LMFDB - Number field 21.7.370406424218316848968019199412507790681604096.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 + 189*x^17 - 1253*x^15 + 7455*x^13 - 2856*x^12 - 32823*x^11 + 34272*x^10 + 80055*x^9 - 154224*x^8 - 53055*x^7 + 311072*x^6 - 146772*x^5 - 247080*x^4 + 270064*x^3 + 23616*x^2 - 145344*x + 58752)

gp: K = bnfinit(y^21 - 21*y^19 + 189*y^17 - 1253*y^15 + 7455*y^13 - 2856*y^12 - 32823*y^11 + 34272*y^10 + 80055*y^9 - 154224*y^8 - 53055*y^7 + 311072*y^6 - 146772*y^5 - 247080*y^4 + 270064*y^3 + 23616*y^2 - 145344*y + 58752, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^19 + 189*x^17 - 1253*x^15 + 7455*x^13 - 2856*x^12 - 32823*x^11 + 34272*x^10 + 80055*x^9 - 154224*x^8 - 53055*x^7 + 311072*x^6 - 146772*x^5 - 247080*x^4 + 270064*x^3 + 23616*x^2 - 145344*x + 58752);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 21*x^19 + 189*x^17 - 1253*x^15 + 7455*x^13 - 2856*x^12 - 32823*x^11 + 34272*x^10 + 80055*x^9 - 154224*x^8 - 53055*x^7 + 311072*x^6 - 146772*x^5 - 247080*x^4 + 270064*x^3 + 23616*x^2 - 145344*x + 58752)

$ x^{21} - 21 x^{19} + 189 x^{17} - 1253 x^{15} + 7455 x^{13} - 2856 x^{12} - 32823 x^{11} + 34272 x^{10} + \cdots + 58752 $ x^21 - 21*x^19 + 189*x^17 - 1253*x^15 + 7455*x^13 - 2856*x^12 - 32823*x^11 + 34272*x^10 + 80055*x^9 - 154224*x^8 - 53055*x^7 + 311072*x^6 - 146772*x^5 - 247080*x^4 + 270064*x^3 + 23616*x^2 - 145344*x + 58752

