LMFDB - Number field 21.3.7655337671796811085236857471811.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*x^2 - 288*x - 64)

gp: K = bnfinit(y^21 - 6*y^20 + 19*y^19 - 35*y^18 + 34*y^17 - 27*y^16 + 33*y^15 - 73*y^14 + 37*y^13 + 44*y^12 + 137*y^11 - 33*y^10 + 60*y^9 + 177*y^8 + 243*y^7 - 219*y^6 - 876*y^5 - 1332*y^4 - 1192*y^3 - 736*y^2 - 288*y - 64, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 6*x^20 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*x^2 - 288*x - 64);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 6*x^20 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*x^2 - 288*x - 64)

$ x^{21} - 6 x^{20} + 19 x^{19} - 35 x^{18} + 34 x^{17} - 27 x^{16} + 33 x^{15} - 73 x^{14} + 37 x^{13} + \cdots - 64 $ x^21 - 6*x^20 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*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:	$[3, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-7655337671796811085236857471811$ -7655337671796811085236857471811 $\medspace = -\,7^{14}\cdot 29^{3}\cdot 77351^{3}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$29.56$	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:	$7^{2/3}29^{1/2}77351^{1/2}\approx 5480.632208375353$
Ramified primes:	$7$, $29$, $77351$ 7, 29, 77351	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-2243179}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{12}+\frac{1}{8}a^{9}+\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{3}{8}a^{7}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}+\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{13}+\frac{1}{16}a^{11}-\frac{3}{16}a^{9}+\frac{3}{16}a^{8}-\frac{1}{16}a^{7}-\frac{7}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{1}{16}a^{11}+\frac{1}{16}a^{10}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{7}{16}a^{6}+\frac{7}{16}a^{5}-\frac{3}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{16}-\frac{1}{32}a^{14}+\frac{1}{32}a^{13}+\frac{1}{32}a^{12}-\frac{3}{32}a^{11}+\frac{1}{16}a^{9}-\frac{1}{4}a^{8}+\frac{5}{32}a^{7}+\frac{11}{32}a^{6}-\frac{9}{32}a^{5}-\frac{3}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{15}-\frac{1}{32}a^{14}-\frac{1}{32}a^{13}+\frac{1}{32}a^{12}-\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{5}{32}a^{8}+\frac{1}{32}a^{7}-\frac{7}{32}a^{6}+\frac{1}{4}a^{5}+\frac{3}{16}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{832}a^{18}+\frac{3}{832}a^{17}+\frac{5}{832}a^{16}-\frac{9}{416}a^{15}-\frac{11}{416}a^{14}+\frac{1}{32}a^{13}+\frac{5}{832}a^{12}+\frac{23}{208}a^{11}+\frac{31}{416}a^{10}+\frac{61}{832}a^{9}+\frac{75}{416}a^{8}-\frac{15}{32}a^{7}+\frac{303}{832}a^{6}-\frac{3}{416}a^{5}+\frac{21}{104}a^{4}-\frac{51}{104}a^{3}+\frac{1}{26}a^{2}-\frac{2}{13}a-\frac{3}{13}$, $\frac{1}{3328}a^{19}+\frac{11}