LMFDB - Number field 22.14.7198079267989980836471065337135104.6

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 7*x^20 - 7*x^18 + 97*x^16 - 67*x^14 - 338*x^12 + 512*x^10 + 14*x^8 - 362*x^6 + 166*x^4 - 9*x^2 - 1)

gp:K = bnfinit(y^22 - 7*y^20 - 7*y^18 + 97*y^16 - 67*y^14 - 338*y^12 + 512*y^10 + 14*y^8 - 362*y^6 + 166*y^4 - 9*y^2 - 1, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 7*x^20 - 7*x^18 + 97*x^16 - 67*x^14 - 338*x^12 + 512*x^10 + 14*x^8 - 362*x^6 + 166*x^4 - 9*x^2 - 1);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 7*x^20 - 7*x^18 + 97*x^16 - 67*x^14 - 338*x^12 + 512*x^10 + 14*x^8 - 362*x^6 + 166*x^4 - 9*x^2 - 1)

$ x^{22} - 7 x^{20} - 7 x^{18} + 97 x^{16} - 67 x^{14} - 338 x^{12} + 512 x^{10} + 14 x^{8} - 362 x^{6} + \cdots - 1 $ x^22 - 7*x^20 - 7*x^18 + 97*x^16 - 67*x^14 - 338*x^12 + 512*x^10 + 14*x^8 - 362*x^6 + 166*x^4 - 9*x^2 - 1

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$22$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[14, 4]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$7198079267989980836471065337135104$ 7198079267989980836471065337135104 $\medspace = 2^{22}\cdot 23^{20}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$34.59$	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:	$2$, $23$ 2, 23	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:	$\Q$
$\Aut(K/\Q)$:	$C_2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23}a^{12}-\frac{8}{23}a^{10}-\frac{4}{23}a^{8}+\frac{2}{23}a^{6}-\frac{2}{23}a^{4}-\frac{2}{23}a^{2}+\frac{3}{23}$, $\frac{1}{23}a^{13}-\frac{8}{23}a^{11}-\frac{4}{23}a^{9}+\frac{2}{23}a^{7}-\frac{2}{23}a^{5}-\frac{2}{23}a^{3}+\frac{3}{23}a$, $\frac{1}{23}a^{14}+\frac{1}{23}a^{10}-\frac{7}{23}a^{8}-\frac{9}{23}a^{6}+\frac{5}{23}a^{4}+\frac{10}{23}a^{2}+\frac{1}{23}$, $\frac{1}{23}a^{15}+\frac{1}{23}a^{11}-\frac{7}{23}a^{9}-\frac{9}{23}a^{7}+\frac{5}{23}a^{5}+\frac{10}{23}a^{3}+\frac{1}{23}a$, $\frac{1}{23}a^{16}+\frac{1}{23}a^{10}-\frac{5}{23}a^{8}+\frac{3}{23}a^{6}-\frac{11}{23}a^{4}+\frac{3}{23}a^{2}-\frac{3}{23}$, $\frac{1}{23}a^{17}+\frac{1}{23}a^{11}-\frac{5}{23}a^{9}+\frac{3}{23}a^{7}-\frac{11}{23}a^{5}+\frac{3}{23}a^{3}-\frac{3}{23}a$, $\frac{1}{23}a^{18}+\frac{3}{23}a^{10}+\frac{7}{23}a^{8}+\frac{10}{23}a^{6}+\frac{5}{23}a^{4}-\frac{1}{23}a^{2}-\frac{3}{23}$, $\frac{1}{23}a^{19}+\frac{3}{23}a^{11}+\frac{7}{23}a^{9}+\frac{10}{23}a^{7}+\frac{5}{23}a^{5}-\frac{1}{23}a^{3}-\frac{3}{23}a$, $\frac{1}{1081}a^{20}+\frac{18}{1081}a^{18}+\frac{20}{1081}a^{