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:	$[7, 7]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-370406424218316848968019199412507790681604096$ -370406424218316848968019199412507790681604096 $\medspace = -\,2^{15}\cdot 3^{19}\cdot 11\cdot 313^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$132.53$	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:	not computed
Ramified primes:	$2$, $3$, $11$, $313$ 2, 3, 11, 313	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-66}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{5}+\frac{1}{8}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{1}{8}a^{4}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{12}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{3}{16}a^{5}-\frac{3}{32}a^{4}+\frac{3}{16}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{32}a^{13}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{1}{16}a^{6}+\frac{1}{32}a^{5}+\frac{3}{16}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{13}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}-\frac{3}{32}a^{8}+\frac{3}{32}a^{7}-\frac{5}{64}a^{6}+\frac{9}{64}a^{5}-\frac{1}{32}a^{4}+\frac{1}{16}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{128}a^{15}+\frac{1}{128}a^{13}+\frac{1}{64}a^{11}-\frac{1}{16}a^{10}-\frac{3}{64}a^{9}+\frac{1}{16}a^{8}-\frac{3}{128}a^{7}-\frac{11}{128}a^{5}-\frac{3}{16}a^{4}-\frac{1}{4}a^{3}+\frac{7}{16}a^{2}-\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{128}a^{16}-\frac{1}{128}a^{14}-\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{32}a^{11}-\frac{3}{64}a^{10}-\frac{1}{16}a^{9}+\frac{9}{128}a^{8}+\frac{3}{32}a^{7}-\frac{9}{128}a^{6}+\frac{5}{64}a^{5}+\frac{1}{16}a^{4}-\frac{1}{16}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{256}a^{17}-\frac{1}{256}a^{16}-\frac{1}{256}a^{15}-\frac{1}{256}a^{14}-\frac{1}{128}a^{12}-\frac{1}{128}a^{11}-\frac{1}{128}a^{10}-\frac{15}{256}a^{9}-\frac{29}{256}a^{8}+\frac{11}{256}a^{7}-\frac{13}{256}a^{6}-\frac{17}{128}a^{5}+\frac{1}{16}a^{4}+\frac{5}{32}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{512}a^{18}-\frac{1}{256}a^{16}-\frac{1}{256}a^{15}-\frac{1}{512}a^{14}+\frac{3}{256}a^{13}+\frac{1}{128}a^{12}-\frac{1}{128}a^{11}+\frac{31}{512}a^{10}+\frac{5}{128}a^{9}-\frac{25}{256}a^{8}-\frac{9}{256}a^{7}+\frac{1}{512}a^{6}-\frac{13}{256}a^{5}+\frac{1}{32}a^{4}+\frac{11}{64}a^{3}-\frac{3}{8}a^{2}+\frac{3}{8}a-\frac{1}{8}$, $\frac{1}{512}a^{19}+\frac{1}{512}a^{15}-\frac{13}{512}a^{11}-\frac{1}{64}a^{10}-\frac{1}{64}a^{9}-\frac{1}{64}a^{8}-\frac{5}{512}a^{7}-\frac{7}{64}a^{6}-\frac{7}{64}a^{5}+\frac{5}{64}a^{4}-\frac{3}{32}a^{3}+\frac{1}{16}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{3072}a^{20}-\frac{1}{1024}a^{19}-\frac{1}{1024}a^{16}+\frac{3}{1024}a^{15}-\frac{5}{768}a^{14}-\frac{3}{256}a^{13}-\frac{7}{1024}a^{12}-\frac{3}{1024}a^{11}+\frac{3}{128}a^{10}-\frac{1}{128}a^{9}+\frac{53}{1024}a^{8}+\frac{49}{1024}a^{7}-\frac{7}{256}a^{6}+\frac{71}{768}a^{5}-\frac{31}{128}a^{4}+\frac{7}{64}a^{3}-\frac{1}{6}a^{2}-\frac{1}{16}a-\frac{1}{8}$ 1, a, a^2, 1/2*a^3 - 1/2*a, 1/2*a^4 - 1/2*a^2, 1/2*a^5 - 1/2*a, 1/4*a^6 - 1/4*a^2, 1/4*a^7 - 1/4*a^3, 1/4*a^8 - 1/4*a^4, 1/8*a^9 - 1/8*a^7 - 1/8*a^5 + 1/8*a^3, 1/8*a^10 - 1/8*a^8 - 1/8*a^6 + 1/8*a^4, 1/16*a^11 - 1/16*a^10 - 1/16*a^9 + 1/16*a^8 - 1/16*a^7 + 1/16*a^6 + 1/16*a^5 - 1/16*a^4 - 1/2*a^2 - 1/2*a, 1/32*a^12 - 1/16*a^9 + 1/16*a^8 + 1/16*a^7 - 1/8*a^6 - 3/16*a^5 - 3/32*a^4 + 3/16*a^3 + 1/8*a^2 - 1/2*a - 1/2, 1/32*a^13 - 1/16*a^10 - 1/16*a^9 + 1/16*a^8 + 1/16*a^6 + 1/32*a^5 + 3/16*a^4 + 1/4*a^2 - 1/2*a, 1/64*a^14 - 1/64*a^13 - 1/32*a^11 - 1/16*a^10 - 3/32*a^8 + 3/32*a^7 - 5/64*a^6 + 9/64*a^5 - 1/32*a^4 + 1/16*a^3 - 1/4*a^2 + 1/4*a, 1/128*a^15 + 1/128*a^13 + 1/64*a^11 - 1/16*a^10 - 3/64*a^9 + 1/16*a^8 - 3/128*a^7 - 11/128*a^5 - 3/16*a^4 - 1/4*a^3 + 7/16*a^2 - 1/8*a + 1/4, 1/128*a^16 - 1/128*a^14 - 1/64*a^13 - 1/64*a^12 - 1/32*a^11 - 3/64*a^10 - 1/16*a^9 + 9/128*a^8 + 3/32*a^7 - 9/128*a^6 + 5/64*a^5 + 1/16*a^4 - 1/16*a^3 - 1/2*a^2 - 1/2, 1/256*a^17 - 1/256*a^16 - 1/256*a^15 - 1/256*a^14 - 1/128*a^12 - 1/128*a^11 - 1/128*a^10 - 15/256*a^9 - 29/256*a^8 + 11/256*a^7 - 13/256*a^6 - 17/128*a^5 + 1/16*a^4 + 5/32*a^3 + 3/8*a^2 - 1/2*a + 1/4, 1/512*a^18 - 1/256*a^16 - 1/256*a^15 - 1/512*a^14 + 3/256*a^13 + 1/128*a^12 - 1/128*a^11 + 31/512*a^10 + 5/128*a^9 - 25/256*a^8 - 9/256*a^7 + 1/512*a^6 - 13/256*a^5 + 1/32*a^4 + 11/64*a^3 - 3/8*a^2 + 3/8*a - 1/8, 1/512*a^19 + 1/512*a^15 - 13/512*a^11 - 1/64*a^10 - 1/64*a^9 - 1/64*a^8 - 5/512*a^7 - 7/64*a^6 - 7/64*a^5 + 5/64*a^4 - 3/32*a^3 + 1/16*a^2 - 1/4*a - 1/2, 1/3072*a^20 - 1/1024*a^19 - 1/1024*a^16 + 3/1024*a^15 - 5/768*a^14 - 3/256*a^13 - 7/1024*a^12 - 3/1024*a^11 + 3/128*a^10 - 1/128*a^9 + 53/1024*a^8 + 49/1024*a^7 - 7/256*a^6 + 71/768*a^5 - 31/128*a^4 + 7/64*a^3 - 1/6*a^2 - 1/16*a - 1/8

