LMFDB - Number field 21.1.2421032603211455207440720697704448.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 16*x^19 - 26*x^18 + 29*x^17 - 36*x^16 + 50*x^15 - 50*x^14 + 35*x^13 - 14*x^12 + 100*x^11 - 22*x^10 - 25*x^9 - 112*x^8 - 78*x^7 - 286*x^6 - 632*x^5 - 1000*x^4 - 1000*x^3 - 688*x^2 - 288*x - 64)

gp: K = bnfinit(y^21 - 6*y^20 + 16*y^19 - 26*y^18 + 29*y^17 - 36*y^16 + 50*y^15 - 50*y^14 + 35*y^13 - 14*y^12 + 100*y^11 - 22*y^10 - 25*y^9 - 112*y^8 - 78*y^7 - 286*y^6 - 632*y^5 - 1000*y^4 - 1000*y^3 - 688*y^2 - 288*y - 64, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 6*x^20 + 16*x^19 - 26*x^18 + 29*x^17 - 36*x^16 + 50*x^15 - 50*x^14 + 35*x^13 - 14*x^12 + 100*x^11 - 22*x^10 - 25*x^9 - 112*x^8 - 78*x^7 - 286*x^6 - 632*x^5 - 1000*x^4 - 1000*x^3 - 688*x^2 - 288*x - 64);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 6*x^20 + 16*x^19 - 26*x^18 + 29*x^17 - 36*x^16 + 50*x^15 - 50*x^14 + 35*x^13 - 14*x^12 + 100*x^11 - 22*x^10 - 25*x^9 - 112*x^8 - 78*x^7 - 286*x^6 - 632*x^5 - 1000*x^4 - 1000*x^3 - 688*x^2 - 288*x - 64)

$ x^{21} - 6 x^{20} + 16 x^{19} - 26 x^{18} + 29 x^{17} - 36 x^{16} + 50 x^{15} - 50 x^{14} + 35 x^{13} + \cdots - 64 $ x^21 - 6*x^20 + 16*x^19 - 26*x^18 + 29*x^17 - 36*x^16 + 50*x^15 - 50*x^14 + 35*x^13 - 14*x^12 + 100*x^11 - 22*x^10 - 25*x^9 - 112*x^8 - 78*x^7 - 286*x^6 - 632*x^5 - 1000*x^4 - 1000*x^3 - 688*x^2 - 288*x - 64

