-\frac{12428}{9}a^{14}+\frac{5473}{3}a^{13}+\frac{33097}{9}a^{12}-\frac{44569}{9}a^{11}-\frac{52868}{9}a^{10}+\frac{75172}{9}a^{9}+\frac{49873}{9}a^{8}-\frac{75074}{9}a^{7}-\frac{8857}{3}a^{6}+\frac{40201}{9}a^{5}+\frac{7571}{9}a^{4}-\frac{9677}{9}a^{3}-\frac{983}{9}a^{2}+\frac{235}{3}a+\frac{62}{9}$, $\frac{20}{9}a^{20}-\frac{4}{9}a^{19}-\frac{419}{9}a^{18}+\frac{25}{9}a^{17}+\frac{3709}{9}a^{16}+\frac{136}{9}a^{15}-\frac{18100}{9}a^{14}-\frac{1612}{9}a^{13}+\frac{53632}{9}a^{12}+\frac{5692}{9}a^{11}-\frac{100912}{9}a^{10}-1272a^{9}+\frac{122032}{9}a^{8}+\frac{6269}{3}a^{7}-\frac{89603}{9}a^{6}-\frac{21838}{9}a^{5}+\frac{33527}{9}a^{4}+\frac{4042}{3}a^{3}-\frac{1174}{3}a^{2}-189a-\frac{115}{9}$, $a^{20}-\frac{2}{3}a^{19}-\frac{61}{3}a^{18}+\frac{31}{3}a^{17}+\frac{524}{3}a^{16}-\frac{203}{3}a^{15}-\frac{2473}{3}a^{14}+252a^{13}+2343a^{12}-594a^{11}-\frac{12529}{3}a^{10}+\frac{2515}{3}a^{9}+\frac{14204}{3}a^{8}-\frac{1348}{3}a^{7}-\frac{9782}{3}a^{6}-\frac{797}{3}a^{5}+\frac{3437}{3}a^{4}+\frac{1010}{3}a^{3}-\frac{289}{3}a^{2}-\frac{176}{3}a-\frac{13}{3}$, $\frac{43}{9}a^{20}-\frac{38}{9}a^{19}-\frac{869}{9}a^{18}+71a^{17}+\frac{7424}{9}a^{16}-\frac{4627}{9}a^{15}-\frac{34825}{9}a^{14}+\frac{6392}{3}a^{13}+\frac{98060}{9}a^{12}-\frac{49963}{9}a^{11}-\frac{171805}{9}a^{10}+\frac{80696}{9}a^{9}+\frac{189227}{9}a^{8}-\frac{70307}{9}a^{7}-\frac{126041}{9}a^{6}+\frac{22838}{9}a^{5}+\frac{44245}{9}a^{4}+\frac{2833}{9}a^{3}-\frac{1649}{3}a^{2}-\frac{1096}{9}a-\frac{14}{3}$, $\frac{2}{3}a^{20}+\frac{8}{9}a^{19}-15a^{18}-\frac{178}{9}a^{17}+\frac{1274}{9}a^{16}+\frac{541}{3}a^{15}-\frac{6638}{9}a^{14}-\frac{7894}{9}a^{13}+\frac{21256}{9}a^{12}+\frac{22510}{9}a^{11}-\frac{14699}{3}a^{10}-\frac{39604}{9}a^{9}+\frac{59669}{9}a^{8}+\frac{44849}{9}a^{7}-\frac{48664}{9}a^{6}-\frac{32056}{9}a^{5}+2174a^{4}+\frac{4129}{3}a^{3}-\frac{674}{3}a^{2}-\frac{496}{3}a-\frac{37}{3}$, $\frac{29}{9}a^{20}-\frac{26}{9}a^{19}-\frac{587}{9}a^{18}+\frac{440}{9}a^{17}+\frac{5027}{9}a^{16}-\frac{3214}{9}a^{15}-\frac{23665}{9}a^{14}+\frac{13475}{9}a^{13}+\frac{22324}{3}a^{12}-\frac{11888}{3}a^{11}-\frac{39374}{3}a^{10}+\frac{59084}{9}a^{9}+\frac{43700}{3}a^{8}-\frac{54533}{9}a^{7}-\frac{29320}{3}a^{6}+\frac{21820}{9}a^{5}+\frac{10342}{3}a^{4}-\frac{896}{9}a^{3}-\frac{3496}{9}a^{2}-\frac{136}{3}a-\frac{2}{3}$, $\frac{5}{3}a^{20}-\frac{13}{9}a^{19}-\frac{304}{9}a^{18}+\frac{73}{3}a^{17}+\frac{2606}{9}a^{16}-\frac{1594}{9}a^{15}-\frac{12277}{9}a^{14}+\frac{2225}{3}a^{13}+\frac{11587}{3}a^{12}-\frac{17687}{9}a^{11}-\frac{61315}{9}a^{10}+3255a^{9}+7553a^{8}-\frac{26720}{9}a^{7}-\frac{45416}{9}a^{6}+\