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:	$13$	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:	$\frac{19}{64}a^{18}-\frac{171}{32}a^{16}-\frac{135}{128}a^{15}+\frac{2565}{64}a^{14}+\frac{2025}{128}a^{13}-248a^{12}-\frac{6075}{64}a^{11}+\frac{90429}{64}a^{10}-\frac{16103}{64}a^{9}-\frac{165375}{32}a^{8}+\frac{563229}{128}a^{7}+\frac{585895}{64}a^{6}-\frac{1820907}{128}a^{5}-\frac{62565}{16}a^{4}+\frac{34297}{2}a^{3}-\frac{76617}{16}a^{2}-\frac{63993}{8}a+\frac{19661}{4}$, $\frac{233}{512}a^{18}-\frac{2097}{256}a^{16}-\frac{411}{256}a^{15}+\frac{31455}{512}a^{14}+\frac{6165}{256}a^{13}-\frac{48671}{128}a^{12}-\frac{18495}{128}a^{11}+\frac{1109463}{512}a^{10}-\frac{50141}{128}a^{9}-\frac{2029185}{256}a^{8}+\frac{1734813}{256}a^{7}+\frac{7185249}{512}a^{6}-\frac{5603931}{256}a^{5}-\frac{47601}{8}a^{4}+\frac{1688431}{64}a^{3}-\frac{118665}{16}a^{2}-\frac{49215}{4}a+\frac{60629}{8}$, $\frac{1}{64}a^{18}-\frac{9}{32}a^{16}-\frac{1}{16}a^{15}+\frac{135}{64}a^{14}+\frac{15}{16}a^{13}-13a^{12}-\frac{45}{8}a^{11}+\frac{4719}{64}a^{10}-\frac{19}{2}a^{9}-\frac{8613}{32}a^{8}+\frac{3393}{16}a^{7}+\frac{30809}{64}a^{6}-\frac{11151}{16}a^{5}-\frac{3729}{16}a^{4}+\frac{6753}{8}a^{3}-\frac{819}{4}a^{2}-396a+229$, $\frac{85}{256}a^{18}-\frac{765}{128}a^{16}-\frac{73}{64}a^{15}+\frac{11475}{256}a^{14}+\frac{1095}{64}a^{13}-\frac{17769}{64}a^{12}-\frac{3285}{32}a^{11}+\frac{405387}{256}a^{10}-\frac{605}{2}a^{9}-\frac{741717}{128}a^{8}+\frac{322065}{64}a^{7}+\frac{2619933}{256}a^{6}-\frac{1037151}{64}a^{5}-\frac{134127}{32}a^{4}+\frac{624385}{32}a^{3}-\frac{90585}{16}a^{2}-\frac{72705}{8}a+5683$, $\frac{201}{256}a^{18}-\frac{1809}{128}a^{16}-\frac{357}{128}a^{15}+\frac{27135}{256}a^{14}+\frac{5355}{128}a^{13}-\frac{41977}{64}a^{12}-\frac{16065}{64}a^{11}+\frac{956631}{256}a^{10}-\frac{42603}{64}a^{9}-\frac{1749465}{128}a^{8}+\frac{1489779}{128}a^{7}+\frac{6197601}{256}a^{6}-\frac{4816341}{128}a^{5}-\frac{330597}{32}a^{4}+\frac{1451429}{32}a^{3}-\frac{101439}{8}a^{2}-\frac{84621}{4}a+\frac{52025}{4}$, $\frac{321}{512}a^{18}-\frac{2889}{256}a^{16}-\frac{579}{256}a^{15}+\frac{43335}{512}a^{14}+\frac{8685}{256}a^{13}-\frac{67007}{128}a^{12}-\frac{26055}{128}a^{11}+\frac{1526271}{512}a^{10}-\frac{65637}{128}a^{9}-\frac{2790585}{256}a^{8}+\frac{2353941}{256}a^{7}+\frac{9899897}{512}a^{6}-\frac{7624611}{256}a^{5}-\frac{134601}{16}a^{4}+\frac{2299007}{64}a^{3}-\frac{157617}{16}a^{2}-\frac{67065}{4}a+\frac{81749}{8}$, $\frac{594003}{1024}a^{20}-\frac{1301933}{1024}a^{19}-\frac{605711}{64}a^{18}+\frac{5328729}{256}a^{17}+\frac{66825931}{1024}a^{16}-\frac{147535213}{1024}a^{15}-\frac{53846697}{128}a^{14}+\frac{237561189}{256}a^{13}+\frac{2408838489}{1024}a^{12}-\frac{7014391759}{1024}a^{11}-\frac{280283369}{64}a