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$21$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[1, 10]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$2421032603211455207440720697704448$ 2421032603211455207440720697704448 $\medspace = 2^{14}\cdot 11^{10}\cdot 283^{3}\cdot 6311^{3}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$38.88$	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:	$2^{2/3}11^{3/4}283^{1/2}6311^{1/2}\approx 12813.668536692152$
Ramified primes:	$2$, $11$, $283$, $6311$ 2, 11, 283, 6311	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{1786013}) $
$\card{ \Aut(K/\Q) }$:	$1$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}+\frac{1}{16}a^{10}-\frac{3}{16}a^{9}+\frac{3}{16}a^{8}-\frac{3}{16}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{15}-\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{3}{16}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{16}-\frac{1}{16}a^{12}+\frac{1}{16}a^{10}+\frac{3}{16}a^{9}+\frac{5}{32}a^{8}+\frac{1}{8}a^{7}-\frac{3}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{13}+\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{3}{32}a^{9}+\frac{1}{8}a^{8}-\frac{3}{16}a^{7}+\frac{3}{16}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{128}a^{18}+\frac{1}{128}a^{17}-\frac{1}{32}a^{15}+\frac{1}{64}a^{14}-\frac{3}{64}a^{13}+\frac{3}{64}a^{12}+\frac{3}{32}a^{11}+\frac{7}{128}a^{10}-\frac{27}{128}a^{9}+\frac{13}{64}a^{8}-\frac{1}{16}a^{7}-\frac{13}{64}a^{6}-\frac{3}{16}a^{5}-\frac{1}{4}a^{4}+\frac{7}{16}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{256}a^{19}+\frac{3}{256}a^{17}+\frac{3}{128}a^{15}-\frac{1}{32}a^{14}+\frac{1}{64}a^{13}-\frac{1}{128}a^{12}+\frac{3}{256}a^{11}-\frac{1}{128}a^{10}+\frac{33}{256}a^{9}+\frac{1}{128}a^{8}-\frac{13}{128}a^{7}-\frac{15}{128}a^{6}+\frac{1}{16}a^{5}+\frac{11}{32}a^{4}-\frac{9}{32}a^{3}+\frac{5}{16}a^{2}-\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{1024}a^{20}-\frac{1}{1024}a^{19}+\frac{3}{1024}a^{18}-\frac{11}{1024}a^{17}+\frac{7}{512}a^{16}-\frac{15}{512}a^{15}-\frac{5}{256}a^{14}+\frac{21}{512}a^{13}-\frac{43}{1024}a^{12}+\frac{43}{1024}a^{11}+\frac{35}{1024}a^{10}-\frac{215}{1024}a^{9}-\frac{5}{256}a^{8}+\frac{23}{256}a^{7}+\frac{103}{512}a^{6}+\frac{27}{128}a^{5}-\frac{5}{16}a^{4}-\frac{29}{128}a^{3}+\frac{29}{64}a^{2}+\frac{15}{32}a+\frac{5}{16}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3, 1/2*a^7 - 1/2*a^3, 1/2*a^8 - 1/2*a^4, 1/2*a^9 - 1/2*a^5, 1/4*a^10 - 1/4*a^9 - 1/4*a^6 + 1/4*a^5 - 1/2*a^4 - 1/2*a^2, 1/4*a^11 - 1/4*a^9 - 1/4*a^7 - 1/4*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2, 1/8*a^12 - 1/8*a^10 - 1/4*a^9 - 1/8*a^8 - 1/8*a^6 - 1/2*a^5 - 1/4*a^4 - 1/4*a^3 - 1/2*a^2, 1/8*a^13 - 1/8*a^11 + 1/8*a^9 - 1/8*a^7 - 1/4*a^6 - 1/4*a^4 - 1/2*a^2, 1/16*a^14 - 1/16*a^13 - 1/16*a^12 - 1/16*a^11 + 1/16*a^10 - 3/16*a^9 + 3/16*a^8 - 3/16*a^7 + 1/8*a^6 + 1/4*a^5 - 3/8*a^4 + 1/4*a^3 - 1/2*a^2 - 1/2*a, 1/16*a^15 - 1/8*a^11 + 1/8*a^9 - 1/8*a^8 - 3/16*a^7 - 1/4*a^6 + 1/8*a^5 - 1/8*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/32*a^16 - 1/16*a^12 + 1/16*a^10 + 3/16*a^9 + 5/32*a^8 + 1/8*a^7 - 3/16*a^6 - 1/16*a^5 - 1/4*a^4 + 1/4*a^3 + 1/4*a^2 - 1/2*a, 1/32*a^17 - 1/16*a^13 + 1/16*a^11 - 1/16*a^10 - 3/32*a^9 + 1/8*a^8 - 3/16*a^7 + 3/16*a^6 - 1/4*a^4 + 1/4*a^3, 1/128*a^18 + 1/128*a^17 - 1/32*a^15 + 1/64*a^14 - 3/64*a^13 + 3/64*a^12 + 3/32*a^11 + 7/128*a^10 - 27/128*a^9 + 13/64*a^8 - 1/16*a^7 - 13/64*a^6 - 3/16*a^5 - 1/4*a^4 + 7/16*a^3 + 3/8*a^2 + 1/4*a - 1/2, 1/256*a^19 + 3/256*a^17 + 3/128*a^15 - 1/32*a^14 + 1/64*a^13 - 1/128*a^12 + 3/256*a^11 - 1/128*a^10 + 33/256*a^9 + 1/128*a^8 - 13/128*a^7 - 15/128*a^6 + 1/16*a^5 + 11/32*a^4 - 9/32*a^3 + 5/16*a^2 - 1/8*a - 1/4, 1/1024*a^20 - 1/1024*a^19 + 3/1024*a^18 - 11/1024*a^17 + 7/512*a^16 - 15/512*a^15 - 5/256*a^14 + 21/512*a^13 - 43/1024*a^12 + 43/1024*a^11 + 35/1024*a^10 - 215/1024*a^9 - 5/256*a^8 + 23/256*a^7 + 103/512*a^6 + 27/128*a^5 - 5/16*a^4 - 29/128*a^3 + 29/64*a^2 + 15/32*a + 5/16