frac{10088}{9}a^{5}+\frac{15887}{9}a^{4}+\frac{17}{3}a^{3}-\frac{1753}{9}a^{2}-\frac{269}{9}a-\frac{10}{9}$, $\frac{125}{9}a^{20}-\frac{37}{9}a^{19}-\frac{2606}{9}a^{18}+\frac{388}{9}a^{17}+\frac{7658}{3}a^{16}-\frac{1031}{9}a^{15}-\frac{37235}{3}a^{14}-\frac{1757}{9}a^{13}+\frac{109934}{3}a^{12}+\frac{13663}{9}a^{11}-\frac{618269}{9}a^{10}-\frac{35590}{9}a^{9}+\frac{745645}{9}a^{8}+\frac{79447}{9}a^{7}-\frac{548135}{9}a^{6}-\frac{110948}{9}a^{5}+\frac{68920}{3}a^{4}+\frac{66514}{9}a^{3}-\frac{22463}{9}a^{2}-\frac{9466}{9}a-\frac{629}{9}$, $\frac{22}{9}a^{20}+\frac{2}{9}a^{19}-\frac{466}{9}a^{18}-\frac{34}{3}a^{17}+\frac{1388}{3}a^{16}+\frac{419}{3}a^{15}-2279a^{14}-\frac{2324}{3}a^{13}+\frac{61490}{9}a^{12}+\frac{20914}{9}a^{11}-\frac{39212}{3}a^{10}-\frac{38287}{9}a^{9}+\frac{145289}{9}a^{8}+5434a^{7}-12057a^{6}-4724a^{5}+\frac{40352}{9}a^{4}+\frac{19663}{9}a^{3}-\frac{3671}{9}a^{2}-\frac{2564}{9}a-\frac{245}{9}$, $\frac{22}{9}a^{20}-\frac{16}{9}a^{19}-\frac{452}{9}a^{18}+\frac{89}{3}a^{17}+\frac{3922}{9}a^{16}-\frac{643}{3}a^{15}-\frac{6229}{3}a^{14}+\frac{8074}{9}a^{13}+\frac{17840}{3}a^{12}-\frac{21530}{9}a^{11}-\frac{95614}{9}a^{10}+\frac{11876}{3}a^{9}+\frac{107657}{9}a^{8}-3439a^{7}-\frac{24487}{3}a^{6}+\frac{8762}{9}a^{5}+\frac{26435}{9}a^{4}+\frac{2414}{9}a^{3}-\frac{3001}{9}a^{2}-\frac{223}{3}a-\frac{31}{9}$ 1/3a^19 - 1/3a^18 - 20/3a^17 + 17/3a^16 + 169/3a^15 - 124/3a^14 - 261a^13 + 171a^12 + 724a^11 - 1318/3a^10 - 3737/3a^9 + 2084/3a^8 + 4040/3a^7 - 1796/3a^6 - 2663/3a^5 + 623/3a^4 + 944/3a^3 + 17/3a^2 - 110/3a - 16/3, 1/9a^20 + 7/9a^19 - 28/9a^18 - 16a^17 + 307/9a^16 + 1246/9a^15 - 1810/9a^14 - 5885/9a^13 + 726a^12 + 16583/9a^11 - 15320/9a^10 - 3216a^9 + 2575a^8 + 31787/9a^7 - 20441/9a^6 - 21256/9a^5 + 2813/3a^4 + 7565/9a^3 - 746/9a^2 - 862/9a - 82/9, a, 4/9a^20 - 4/9a^19 - 9a^18 + 70/9a^17 + 694/9a^16 - 59a^15 - 3266/9a^14 + 770/3a^13 + 3071/3a^12 - 6311/9a^11 - 5360/3a^10 + 10799/9a^9 + 17423/9a^8 - 10595/9a^7 - 11227/9a^6 + 1670/3a^5 + 3773/9a^4 - 239/3a^3 - 409/9a^2 - 22/9a - 8/9, a + 1, 3a^20 + 2/9a^19 - 571/9a^18 - 121/9a^17 + 5101/9a^16 + 1552/9a^15 - 25159/9a^14 - 8830/9a^13 + 25226/3a^12 + 9046/3a^11 - 145720/9a^10 - 16931/3a^9 + 60674/3a^8 + 65335/9a^7 - 46204/3a^6 - 18770/3a^5 + 53405/9a^4 + 8570/3a^3 - 1861/3a^2 - 1105/3a - 236/9, 17/9a^20 - 26/9a^19 - 37a^18 + 158/3a^17 + 306a^16 - 413a^15 - 12428/9a^14 + 5473/3a^13 + 33097/9a^12 - 44569/9a^11 - 52868/9a^10 + 75172/9a^9 + 49873/9a^8 - 75074/9a^7 - 8857/3a^6 + 40201/9a^5 + 7571/9a^4 - 9677/9a^3 - 983/9a^2 + 235/3a + 62/9, 20/9a^20 - 4/9a^19 - 419/9a^18 + 25/9a^17 + 3709/9a^16 + 136/9a^15 - 18100/9a^14 - 1612/9a^13 + 53632/9a^12 + 5692/9a^11 - 100912/9a^10 - 1272a^9 + 122032/9a^8 + 6269/3a^7 - 89603/9a^6 - 21838/9a^5 + 33527/9a^4 + 4042/3a^3 - 1174/3a^2 - 189a - 115/9, a^20 - 2/3a^19 - 61/3a^18 + 31/3a^17 + 524/3a^16 - 203/3a^15 - 2473/3a^14 + 252a^13 + 2343a^12 - 594a^11 - 12529/3a^10 + 2515/3a^9 + 14204/3a^8 - 1348/3a^7 - 9782/3a^6 - 797/3a^5 + 3437/3a^4 + 1010/3a^3 - 289/3a^2 - 176/3a - 13/3, 43/9a^20 - 38/9a^19 - 869/9a^18 + 71a^17 + 7424/9a^16 - 4627/9a^15 - 34825/9a^14 + 6392/3a^13 + 98060/9a^12 - 49963/9a^11 - 171805/9a^10 + 80696/9a^9 + 189227/9a^8 - 70307/9a^7 - 126041/9a^6 + 22838/9a^5 + 44245/9a^4 + 2833/9a^3 - 1649/3a^2 - 1096/9a - 14/3, 2/3a^20 + 8/9a^19 - 15a^18 - 178/9a^17 + 1274/9a^16 + 541/3a^15 - 6638/9a^14 - 7894/9a^13 + 21256/9a^12 + 22510/9a^11 - 14699/3a^10 - 39604/9a^9 + 59669/9a^8 + 44849/9a^7 - 48664/9a^6 - 32056/9a^5 + 2174a^4 + 4129/3a^3 - 674/3a^2 - 496/3a - 37/3, 29/9a^20 - 26/9a^19 - 587/9a^18 + 440/9a^17 + 5027/9a^16 - 3214/9a^15 - 23665/9a^14 + 13475/9a^13 + 22324/3a^12 - 11888/3a^11 - 39374/3a^10 + 59084/9a^9 + 43700/3a^8 - 54533/9a^7 - 29320/3a^6 + 21820/9a^5 + 10342/3a^4 - 896/9a^3 - 3496/9a^2 - 136/3a - 2/3, 5/3a^20 - 13/9a^19 - 304/9a^18 + 73/3a^17 + 2606/9a^16 - 1594/9a^15 - 12277/9a^14 + 2225/3a^13 + 11587/3a^12 - 17687/9a^11 - 61315/9a^10 + 3255a^9 + 7553a^8 - 26720/9a^7 - 45416/9a^6 + 10088/9a^5 + 15887/9a^4 + 17/3a^3 - 1753/9a^2 - 269/9a - 10/9, 125/9a^20 - 37/9a^19 - 2606/9a^18 + 388/9a^17 + 7658/3a^16 - 1031/9a^15 - 37235/3a^14 - 1757/9a^13 + 109934/3a^12 + 13663/9a^11 - 618269/9a^10 - 35590/9a^9 + 745645/9a^8 + 79447/9a^7 - 548135/9a^6 - 110948/9a^5 + 68920/3a^4 + 66514/9a^3 - 22463/9a^2 - 9466/9a - 629/9, 22/9a^20 + 2/9a^19 - 466/9a^18 - 34/3a^17 + 1388/3a^16 + 419/3a^15 - 2279a^14 - 2324/3a^13 + 61490/9a^12 + 20914/9a^11 - 39212/3a^10 - 38287/9a^9 + 145289/9a^8 + 5434a^7 - 12057a^6 - 4724a^5 + 40352/9a^4 + 19663/9a^3 - 3671/9a^2 - 2564/9a - 245/9, 22/9a^20 - 16/9a^19 - 452/9a^18 + 89/3a^17 + 3922/9a^16 - 643/3a^15 - 6229/3a^14 + 8074/9a^13 + 17840/3a^12 - 21530/9a^11 - 95614/9a^10 + 11876/3a^9 + 107657/9a^8 - 3439a^7 - 24487/3a^6 + 8762/9a^5 + 26435/9a^4 + 2414/9a^3 - 3001/9a^2 - 223/3a - 31/9 (assuming GRH)	comment:Fundamental units sage:UK.fundamental_units() gp:K.fu magma:[K\|fUK(g): g in Generators(UK)]; oscar:[K(fUK(a)) for a in gens(UK)]
Regulator:	$ 295522286712 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{4}\cdot 295522286712 \cdot 1}{2\cdot\sqrt{40453417591156198845459427294648145741}}\cr\approx \mathstrut & 0.296614277015826 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