{1664}a^{17}-\frac{7}{3328}a^{16}+\frac{29}{1664}a^{15}+\frac{23}{832}a^{14}-\frac{177}{3328}a^{13}+\frac{181}{3328}a^{12}-\frac{15}{208}a^{11}-\frac{333}{3328}a^{10}+\frac{643}{3328}a^{9}+\frac{165}{1664}a^{8}+\frac{1161}{3328}a^{7}-\frac{499}{3328}a^{6}-\frac{181}{832}a^{5}+\frac{19}{832}a^{4}+\frac{79}{416}a^{3}+\frac{45}{104}a^{2}-\frac{33}{104}a-\frac{17}{52}$, $\frac{1}{13312}a^{20}-\frac{1}{13312}a^{19}-\frac{1}{6656}a^{18}-\frac{205}{13312}a^{17}-\frac{159}{13312}a^{16}+\frac{77}{6656}a^{15}+\frac{259}{13312}a^{14}-\frac{29}{6656}a^{13}-\frac{749}{13312}a^{12}-\frac{121}{1024}a^{11}-\frac{71}{832}a^{10}+\frac{2175}{13312}a^{9}+\frac{67}{1024}a^{8}+\frac{469}{3328}a^{7}+\frac{3527}{13312}a^{6}-\frac{493}{1664}a^{5}+\frac{275}{3328}a^{4}-\frac{391}{1664}a^{3}-\frac{25}{208}a^{2}-\frac{9}{416}a-\frac{67}{208}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a^6 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^9 - 1/2*a^7 - 1/2*a^6 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2, 1/4*a^10 - 1/4*a^9 - 1/4*a^8 - 1/4*a^6 - 1/4*a^5 - 1/2*a^3 - 1/4*a^2 - 1/2*a, 1/4*a^11 - 1/4*a^8 + 1/4*a^7 - 1/4*a^5 - 1/4*a^3 - 1/4*a^2 - 1/2*a, 1/8*a^12 + 1/8*a^9 + 1/8*a^8 - 1/4*a^7 + 1/8*a^6 + 1/8*a^4 - 3/8*a^3 - 1/2*a^2 - 1/2*a, 1/8*a^13 - 1/8*a^10 - 1/8*a^9 - 3/8*a^7 - 1/4*a^6 + 3/8*a^5 + 1/8*a^4 - 1/2*a^3 + 1/4*a^2 - 1/2*a, 1/16*a^14 - 1/16*a^13 + 1/16*a^11 - 3/16*a^9 + 3/16*a^8 - 1/16*a^7 - 7/16*a^6 - 1/4*a^5 - 1/16*a^4 - 1/2*a^3 - 1/4*a^2 - 1/2*a, 1/16*a^15 - 1/16*a^13 - 1/16*a^12 + 1/16*a^11 + 1/16*a^10 + 1/8*a^9 - 1/4*a^8 + 1/4*a^7 + 7/16*a^6 + 7/16*a^5 - 3/16*a^4 - 3/8*a^3 - 1/2*a, 1/32*a^16 - 1/32*a^14 + 1/32*a^13 + 1/32*a^12 - 3/32*a^11 + 1/16*a^9 - 1/4*a^8 + 5/32*a^7 + 11/32*a^6 - 9/32*a^5 - 3/8*a^4 - 3/8*a^3 + 1/4*a^2 - 1/2*a, 1/32*a^17 - 1/32*a^15 - 1/32*a^14 - 1/32*a^13 + 1/32*a^12 - 1/16*a^11 - 1/16*a^10 - 1/16*a^9 - 5/32*a^8 + 1/32*a^7 - 7/32*a^6 + 1/4*a^5 + 3/16*a^4 + 3/8*a^3 + 1/4*a^2 - 1/2*a, 1/832*a^18 + 3/832*a^17 + 5/832*a^16 - 9/416*a^15 - 11/416*a^14 + 1/32*a^13 + 5/832*a^12 + 23/208*a^11 + 31/416*a^10 + 61/832*a^9 + 75/416*a^8 - 15/32*a^7 + 303/832*a^6 - 3/416*a^5 + 21/104*a^4 - 51/104*a^3 + 1/26*a^2 - 2/13*a - 3/13, 1/3328*a^19 + 11/1664*a^17 - 7/3328*a^16 + 29/1664*a^15 + 23/832*a^14 - 177/3328*a^13 + 181/3328*a^12 - 15/208*a^11 - 333/3328*a^10 + 643/3328*a^9 + 165/1664*a^8 + 1161/3328*a^7 - 499/3328*a^6 - 181/832*a^5 + 19/832*a^4 + 79/416*a^3 + 45/104*a^2 - 33/104*a - 17/52, 1/13312*a^20 - 1/13312*a^19 - 1/6656*a^18 - 205/13312*a^17 - 159/13312*a^16 + 77/6656*a^15 + 259/13312*a^14 - 29/6656*a^13 - 749/13312*a^12 - 121/1024*a^11 - 71/832*a^10 + 2175/13312*a^9 + 67/1024*a^8 + 469/3328*a^7 + 3527/13312*a^6 - 493/1664*a^5 + 275/3328*a^4 - 391/1664*a^3 - 25/208*a^2 - 9/416*a - 67/208