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$10$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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:	$\frac{173}{1024}a^{20}-\frac{1025}{1024}a^{19}+\frac{2623}{1024}a^{18}-\frac{3827}{1024}a^{17}+\frac{1595}{512}a^{16}-\frac{1391}{512}a^{15}+\frac{883}{256}a^{14}-\frac{767}{512}a^{13}-\frac{3543}{1024}a^{12}+\frac{9075}{1024}a^{11}+\frac{5399}{1024}a^{10}+\frac{8185}{1024}a^{9}-\frac{4975}{256}a^{8}-\frac{1411}{256}a^{7}-\frac{12049}{512}a^{6}-\frac{4577}{128}a^{5}-\frac{3689}{32}a^{4}-\frac{20061}{128}a^{3}-\frac{10051}{64}a^{2}-\frac{2833}{32}a-\frac{483}{16}$, $\frac{55}{64}a^{20}-\frac{777}{128}a^{19}+\frac{159}{8}a^{18}-\frac{5257}{128}a^{17}+\frac{1933}{32}a^{16}-\frac{4959}{64}a^{15}+\frac{3135}{32}a^{14}-\frac{3547}{32}a^{13}+\frac{815}{8}a^{12}-\frac{8647}{128}a^{11}+\frac{6849}{64}a^{10}-\frac{12195}{128}a^{9}+\frac{1935}{64}a^{8}-\frac{5249}{64}a^{7}-\frac{83}{64}a^{6}-\frac{1653}{8}a^{5}-\frac{5289}{16}a^{4}-\frac{6845}{16}a^{3}-\frac{2343}{8}a^{2}-\frac{537}{4}a-\frac{35}{2}$, $\frac{657}{1024}a^{20}-\frac{3081}{1024}a^{19}+\frac{4843}{1024}a^{18}-\frac{1163}{1024}a^{17}-\frac{2361}{512}a^{16}-\frac{4247}{512}a^{15}+\frac{8551}{256}a^{14}-\frac{16755}{512}a^{13}+\frac{14437}{1024}a^{12}-\frac{12141}{1024}a^{11}+\frac{100107}{1024}a^{10}+\frac{26041}{1024}a^{9}-\frac{20237}{256}a^{8}-\frac{28989}{256}a^{7}-\frac{24633}{512}a^{6}-\frac{27221}{128}a^{5}-607a^{4}-\frac{128493}{128}a^{3}-\frac{64915}{64}a^{2}-\frac{18545}{32}a-\frac{2427}{16}$, $\frac{321}{1024}a^{20}-\frac{1833}{1024}a^{19}+\frac{4331}{1024}a^{18}-\frac{5083}{1024}a^{17}+\frac{439}{512}a^{16}+\frac{2153}{512}a^{15}-\frac{2097}{256}a^{14}+\frac{8877}{512}a^{13}-\frac{30763}{1024}a^{12}+\frac{40531}{1024}a^{11}-\frac{8501}{1024}a^{10}+\frac{31177}{1024}a^{9}-\frac{11885}{256}a^{8}-\frac{1957}{256}a^{7}-\frac{24809}{512}a^{6}-\frac{8517}{128}a^{5}-\frac{1827}{8}a^{4}-\frac{39805}{128}a^{3}-\frac{20195}{64}a^{2}-\frac{5825}{32}a-\frac{971}{16}$, $\frac{223}{256}a^{20}-\frac{703}{128}a^{19}+\frac{3937}{256}a^{18}-\frac{3307}{128}a^{17}+\frac{3689}{128}a^{16}-\frac{2079}{64}a^{15}+\frac{2777}{64}a^{14}-\frac{5359}{128}a^{13}+\frac{4817}{256}a^{12}+\frac{397}{32}a^{11}+\frac{14583}{256}a^{10}-\frac{1179}{64}a^{9}-\frac{6937}{128}a^{8}-\frac{6659}{128}a^{7}-\frac{3129}{64}a^{6}-\frac{6849}{32}a^{5}-\frac{15813}{32}a^{4}-\frac{1295}{2}a^{3}-\frac{2203}{4}a^{2}-\frac{561}{2}a-\frac{143}{2}$, $\frac{35045}{1024}a^{20}-\frac{241925}{1024}a^{19}+\frac{768903}{1024}a^{18}-\frac{1530271}{1024}a^{17}+\frac{1071107}{512}a^{16}-\frac{1322555}{512}a^{15}+\frac{823699}{256}a^{14}-\frac{1809727}{512}a^{13}+\frac{3095001}{1024}a^{12}-\frac{1677641}{1024}a^{11}+\frac{3487479}{1024}a^{10}-\frac{2839931}{1024