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 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*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))))

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 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*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)))]

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 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*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)));

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 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*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):

comment:Galois group

sage:K.galois_group(type='pari')

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

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.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(b)]

magma:L := Subfields(K); L[2..#L];

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	$18{,}\,{\href{/padicField/2.3.0.1}{3} }$	${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$	$21$	R	${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$	${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$	$15{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$	$18{,}\,{\href{/padicField/23.3.0.1}{3} }$	${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$	$21$	$15{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$	${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$	${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$	$21$

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

comment:Frobenius cycle types

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)]

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]])

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))];

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$7$ 7	7.1.3.2a1.1	$x^{3} + 7$	$3$	$1$	$2$	$C_3$	$$[\ ]_{3}$$
	7.1.3.2a1.1	$x^{3} + 7$	$3$	$1$	$2$	$C_3$	$$[\ ]_{3}$$
	7.5.3.10a1.3	$x^{15} + 3 x^{11} + 12 x^{10} + 3 x^{7} + 24 x^{6} + 48 x^{5} + x^{3} + 12 x^{2} + 48 x + 71$	$3$	$5$	$10$	$C_{15}$	$$[\ ]_{3}^{5}$$
$596\!\cdots\!909$ 59646183107903161080974909	$\Q_{59\!\cdots\!09}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{59\!\cdots\!09}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{59\!\cdots\!09}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $3$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
		Deg $3$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $6$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$

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