16}-\frac{14}{1081}a^{14}+\frac{6}{1081}a^{12}-\frac{3}{23}a^{10}-\frac{522}{1081}a^{8}+\frac{124}{1081}a^{6}+\frac{23}{47}a^{4}-\frac{521}{1081}a^{2}+\frac{502}{1081}$, $\frac{1}{1081}a^{21}+\frac{18}{1081}a^{19}+\frac{20}{1081}a^{17}-\frac{14}{1081}a^{15}+\frac{6}{1081}a^{13}-\frac{3}{23}a^{11}-\frac{522}{1081}a^{9}+\frac{124}{1081}a^{7}+\frac{23}{47}a^{5}-\frac{521}{1081}a^{3}+\frac{502}{1081}a$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/23*a^12 - 8/23*a^10 - 4/23*a^8 + 2/23*a^6 - 2/23*a^4 - 2/23*a^2 + 3/23, 1/23*a^13 - 8/23*a^11 - 4/23*a^9 + 2/23*a^7 - 2/23*a^5 - 2/23*a^3 + 3/23*a, 1/23*a^14 + 1/23*a^10 - 7/23*a^8 - 9/23*a^6 + 5/23*a^4 + 10/23*a^2 + 1/23, 1/23*a^15 + 1/23*a^11 - 7/23*a^9 - 9/23*a^7 + 5/23*a^5 + 10/23*a^3 + 1/23*a, 1/23*a^16 + 1/23*a^10 - 5/23*a^8 + 3/23*a^6 - 11/23*a^4 + 3/23*a^2 - 3/23, 1/23*a^17 + 1/23*a^11 - 5/23*a^9 + 3/23*a^7 - 11/23*a^5 + 3/23*a^3 - 3/23*a, 1/23*a^18 + 3/23*a^10 + 7/23*a^8 + 10/23*a^6 + 5/23*a^4 - 1/23*a^2 - 3/23, 1/23*a^19 + 3/23*a^11 + 7/23*a^9 + 10/23*a^7 + 5/23*a^5 - 1/23*a^3 - 3/23*a, 1/1081*a^20 + 18/1081*a^18 + 20/1081*a^16 - 14/1081*a^14 + 6/1081*a^12 - 3/23*a^10 - 522/1081*a^8 + 124/1081*a^6 + 23/47*a^4 - 521/1081*a^2 + 502/1081, 1/1081*a^21 + 18/1081*a^19 + 20/1081*a^17 - 14/1081*a^15 + 6/1081*a^13 - 3/23*a^11 - 522/1081*a^9 + 124/1081*a^7 + 23/47*a^5 - 521/1081*a^3 + 502/1081*a

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:	$17$	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:	$a$, $\frac{239}{1081}a^{20}-\frac{1244}{1081}a^{18}-\frac{4103}{1081}a^{16}+\frac{16911}{1081}a^{14}+\frac{17320}{1081}a^{12}-\frac{1371}{23}a^{10}-\frac{2370}{1081}a^{8}+\frac{53183}{1081}a^{6}-\frac{6485}{1081}a^{4}-\frac{6173}{1081}a^{2}-\frac{154}{1081}$, $\frac{614}{1081}a^{21}-\frac{4552}{1081}a^{19}-\frac{2807}{1081}a^{17}+\frac{62985}{1081}a^{15}-\frac{61693}{1081}a^{13}-\frac{4538}{23}a^{11}+\frac{16997}{47}a^{9}-\frac{35348}{1081}a^{7}-\frac{266078}{1081}a^{5}+\frac{137040}{1081}a^{3}-\frac{9633}{1081}a$, $\frac{686}{1081}a^{21}-\frac{4478}{1081}a^{19}-\frac{6631}{1081}a^{17}+\frac{61742}{1081}a^{15}-\frac{900}{47}a^{13}-\frac{4654}{23}a^{11}+\frac{254835}{1081}a^{9}+\frac{45020}{1081}a^{7}-\frac{178499}{1081}a^{5}+\frac{78237}{1081}a^{3}-\frac{476}{47}a$, $\frac{278}{1081}a^{20}-\frac{1717}{1081}a^{18}-\frac{3276}{1081}a^{16}+\frac{23791}{1081}a^{14}-\frac{306}{1081}a^{12}-\frac{1860}{23}a^{10}+\frac{3203}{47}a^{8}+\frac{37903}{1081}a^{6}-\frac{57764}{1081}a^{4}+\frac{15620}{1