}a^{9}+\frac{63123}{256}a^{8}-\frac{691853}{256}a^{7}-\frac{418805}{512}a^{6}-\frac{1028321}{128}a^{5}-\frac{232297}{16}a^{4}-\frac{2392073}{128}a^{3}-\frac{896231}{64}a^{2}-\frac{214333}{32}a-\frac{21055}{16}$, $\frac{78905}{1024}a^{20}-\frac{554649}{1024}a^{19}+\frac{1804595}{1024}a^{18}-\frac{3700043}{1024}a^{17}+\frac{2695215}{512}a^{16}-\frac{3438471}{512}a^{15}+\frac{2177271}{256}a^{14}-\frac{4932939}{512}a^{13}+\frac{9034909}{1024}a^{12}-\frac{5908333}{1024}a^{11}+\frac{9638531}{1024}a^{10}-\frac{8499095}{1024}a^{9}+\frac{630759}{256}a^{8}-\frac{1838705}{256}a^{7}-\frac{220057}{512}a^{6}-\frac{2374053}{128}a^{5}-\frac{482535}{16}a^{4}-\frac{5006781}{128}a^{3}-\frac{1750483}{64}a^{2}-\frac{406961}{32}a-\frac{30139}{16}$, $\frac{45}{64}a^{20}-\frac{297}{64}a^{19}+\frac{55}{4}a^{18}-\frac{789}{32}a^{17}+\frac{477}{16}a^{16}-\frac{1105}{32}a^{15}+\frac{1531}{32}a^{14}-\frac{909}{16}a^{13}+\frac{2899}{64}a^{12}-\frac{953}{64}a^{11}+\frac{1885}{32}a^{10}-\frac{1435}{32}a^{9}-\frac{231}{16}a^{8}-\frac{683}{16}a^{7}-\frac{221}{8}a^{6}-\frac{2739}{16}a^{5}-339a^{4}-\frac{1691}{4}a^{3}-\frac{1299}{4}a^{2}-\frac{307}{2}a-35$, $\frac{1627}{1024}a^{20}-\frac{9627}{1024}a^{19}+\frac{24377}{1024}a^{18}-\frac{34177}{1024}a^{17}+\frac{11965}{512}a^{16}-\frac{6309}{512}a^{15}+\frac{2813}{256}a^{14}+\frac{6879}{512}a^{13}-\frac{62873}{1024}a^{12}+\frac{110377}{1024}a^{11}+\frac{32969}{1024}a^{10}+\frac{85339}{1024}a^{9}-\frac{48243}{256}a^{8}-\frac{16595}{256}a^{7}-\frac{98667}{512}a^{6}-\frac{43839}{128}a^{5}-\frac{17077}{16}a^{4}-\frac{187863}{128}a^{3}-\frac{89657}{64}a^{2}-\frac{23987}{32}a-\frac{3697}{16}$, $\frac{65}{64}a^{20}-\frac{219}{32}a^{19}+\frac{673}{32}a^{18}-\frac{2557}{64}a^{17}+\frac{1673}{32}a^{16}-\frac{975}{16}a^{15}+\frac{2367}{32}a^{14}-\frac{1223}{16}a^{13}+\frac{3501}{64}a^{12}-\frac{99}{8}a^{11}+\frac{2291}{32}a^{10}-\frac{3135}{64}a^{9}-\frac{213}{8}a^{8}-\frac{1923}{32}a^{7}-\frac{1403}{32}a^{6}-\frac{927}{4}a^{5}-\frac{3783}{8}a^{4}-\frac{4769}{8}a^{3}-\frac{1877}{4}a^{2}-\frac{451}{2}a-52$ 173/1024a^20 - 1025/1024a^19 + 2623/1024a^18 - 3827/1024a^17 + 1595/512a^16 - 1391/512a^15 + 883/256a^14 - 767/512a^13 - 3543/1024a^12 + 9075/1024a^11 + 5399/1024a^10 + 8185/1024a^9 - 4975/256a^8 - 1411/256a^7 - 12049/512a^6 - 4577/128a^5 - 3689/32a^4 - 20061/128a^3 - 10051/64a^2 - 2833/32a - 483/16, 55/64a^20 - 777/128a^19 + 159/8a^18 - 5257/128a^17 + 1933/32a^16 - 4959/64a^15 + 3135/32a^14 - 3547/32a^13 + 815/8a^12 - 8647/128a^11 + 6849/64a^10 - 12195/128a^9 + 1935/64a^8 - 5249/64a^7 - 83/64a^6 - 1653/8a^5 - 5289/16a^4 - 6845/16a^3 - 2343/8a^2 - 537/4a - 35/2, 657/1024a^20 - 3081/1024a^19 + 4843/1024a^18 - 1163/1024a^17 - 2361/512a^16 - 4247/512a^15 + 8551/256a^14 - 16755/512a^13 + 14437/1024a^12 - 12141/1024a^11 + 