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

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:	$11$	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:	$\frac{173}{1024}a^{20}-\frac{1025}{1024}a^{19}+\frac{1571}{512}a^{18}-\frac{5345}{1024}a^{17}+\frac{3737}{1024}a^{16}-\frac{187}{512}a^{15}-\frac{329}{1024}a^{14}-\frac{2719}{512}a^{13}-\frac{2573}{1024}a^{12}+\frac{19807}{1024}a^{11}+\frac{2949}{256}a^{10}+\frac{3447}{1024}a^{9}-\frac{4781}{1024}a^{8}+\frac{2863}{64}a^{7}+\frac{29303}{1024}a^{6}-\frac{4347}{128}a^{5}-\frac{41925}{256}a^{4}-\frac{27095}{128}a^{3}-\frac{2879}{16}a^{2}-\frac{2833}{32}a-\frac{467}{16}$, $\frac{15}{256}a^{20}+\frac{3}{16}a^{19}-\frac{373}{128}a^{18}+\frac{3491}{256}a^{17}-\frac{4663}{128}a^{16}+\frac{3825}{64}a^{15}-\frac{18047}{256}a^{14}+\frac{17931}{256}a^{13}-\frac{5735}{64}a^{12}+\frac{25677}{256}a^{11}-\frac{9931}{256}a^{10}+\frac{8097}{128}a^{9}-\frac{20233}{256}a^{8}+\frac{28739}{256}a^{7}+\frac{435}{16}a^{6}+\frac{1043}{64}a^{5}-\frac{7539}{32}a^{4}-\frac{1215}{4}a^{3}-\frac{2337}{8}a^{2}-\frac{569}{4}a-55$, $\frac{5191}{6656}a^{20}-\frac{35617}{6656}a^{19}+\frac{63721}{3328}a^{18}-\frac{277767}{6656}a^{17}+\frac{363773}{6656}a^{16}-\frac{165663}{3328}a^{15}+\frac{276141}{6656}a^{14}-\frac{107075}{1664}a^{13}+\frac{387091}{6656}a^{12}+\frac{132413}{6656}a^{11}+\frac{165377}{3328}a^{10}-\frac{378741}{6656}a^{9}+\frac{494333}{6656}a^{8}+\frac{365881}{3328}a^{7}+\frac{325103}{6656}a^{6}-\frac{97463}{416}a^{5}-\frac{804513}{1664}a^{4}-\frac{436695}{832}a^{3}-\frac{18539}{52}a^{2}-\frac{33253}{208}a-\frac{4099}{104}$, $\frac{5113}{3328}a^{20}-\frac{17609}{1664}a^{19}+\frac{63587}{1664}a^{18}-\frac{282507}{3328}a^{17}+\frac{24239}{208}a^{16}-\frac{48853}{416}a^{15}+\frac{366039}{3328}a^{14}-\frac{516057}{3328}a^{13}+\frac{228231}{1664}a^{12}+\frac{50451}{3328}a^{11}+\frac{412021}{3328}a^{10}-\frac{101895}{832}a^{9}+\frac{508141}{3328}a^{8}+\frac{644555}{3328}a^{7}+\frac{16561}{128}a^{6}-\frac{29209}{64}a^{5}-\frac{200045}{208}a^{4}-\frac{222325}{208}a^{3}-\frac{76047}{104}a^{2}-\frac{8555}{26}a-\frac{1997}{26}$, $\frac{181}{416}a^{20}-\frac{145}{64}a^{19}+\frac{4567}{832}a^{18}-\frac{3131}{832}a^{17}-\frac{6295}{416}a^{16}+\frac{8331}{208}a^{15}-\frac{819}{16}a^{14}+\frac{32229}{832}a^{13}-\frac{28461}{416}a^{12}+\frac{48353}{416}a^{11}-\frac{335}{64}a^{10}+\frac{9017}{208}a^{9}-\frac{24367}{416}a^{8}+\frac{146039}{832}a^{7}+\frac{16699}{208}a^{6}-\frac{36191}{416}a^{5}-\frac{54569}{104}a^{4}-\frac{5215}{8}a^{3}-\frac{7145}{13}a^{2}-\frac{6767}{26}a-\frac{1106}{13}$, $\frac{13033}{6656}a^{20}-\frac{93939}{6656}a^{19}+\frac{177435}{3328}a^{18}-\frac{838681}{6656}a^{17}+\frac{98259}{512}a^{16}-\frac{729093}{3328}a^{15}+\frac{114879}{512}a^{14}-\frac{119955}{416}a^{13}+\frac{1921089}{6656}a^{12}-\frac{50433}{512}a^{11}+\frac{720241}{3328}a^{10}-\frac{1627615}{6656}a^{9}+\frac{1956171}{6656}a^{8}+\frac{33961}{256}a^{7}+\frac{974453}{6656}a^{6}-\frac{264071}{416}a^{5}-\frac{1683147}{1664}a^{4}-\frac{891349}{832}a^{3}-\frac{16249}{26}a^{2}-\frac{55135}{208}a-\frac{3721}{104}$, $\frac{32527}{13312}a^{20}-\frac{206187}{13312}a^{19}+\frac{26029}{512}a^{18}-\frac{1285979}{13312}a^{17}+\frac{1228003}{13312}a^{16}-\frac{270489}{6656}a^{15}+\frac{87933}{13312}a^{14}-\frac{532277}{6656}a^{13}+\frac{214209}{13312}a^{12}+\frac{3059029}{13312}a^{11}+\frac{412847}{3328}a^{10}-\frac{930931