081}a^{2}+\frac{436}{1081}$, $\frac{467}{1081}a^{21}-\frac{3015}{1081}a^{19}-\frac{4760}{1081}a^{17}+\frac{41872}{1081}a^{15}-\frac{10734}{1081}a^{13}-\frac{3236}{23}a^{11}+\frac{7067}{47}a^{9}+\frac{50482}{1081}a^{7}-\frac{125244}{1081}a^{5}+\frac{42406}{1081}a^{3}-\frac{96}{1081}a$, $\frac{427}{1081}a^{21}-\frac{2560}{1081}a^{19}-\frac{5513}{1081}a^{17}+\frac{35617}{1081}a^{15}+\frac{6416}{1081}a^{13}-\frac{2856}{23}a^{11}+\frac{3753}{47}a^{9}+\frac{78939}{1081}a^{7}-\frac{72614}{1081}a^{5}+\frac{6846}{1081}a^{3}+\frac{3136}{1081}a$, $\frac{220}{1081}a^{20}-\frac{1398}{1081}a^{18}-\frac{2321}{1081}a^{16}+\frac{19198}{1081}a^{14}-\frac{4320}{1081}a^{12}-\frac{1459}{23}a^{10}+\frac{75416}{1081}a^{8}+\frac{20653}{1081}a^{6}-\frac{59729}{1081}a^{4}+\frac{18155}{1081}a^{2}+\frac{695}{1081}$, $\frac{468}{1081}a^{20}-\frac{2997}{1081}a^{18}-\frac{4975}{1081}a^{16}+\frac{41905}{1081}a^{14}-\frac{7532}{1081}a^{12}-\frac{3304}{23}a^{10}+\frac{149000}{1081}a^{8}+\frac{66680}{1081}a^{6}-\frac{115315}{1081}a^{4}+\frac{29853}{1081}a^{2}-\frac{64}{1081}$, $\frac{213}{1081}a^{21}-\frac{1524}{1081}a^{19}-\frac{1333}{1081}a^{17}+\frac{21317}{1081}a^{15}-\frac{16488}{1081}a^{13}-\frac{1607}{23}a^{11}+\frac{118597}{1081}a^{9}+\frac{6672}{1081}a^{7}-\frac{90128}{1081}a^{5}+\frac{37218}{1081}a^{3}+\frac{518}{1081}a$, $a+1$, $\frac{241}{1081}a^{21}-\frac{1}{1081}a^{20}-\frac{1772}{1081}a^{19}+\frac{76}{1081}a^{18}-\frac{1102}{1081}a^{17}-\frac{255}{1081}a^{16}+\frac{24074}{1081}a^{15}-\frac{75}{47}a^{14}-\frac{24498}{1081}a^{13}+\frac{3049}{1081}a^{12}-\frac{1644}{23}a^{11}+\frac{229}{23}a^{10}+\frac{155352}{1081}a^{9}-\frac{9583}{1081}a^{8}-\frac{34412}{1081}a^{7}-\frac{18125}{1081}a^{6}-\frac{103892}{1081}a^{5}+\frac{8260}{1081}a^{4}+\frac{69160}{1081}a^{3}+\frac{7571}{1081}a^{2}-\frac{4649}{1081}a-\frac{220}{1081}$, $\frac{220}{1081}a^{20}-\frac{1398}{1081}a^{18}-\frac{2321}{1081}a^{16}+\frac{19198}{1081}a^{14}-\frac{4320}{1081}a^{12}-\frac{1459}{23}a^{10}+\frac{75416}{1081}a^{8}+\frac{20653}{1081}a^{6}-\frac{59729}{1081}a^{4}+\frac{18155}{1081}a^{2}+a+\frac{695}{1081}$, $\frac{573}{1081}a^{21}-\frac{459}{1081}a^{20}-\frac{3551}{1081}a^{19}+\frac{2642}{1081}a^{18}-\frac{6635}{1081}a^{17}+\frac{6424}{1081}a^{16}+\frac{48942}{1081}a^{15}-\frac{36109}{1081}a^{14}-\frac{2249}{1081}a^{13}-\frac{13000}{1081}a^{12}-\frac{3776}{23}a^{11}+\frac{2830}{23}a^{10}+\frac{157170}{1081}a^{9}-\frac{73046}{1081}a^{8}+\frac{67527}{1081}a^{7}-\frac{73836}{1081}a^{6}-\frac{5104