^{10}+\frac{7648449503}{256}a^{9}-\frac{18325802655}{1024}a^{8}-\frac{53761602951}{1024}a^{7}+\frac{10649986661}{128}a^{6}+\frac{920236027}{256}a^{5}-\frac{12222411863}{128}a^{4}+\frac{3905114165}{64}a^{3}+28117993a^{2}-\frac{755128357}{16}a+\frac{128659013}{8}$, $\frac{4955}{3072}a^{20}-\frac{3253}{1024}a^{19}-\frac{13369}{512}a^{18}+\frac{3261}{64}a^{17}+\frac{181881}{1024}a^{16}-\frac{354877}{1024}a^{15}-\frac{1745963}{1536}a^{14}+\frac{288321}{128}a^{13}+\frac{6475739}{1024}a^{12}-\frac{17535207}{1024}a^{11}-\frac{6147819}{512}a^{10}+\frac{4818241}{64}a^{9}-\frac{45451749}{1024}a^{8}-\frac{133753031}{1024}a^{7}+\frac{107248909}{512}a^{6}+\frac{1132637}{384}a^{5}-\frac{30400291}{128}a^{4}+\frac{1289701}{8}a^{3}+\frac{1562383}{24}a^{2}-\frac{1953811}{16}a+44087$, $\frac{69505}{1536}a^{20}-\frac{25295}{256}a^{19}-\frac{188613}{256}a^{18}+\frac{412997}{256}a^{17}+\frac{2595629}{512}a^{16}-\frac{1426121}{128}a^{15}-\frac{6274249}{192}a^{14}+\frac{9194265}{128}a^{13}+\frac{93455173}{512}a^{12}-\frac{135880031}{256}a^{11}-\frac{86491713}{256}a^{10}+\frac{591749997}{256}a^{9}-\frac{712997853}{512}a^{8}-\frac{64733865}{16}a^{7}+\frac{824254849}{128}a^{6}+\frac{46687765}{192}a^{5}-\frac{117773033}{16}a^{4}+\frac{75776113}{16}a^{3}+\frac{25808663}{12}a^{2}-\frac{14592411}{4}a+\frac{2497285}{2}$, $\frac{61927}{1536}a^{20}+\frac{2591}{256}a^{19}-\frac{449129}{512}a^{18}-\frac{17273}{64}a^{17}+\frac{4209719}{512}a^{16}+\frac{405047}{128}a^{15}-\frac{84888869}{1536}a^{14}-\frac{6180805}{256}a^{13}+\frac{167776419}{512}a^{12}+\frac{8624767}{256}a^{11}-\frac{769438367}{512}a^{10}+\frac{91100949}{128}a^{9}+\frac{2194047417}{512}a^{8}-4489799a^{7}-\frac{3086581121}{512}a^{6}+\frac{8347770349}{768}a^{5}+\frac{14653333}{8}a^{4}-\frac{734766535}{64}a^{3}+\frac{164749337}{48}a^{2}+\frac{19814309}{4}a-\frac{22867613}{8}$, $\frac{3253009}{768}a^{20}-\frac{697643}{512}a^{19}-\frac{13004869}{128}a^{18}+\frac{2317497}{256}a^{17}+\frac{66134755}{64}a^{16}+\frac{71846363}{512}a^{15}-\frac{5402931479}{768}a^{14}-\frac{52436715}{32}a^{13}+\frac{10688053659}{256}a^{12}-\frac{782615717}{512}a^{11}-\frac{6178116645}{32}a^{10}+\frac{27024286085}{256}a^{9}+\frac{70631418101}{128}a^{8}-\frac{313214922123}{512}a^{7}-\frac{198780459073}{256}a^{6}+\frac{560479327711}{384}a^{5}+\frac{12920357917}{64}a^{4}-\frac{24633903405}{16}a^{3}+\frac{23852086969}{48}a^{2}+\frac{2617371925}{4}a-\frac{1639750493}{4}$, $\frac{156559}{1024}a^{20}+\frac{354439}{1024}a^{19}-\frac{345087}{128}a^{18}-\frac{863235}{128}a^{17}+\frac{18900715}{1024}a^{16}+\frac{55551691}{1024}a^{15}-\frac{26306439}{256}a^{14}-\frac{85248925}{256}a^{13}+\frac{591485237}{1024}a^{12}+\frac{1525706325}{1024}a^{11}-\frac{87311837}{32}a^{10}-\frac{240137331}{64}a^{9}+\frac{9237790793