}{13312}a^{9}+\frac{1028817}{13312}a^{8}+\frac{436009}{832}a^{7}+\frac{3624013}{13312}a^{6}-\frac{1107337}{1664}a^{5}-\frac{6545255}{3328}a^{4}-\frac{3834589}{1664}a^{3}-\frac{362729}{208}a^{2}-\frac{340107}{416}a-\frac{48513}{208}$, $\frac{9551}{6656}a^{20}-\frac{60753}{6656}a^{19}+\frac{100085}{3328}a^{18}-\frac{383399}{6656}a^{17}+\frac{376501}{6656}a^{16}-\frac{95943}{3328}a^{15}+\frac{6489}{512}a^{14}-\frac{98765}{1664}a^{13}+\frac{155707}{6656}a^{12}+\frac{822949}{6656}a^{11}+\frac{270941}{3328}a^{10}-\frac{360493}{6656}a^{9}+\frac{340021}{6656}a^{8}+\frac{1010037}{3328}a^{7}+\frac{1090247}{6656}a^{6}-\frac{169877}{416}a^{5}-\frac{1909129}{1664}a^{4}-\frac{1102871}{832}a^{3}-\frac{50925}{52}a^{2}-\frac{93213}{208}a-\frac{12963}{104}$, $\frac{3411}{13312}a^{20}-\frac{25055}{13312}a^{19}+\frac{48189}{6656}a^{18}-\frac{233087}{13312}a^{17}+\frac{367271}{13312}a^{16}-\frac{216597}{6656}a^{15}+\frac{450249}{13312}a^{14}-\frac{283185}{6656}a^{13}+\frac{601005}{13312}a^{12}-\frac{24123}{1024}a^{11}+\frac{128307}{3328}a^{10}-\frac{587351}{13312}a^{9}+\frac{675917}{13312}a^{8}+\frac{3115}{832}a^{7}+\frac{25421}{1024}a^{6}-\frac{148537}{1664}a^{5}-\frac{391227}{3328}a^{4}-\frac{216201}{1664}a^{3}-\frac{13093}{208}a^{2}-\frac{11503}{416}a+\frac{163}{208}$, $\frac{9159}{3328}a^{20}-\frac{15463}{832}a^{19}+\frac{109327}{1664}a^{18}-\frac{470669}{3328}a^{17}+\frac{304399}{1664}a^{16}-\frac{10851}{64}a^{15}+\frac{495009}{3328}a^{14}-\frac{753321}{3328}a^{13}+\frac{74261}{416}a^{12}+\frac{337941}{3328}a^{11}+\frac{601145}{3328}a^{10}-\frac{269205}{1664}a^{9}+\frac{671343}{3328}a^{8}+\frac{1426151}{3328}a^{7}+\frac{197113}{832}a^{6}-\frac{659921}{832}a^{5}-\frac{772063}{416}a^{4}-\frac{215609}{104}a^{3}-\frac{152401}{104}a^{2}-\frac{34269}{52}a-\frac{2144}{13}$, $\frac{29545}{13312}a^{20}-\frac{204169}{13312}a^{19}+\frac{369159}{6656}a^{18}-\frac{1640213}{13312}a^{17}+\frac{2242729}{13312}a^{16}-\frac{1110619}{6656}a^{15}+\frac{155687}{1024}a^{14}-\frac{1431477}{6656}a^{13}+\frac{2530443}{13312}a^{12}+\frac{466099}{13312}a^{11}+\frac{130183}{832}a^{10}-\frac{2220585}{13312}a^{9}+\frac{2818783}{13312}a^{8}+\frac{976613}{3328}a^{7}+\frac{2115999}{13312}a^{6}-\frac{1106169}{1664}a^{5}-\frac{4556293}{3328}a^{4}-\frac{2467871}{1664}a^{3}-\frac{207397}{208}a^{2}-\frac{183329}{416}a-\frac{20971}{208}$ 173/1024a^20 - 1025/1024a^19 + 1571/512a^18 - 5345/1024a^17 + 3737/1024a^16 - 187/512a^15 - 329/1024a^14 - 2719/512a^13 - 2573/1024a^12 + 19807/1024a^11 + 2949/256a^10 + 3447/1024a^9 - 4781/1024a^8 + 2863/64a^7 + 29303/1024a^6 - 4347/128a^5 - 41925/256a^4 - 27095/128a^3 - 2879/16a^2 - 2833/32a - 467/16, 15/256a^20 + 3/16a^19 - 373/128a^18 + 3491/256a^17 - 4663/128a^16 + 3825/64a^15 - 18047/256a^14 + 17931/256a^13 - 5735/64a^12 + 25677/256a^11 - 9931/256a^10 + 8097/128a^9 - 20233/256a^8 + 28739/256a^7 + 435/16a^6 + 1043/64a^5 - 7539/32a^4 - 1215/4a^3 - 2337/8a^2 - 569/4a - 55, 5191/6656a^20 - 35617/6656a^19 + 63721/3328a^18 - 277767/6656a^17 + 363773/6656a^16 - 165663/3328a^15 + 276141/6656a^14 - 107075/1664a^13 + 387091/6656a^12 + 132413/6656a^11 + 165377/3328a^10 - 378741/6656a^9 + 494333/6656a^8 + 365881/3328a^7 + 325103/6656a^6 - 97463/416a^5 - 804513/1664a^4 - 436695/832a^3 - 18539/52a^2 - 33253/208a - 4099/104, 5113/3328a^20 - 17609/1664a^19 + 63587/1664a^18 - 282507/3328a^17 + 24239/208a^16 - 48853/416a^15 + 366039/3328a^14 - 