100107/1024a^10 + 26041/1024a^9 - 20237/256a^8 - 28989/256a^7 - 24633/512a^6 - 27221/128a^5 - 607a^4 - 128493/128a^3 - 64915/64a^2 - 18545/32a - 2427/16, 321/1024a^20 - 1833/1024a^19 + 4331/1024a^18 - 5083/1024a^17 + 439/512a^16 + 2153/512a^15 - 2097/256a^14 + 8877/512a^13 - 30763/1024a^12 + 40531/1024a^11 - 8501/1024a^10 + 31177/1024a^9 - 11885/256a^8 - 1957/256a^7 - 24809/512a^6 - 8517/128a^5 - 1827/8a^4 - 39805/128a^3 - 20195/64a^2 - 5825/32a - 971/16, 223/256a^20 - 703/128a^19 + 3937/256a^18 - 3307/128a^17 + 3689/128a^16 - 2079/64a^15 + 2777/64a^14 - 5359/128a^13 + 4817/256a^12 + 397/32a^11 + 14583/256a^10 - 1179/64a^9 - 6937/128a^8 - 6659/128a^7 - 3129/64a^6 - 6849/32a^5 - 15813/32a^4 - 1295/2a^3 - 2203/4a^2 - 561/2a - 143/2, 35045/1024a^20 - 241925/1024a^19 + 768903/1024a^18 - 1530271/1024a^17 + 1071107/512a^16 - 1322555/512a^15 + 823699/256a^14 - 1809727/512a^13 + 3095001/1024a^12 - 1677641/1024a^11 + 3487479/1024a^10 - 2839931/1024a^9 + 63123/256a^8 - 691853/256a^7 - 418805/512a^6 - 1028321/128a^5 - 232297/16a^4 - 2392073/128a^3 - 896231/64a^2 - 214333/32a - 21055/16, 78905/1024a^20 - 554649/1024a^19 + 1804595/1024a^18 - 3700043/1024a^17 + 2695215/512a^16 - 3438471/512a^15 + 2177271/256a^14 - 4932939/512a^13 + 9034909/1024a^12 - 5908333/1024a^11 + 9638531/1024a^10 - 8499095/1024a^9 + 630759/256a^8 - 1838705/256a^7 - 220057/512a^6 - 2374053/128a^5 - 482535/16a^4 - 5006781/128a^3 - 1750483/64a^2 - 406961/32a - 30139/16, 45/64a^20 - 297/64a^19 + 55/4a^18 - 789/32a^17 + 477/16a^16 - 1105/32a^15 + 1531/32a^14 - 909/16a^13 + 2899/64a^12 - 953/64a^11 + 1885/32a^10 - 1435/32a^9 - 231/16a^8 - 683/16a^7 - 221/8a^6 - 2739/16a^5 - 339a^4 - 1691/4a^3 - 1299/4a^2 - 307/2a - 35, 1627/1024a^20 - 9627/1024a^19 + 24377/1024a^18 - 34177/1024a^17 + 11965/512a^16 - 6309/512a^15 + 2813/256a^14 + 6879/512a^13 - 62873/1024a^12 + 110377/1024a^11 + 32969/1024a^10 + 85339/1024a^9 - 48243/256a^8 - 16595/256a^7 - 98667/512a^6 - 43839/128a^5 - 17077/16a^4 - 187863/128a^3 - 89657/64a^2 - 23987/32a - 3697/16, 65/64a^20 - 219/32a^19 + 673/32a^18 - 2557/64a^17 + 1673/32a^16 - 975/16a^15 + 2367/32a^14 - 1223/16a^13 + 3501/64a^12 - 99/8a^11 + 2291/32a^10 - 3135/64a^9 - 213/8a^8 - 1923/32a^7 - 1403/32a^6 - 927/4a^5 - 3783/8a^4 - 4769/8a^3 - 1877/4a^2 - 451/2a - 52 (assuming GRH)	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:	$ 19772231044.3 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 19772231044.3 \cdot 1}{2\cdot\sqrt{2421032603211455207440720697704448}}\cr\approx \mathstrut & 38.5348826943 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 16*x^19 - 26*x^18 + 29*x^17 - 36*x^16 + 50*x^15 - 50*x^14 + 35*x^13 - 14*x^12 + 100*x^11 - 22*x^10 - 25*x^9 - 112*x^8 - 78*x^7 - 286*x^6 - 632*x^5 - 1000*x^4 - 1000*x^3 - 688*x^2 - 288*x - 64)