}{1024}a^{8}+\frac{4084567497}{1024}a^{7}-\frac{4394144923}{256}a^{6}+\frac{511477015}{256}a^{5}+\frac{2189649933}{128}a^{4}-\frac{441663509}{64}a^{3}-\frac{125955059}{16}a^{2}+\frac{71081541}{16}a+\frac{7944773}{8}$, $\frac{1291919}{3072}a^{20}+\frac{624415}{1024}a^{19}-\frac{4204337}{512}a^{18}-\frac{1600457}{128}a^{17}+\frac{67694629}{1024}a^{16}+\frac{110147199}{1024}a^{15}-\frac{621235931}{1536}a^{14}-\frac{87231297}{128}a^{13}+\frac{2390915599}{1024}a^{12}+\frac{2835928069}{1024}a^{11}-\frac{5553741987}{512}a^{10}-\frac{504542149}{128}a^{9}+\frac{33808061375}{1024}a^{8}-\frac{10704924867}{1024}a^{7}-\frac{27811783219}{512}a^{6}+\frac{17701372529}{384}a^{5}+\frac{4730142517}{128}a^{4}-\frac{458187111}{8}a^{3}+\frac{12131689}{24}a^{2}+\frac{431239565}{16}a-\frac{20748923}{2}$ 19/64a^18 - 171/32a^16 - 135/128a^15 + 2565/64a^14 + 2025/128a^13 - 248a^12 - 6075/64a^11 + 90429/64a^10 - 16103/64a^9 - 165375/32a^8 + 563229/128a^7 + 585895/64a^6 - 1820907/128a^5 - 62565/16a^4 + 34297/2a^3 - 76617/16a^2 - 63993/8a + 19661/4, 233/512a^18 - 2097/256a^16 - 411/256a^15 + 31455/512a^14 + 6165/256a^13 - 48671/128a^12 - 18495/128a^11 + 1109463/512a^10 - 50141/128a^9 - 2029185/256a^8 + 1734813/256a^7 + 7185249/512a^6 - 5603931/256a^5 - 47601/8a^4 + 1688431/64a^3 - 118665/16a^2 - 49215/4a + 60629/8, 1/64a^18 - 9/32a^16 - 1/16a^15 + 135/64a^14 + 15/16a^13 - 13a^12 - 45/8a^11 + 4719/64a^10 - 19/2a^9 - 8613/32a^8 + 3393/16a^7 + 30809/64a^6 - 11151/16a^5 - 3729/16a^4 + 6753/8a^3 - 819/4a^2 - 396a + 229, 85/256a^18 - 765/128a^16 - 73/64a^15 + 11475/256a^14 + 1095/64a^13 - 17769/64a^12 - 3285/32a^11 + 405387/256a^10 - 605/2a^9 - 741717/128a^8 + 322065/64a^7 + 2619933/256a^6 - 1037151/64a^5 - 134127/32a^4 + 624385/32a^3 - 90585/16a^2 - 72705/8a + 5683, 201/256a^18 - 1809/128a^16 - 357/128a^15 + 27135/256a^14 + 5355/128a^13 - 41977/64a^12 - 16065/64a^11 + 956631/256a^10 - 42603/64a^9 - 1749465/128a^8 + 1489779/128a^7 + 6197601/256a^6 - 4816341/128a^5 - 330597/32a^4 + 1451429/32a^3 - 101439/8a^2 - 84621/4a + 52025/4, 321/512a^18 - 2889/256a^16 - 579/256a^15 + 43335/512a^14 + 8685/256a^13 - 67007/128a^12 - 26055/128a^11 + 1526271/512a^10 - 65637/128a^9 - 2790585/256a^8 + 2353941/256a^7 + 9899897/512a^6 - 7624611/256a^5 - 134601/16a^4 + 2299007/64a^3 - 157617/16a^2 - 67065/4a + 81749/8, 594003/1024a^20 - 1301933/1024a^19 - 605711/64a^18 + 5328729/256a^17 + 66825931/1024a^16 - 147535213/1024a^15 - 53846697/128a^14 + 237561189/256a^13 + 2408838489/1024a^12 - 7014391759/1024a^11 - 280283369/64a^10 + 7648449503/256a^9 - 18325802655/1024a^8 - 53761602951/1024a^7 + 10649986661/128a^6 + 920236027/256a^5 - 12222411863/128a^4 + 3905114165/64a^3 + 28117993a^2 - 755128357/16a + 128659013/8, 4955/3072a^20 - 3253/1024a^19 - 13369/512a^18 + 3261/64a^17 + 181881/1024a^16 - 354877/1024a^15 - 1745963/1536a^14 + 288321/128a^13 + 6475739/1024a^12 - 17535207/1024a^11 - 6147819/512a^10 + 4818241/64a^9 - 45451749/1024a^8 - 133753031/1024a^7 + 107248909/512a^6 + 1132637/384a^5 - 30400291/128a^4 + 1289701/8a^3 + 1562383/24a^2 - 1953811/16a + 44087, 69505/1536a^20 - 25295/256a^19 - 188613/256a^18 + 412997/256a^17 + 2595629/512a^16 - 1426121/128a^15 - 6274249/192a^14 + 9194265/128a^13 + 93455173/512a^12 - 135880031/256a^11 - 86491713/256a^10 + 591749997/256a^9 - 712997853/512a^8 - 64733865/16a^7 + 824254849/128a^6 + 46687765/192a^5 - 117773033/16a^4 + 75776113/16a^3 + 25808663/12a^2 - 14592411/4a + 2497285/2, 61927/1536a^20 + 2591/256a^19 - 449129/512a^18 - 17273/64a^17 + 4209719/512a^16 + 405047/128a^15 - 84888869/1536a^14 - 6180805/256a^13 + 167776419/512a^12 + 8624767/256a^11 - 769438367/512a^10 + 91100949/128a^9 + 2194047417/512a^8 - 4489799a^7 - 3086581121/512a^6 + 8347770349/768a^5 + 14653333/8a^4 - 734766535/64a^3 + 164749337/48a^2 + 19814309/4a - 22867613/8, 3253009/768a^20 - 697643/512a^19 - 13004869/128a^18 + 2317497/256a^17 + 66134755/64a^16 + 71846363/512a^15 - 5402931479/768a^14 - 52436715/32a^13 + 10688053659/256a^12 - 782615717/512a^11 - 6178116645/32a^10 + 27024286085/256a^9 + 70631418101/128a^8 - 313214922123/512a^7 - 198780459073/256a^6 + 560479327711/384a^5 + 12920357917/64a^4 - 24633903405/16a^3 + 23852086969/48a^2 + 2617371925/4a - 1639750493/4, 156559/1024a^20 + 354439/1024a^19 - 345087/128a^18 - 863235/128a^17 + 18900715/1024a^16 + 55551691/1024a^15 - 26306439/256a^14 - 85248925/256a^13 + 591485237/1024a^12 + 1525706325/1024a^11 - 87311837/32a^10 - 240137331/64a^9 + 9237790793/1024a^8 + 4084567497/1024a^7 - 4394144923/256a^6 + 511477015/256a^5 + 2189649933/128a^4 - 441663509/64a^3 - 125955059/16a^2 + 71081541/16a + 7944773/8, 1291919/3072a^20 + 624415/1024a^19 - 4204337/512a^18 - 1600457/128a^17 + 67694629/1024a^16 + 110147199/1024a^15 - 621235931/1536a^14 - 87231297/128a^13 + 2390915599/1024a^12 + 2835928069/1024a^11 - 5553741987/512a^10 - 504542149/128a^9 + 33808061375/1024a^8 - 10704924867/1024a^7 - 27811783219/512a^6 + 17701372529/384a^5 + 4730142517/128a^4 - 458187111/8a^3 + 12131689/24a^2 + 431239565/16a - 20748923/2 (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:	$ 2319698880310000 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 2319698880310000 \cdot 1}{2\cdot\sqrt{370406424218316848968019199412507790681604096}}\cr\approx \mathstrut & 2.98216322391208 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 + 189*x^17 - 1253*x^15 + 7455*x^13 - 2856*x^12 - 32823*x^11 + 34272*x^10 + 80055*x^9 - 154224*x^8 - 53055*x^7 + 311072*x^6 - 146772*x^5 - 247080*x^4 + 270064*x^3 + 23616*x^2 - 145344*x + 58752)