516057/3328a^13 + 228231/1664a^12 + 50451/3328a^11 + 412021/3328a^10 - 101895/832a^9 + 508141/3328a^8 + 644555/3328a^7 + 16561/128a^6 - 29209/64a^5 - 200045/208a^4 - 222325/208a^3 - 76047/104a^2 - 8555/26a - 1997/26, 181/416a^20 - 145/64a^19 + 4567/832a^18 - 3131/832a^17 - 6295/416a^16 + 8331/208a^15 - 819/16a^14 + 32229/832a^13 - 28461/416a^12 + 48353/416a^11 - 335/64a^10 + 9017/208a^9 - 24367/416a^8 + 146039/832a^7 + 16699/208a^6 - 36191/416a^5 - 54569/104a^4 - 5215/8a^3 - 7145/13a^2 - 6767/26a - 1106/13, 13033/6656a^20 - 93939/6656a^19 + 177435/3328a^18 - 838681/6656a^17 + 98259/512a^16 - 729093/3328a^15 + 114879/512a^14 - 119955/416a^13 + 1921089/6656a^12 - 50433/512a^11 + 720241/3328a^10 - 1627615/6656a^9 + 1956171/6656a^8 + 33961/256a^7 + 974453/6656a^6 - 264071/416a^5 - 1683147/1664a^4 - 891349/832a^3 - 16249/26a^2 - 55135/208a - 3721/104, 32527/13312a^20 - 206187/13312a^19 + 26029/512a^18 - 1285979/13312a^17 + 1228003/13312a^16 - 270489/6656a^15 + 87933/13312a^14 - 532277/6656a^13 + 214209/13312a^12 + 3059029/13312a^11 + 412847/3328a^10 - 930931/13312a^9 + 1028817/13312a^8 + 436009/832a^7 + 3624013/13312a^6 - 1107337/1664a^5 - 6545255/3328a^4 - 3834589/1664a^3 - 362729/208a^2 - 340107/416a - 48513/208, 9551/6656a^20 - 60753/6656a^19 + 100085/3328a^18 - 383399/6656a^17 + 376501/6656a^16 - 95943/3328a^15 + 6489/512a^14 - 98765/1664a^13 + 155707/6656a^12 + 822949/6656a^11 + 270941/3328a^10 - 360493/6656a^9 + 340021/6656a^8 + 1010037/3328a^7 + 1090247/6656a^6 - 169877/416a^5 - 1909129/1664a^4 - 1102871/832a^3 - 50925/52a^2 - 93213/208a - 12963/104, 3411/13312a^20 - 25055/13312a^19 + 48189/6656a^18 - 233087/13312a^17 + 367271/13312a^16 - 216597/6656a^15 + 450249/13312a^14 - 283185/6656a^13 + 601005/13312a^12 - 24123/1024a^11 + 128307/3328a^10 - 587351/13312a^9 + 675917/13312a^8 + 3115/832a^7 + 25421/1024a^6 - 148537/1664a^5 - 391227/3328a^4 - 216201/1664a^3 - 13093/208a^2 - 11503/416a + 163/208, 9159/3328a^20 - 15463/832a^19 + 109327/1664a^18 - 470669/3328a^17 + 304399/1664a^16 - 10851/64a^15 + 495009/3328a^14 - 753321/3328a^13 + 74261/416a^12 + 337941/3328a^11 + 601145/3328a^10 - 269205/1664a^9 + 671343/3328a^8 + 1426151/3328a^7 + 197113/832a^6 - 659921/832a^5 - 772063/416a^4 - 215609/104a^3 - 152401/104a^2 - 34269/52a - 2144/13, 29545/13312a^20 - 204169/13312a^19 + 369159/6656a^18 - 1640213/13312a^17 + 2242729/13312a^16 - 1110619/6656a^15 + 155687/1024a^14 - 1431477/6656a^13 + 2530443/13312a^12 + 466099/13312a^11 + 130183/832a^10 - 2220585/13312a^9 + 2818783/13312a^8 + 976613/3328a^7 + 2115999/13312a^6 - 1106169/1664a^5 - 4556293/3328a^4 - 2467871/1664a^3 - 207397/208a^2 - 183329/416a - 20971/208 (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:	$ 14306648.6019 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 14306648.6019 \cdot 1}{2\cdot\sqrt{7655337671796811085236857471811}}\cr\approx \mathstrut & 0.315670894428 \end{aligned}\] (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 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*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 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*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 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*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 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*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