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))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^21 - 6*x^20 + 16*x^19 - 26*x^18 + 29*x^17 - 36*x^16 + 50*x^15 - 50*x^14 + 35*x^13 - 14*x^12 + 100*x^11 - 22*x^10 - 25*x^9 - 112*x^8 - 78*x^7 - 286*x^6 - 632*x^5 - 1000*x^4 - 1000*x^3 - 688*x^2 - 288*x - 64, 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)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - 6*x^20 + 16*x^19 - 26*x^18 + 29*x^17 - 36*x^16 + 50*x^15 - 50*x^14 + 35*x^13 - 14*x^12 + 100*x^11 - 22*x^10 - 25*x^9 - 112*x^8 - 78*x^7 - 286*x^6 - 632*x^5 - 1000*x^4 - 1000*x^3 - 688*x^2 - 288*x - 64);

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)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 6*x^20 + 16*x^19 - 26*x^18 + 29*x^17 - 36*x^16 + 50*x^15 - 50*x^14 + 35*x^13 - 14*x^12 + 100*x^11 - 22*x^10 - 25*x^9 - 112*x^8 - 78*x^7 - 286*x^6 - 632*x^5 - 1000*x^4 - 1000*x^3 - 688*x^2 - 288*x - 64);

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

$D_7\wr S_3$ (as 21T62):

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 solvable group of order 16464

The 65 conjugacy class representatives for $D_7\wr S_3$

Character table for $D_7\wr S_3$

Intermediate fields

3.1.44.1

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

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 28 sibling:	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	R	${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$	${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$	${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$	R	${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$	${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$	${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$	${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$	${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.7.0.1}{7} }$	$21$	${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }$	${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }$	${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.2.0.1}{2} }^{9}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$	${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$	${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
$2$ 2	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
$11$ 11	$\Q_{11}$	$x + 9$	$1$	$1$	$0$	Trivial	$[\ ]$
	11.2.0.1	$x^{2} + 7 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.0.1	$x^{2} + 7 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	11.2.0.1	$x^{2} + 7 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	11.4.3.1	$x^{4} + 11$	$4$	$1$	$3$	$D_{4}$	$[\ ]_{4}^{2}$
	11.4.3.1	$x^{4} + 11$	$4$	$1$	$3$	$D_{4}$	$[\ ]_{4}^{2}$
	11.4.3.1	$x^{4} + 11$	$4$	$1$	$3$	$D_{4}$	$[\ ]_{4}^{2}$
$283$ 283	$\Q_{283}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $14$	$1$	$14$	$0$	$C_{14}$	$[\ ]^{14}$
$6311$ 6311	$\Q_{6311}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $14$	$1$	$14$	$0$	$C_{14}$	$[\ ]^{14}$

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