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 - 21*x^19 + 189*x^17 - 1253*x^15 + 7455*x^13 - 2856*x^12 - 32823*x^11 + 34272*x^10 + 80055*x^9 - 154224*x^8 - 53055*x^7 + 311072*x^6 - 146772*x^5 - 247080*x^4 + 270064*x^3 + 23616*x^2 - 145344*x + 58752, 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 - 21*x^19 + 189*x^17 - 1253*x^15 + 7455*x^13 - 2856*x^12 - 32823*x^11 + 34272*x^10 + 80055*x^9 - 154224*x^8 - 53055*x^7 + 311072*x^6 - 146772*x^5 - 247080*x^4 + 270064*x^3 + 23616*x^2 - 145344*x + 58752);

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 - 21*x^19 + 189*x^17 - 1253*x^15 + 7455*x^13 - 2856*x^12 - 32823*x^11 + 34272*x^10 + 80055*x^9 - 154224*x^8 - 53055*x^7 + 311072*x^6 - 146772*x^5 - 247080*x^4 + 270064*x^3 + 23616*x^2 - 145344*x + 58752);

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

$C_3^7.C_2\wr C_7:C_3$ (as 21T137):

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 5878656

The 183 conjugacy class representatives for $C_3^7.C_2\wr C_7:C_3$

Character table for $C_3^7.C_2\wr C_7:C_3$

Intermediate fields

7.7.9597924961.1

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 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	R	$21$	$21$	R	${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$	${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$	$21$	${\href{/padicField/29.9.0.1}{9} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{3}$	${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{3}$	${\href{/padicField/53.7.0.1}{7} }^{3}$	${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{5}$

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	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.2.3.1	$x^{2} + 4 x + 2$	$2$	$1$	$3$	$C_2$	$[3]$
	2.3.0.1	$x^{3} + x + 1$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	2.3.0.1	$x^{3} + x + 1$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	2.6.6.1	$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$	$2$	$3$	$6$	$A_4$	$[2, 2]^{3}$
	2.6.6.4	$x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$	$2$	$3$	$6$	$A_4\times C_2$	$[2, 2, 2]^{3}$
$3$ 3	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.9.9.6	$x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$	$3$	$3$	$9$	$S_3\times C_3$	$[3/2]_{2}^{3}$
	3.9.9.4	$x^{9} - 24 x^{8} + 318 x^{7} - 189 x^{6} + 1080 x^{5} + 2826 x^{4} + 1350 x^{3} - 108 x^{2} - 54 x + 27$	$3$	$3$	$9$	$(C_3^3:C_3):C_2$	$[3/2, 3/2, 3/2]_{2}^{3}$
$11$ 11	$\Q_{11}$	$x + 9$	$1$	$1$	$0$	Trivial	$[\ ]$
	11.2.1.2	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.3.0.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	11.3.0.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	11.3.0.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	11.3.0.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	11.6.0.1	$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$313$ 313	$\Q_{313}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $3$	$3$	$1$	$2$
		Deg $3$	$3$	$1$	$2$
		Deg $3$	$3$	$1$	$2$
		Deg $3$	$3$	$1$	$2$
		Deg $6$	$3$	$2$	$4$

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