$D_7\wr C_3$ (as 21T45):

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 8232

The 55 conjugacy class representatives for $D_7\wr C_3$

Character table for $D_7\wr C_3$

Intermediate fields

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

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

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 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	${\href{/padicField/2.6.0.1}{6} }^{3}{,}\,{\href{/padicField/2.3.0.1}{3} }$	${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$	$21$	R	${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$	${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$	${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$	${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$	R	${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$	${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }$	${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$	$21$	${\href{/padicField/53.3.0.1}{3} }^{7}$	${\href{/padicField/59.3.0.1}{3} }^{7}$

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
$7$ 7	7.3.2.2	$x^{3} + 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.6.4.3	$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
	7.6.4.3	$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
	7.6.4.3	$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
$29$ 29	$\Q_{29}$	$x + 27$	$1$	$1$	$0$	Trivial	$[\ ]$
	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	29.7.0.1	$x^{7} + 2 x + 27$	$1$	$7$	$0$	$C_7$	$[\ ]^{7}$
	29.7.0.1	$x^{7} + 2 x + 27$	$1$	$7$	$0$	$C_7$	$[\ ]^{7}$
$77351$ 77351	$\Q_{77351}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{77351}$	$x$	$1$	$1$	$0$	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 $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $7$	$1$	$7$	$0$	$C_7$	$[\ ]^{7}$

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