}{47}a^{5}+\frac{66214}{1081}a^{4}+\frac{38833}{1081}a^{3}-\frac{11982}{1081}a^{2}-\frac{5728}{1081}a-\frac{541}{1081}$, $\frac{695}{1081}a^{21}+\frac{319}{1081}a^{20}-\frac{4645}{1081}a^{19}-\frac{63}{47}a^{18}-\frac{6263}{1081}a^{17}-\frac{6733}{1081}a^{16}+\frac{65094}{1081}a^{15}+\frac{19739}{1081}a^{14}-\frac{27367}{1081}a^{13}+\frac{40830}{1081}a^{12}-\frac{5090}{23}a^{11}-\frac{1732}{23}a^{10}+\frac{287267}{1081}a^{9}-\frac{72095}{1081}a^{8}+\frac{3702}{47}a^{7}+\frac{108270}{1081}a^{6}-\frac{230937}{1081}a^{5}+\frac{40253}{1081}a^{4}+\frac{55641}{1081}a^{3}-\frac{40615}{1081}a^{2}+\frac{11900}{1081}a-\frac{1213}{1081}$, $\frac{733}{1081}a^{21}+\frac{41}{1081}a^{20}-\frac{4619}{1081}a^{19}-\frac{531}{1081}a^{18}-\frac{8276}{1081}a^{17}+\frac{867}{1081}a^{16}+\frac{64891}{1081}a^{15}+\frac{8638}{1081}a^{14}-\frac{5143}{1081}a^{13}-\frac{17473}{1081}a^{12}-\frac{5224}{23}a^{11}-\frac{786}{23}a^{10}+\frac{209104}{1081}a^{9}+\frac{72081}{1081}a^{8}+\frac{134367}{1081}a^{7}+\frac{19043}{1081}a^{6}-\frac{169757}{1081}a^{5}-\frac{49610}{1081}a^{4}+\frac{19910}{1081}a^{3}+\frac{2750}{1081}a^{2}+\frac{2729}{1081}a-\frac{239}{1081}$, $\frac{1119}{1081}a^{21}-\frac{964}{1081}a^{20}-\frac{6648}{1081}a^{19}+\frac{5349}{1081}a^{18}-\frac{14656}{1081}a^{17}+\frac{14607}{1081}a^{16}+\frac{91870}{1081}a^{15}-\frac{72796}{1081}a^{14}+\frac{18746}{1081}a^{13}-\frac{42773}{1081}a^{12}-\frac{7294}{23}a^{11}+250a^{10}+\frac{223342}{1081}a^{9}-\frac{93598}{1081}a^{8}+\frac{194263}{1081}a^{7}-\frac{173539}{1081}a^{6}-\frac{192761}{1081}a^{5}+\frac{93759}{1081}a^{4}+\frac{24006}{1081}a^{3}-\frac{4510}{1081}a^{2}+\frac{5023}{1081}a-\frac{204}{1081}$ a, 239/1081a^20 - 1244/1081a^18 - 4103/1081a^16 + 16911/1081a^14 + 17320/1081a^12 - 1371/23a^10 - 2370/1081a^8 + 53183/1081a^6 - 6485/1081a^4 - 6173/1081a^2 - 154/1081, 614/1081a^21 - 4552/1081a^19 - 2807/1081a^17 + 62985/1081a^15 - 61693/1081a^13 - 4538/23a^11 + 16997/47a^9 - 35348/1081a^7 - 266078/1081a^5 + 137040/1081a^3 - 9633/1081a, 686/1081a^21 - 4478/1081a^19 - 6631/1081a^17 + 61742/1081a^15 - 900/47a^13 - 4654/23a^11 + 254835/1081a^9 + 45020/1081a^7 - 178499/1081a^5 + 78237/1081a^3 - 476/47a, 278/1081a^20 - 1717/1081a^18 - 3276/1081a^16 + 23791/1081a^14 - 306/1081a^12 - 1860/23a^10 + 3203/47a^8 + 37903/1081a^6 - 57764/1081a^4 + 15620/1081a^2 + 436/1081, 467/1081a^21 - 3015/1081a^19 - 4760/1081a^17 + 41872/1081a^15 - 10734/1081a^13 - 3236/23a^11 + 7067/47a^9 + 50482/1081a^7 - 125244/1081a^5 + 42406/1081a^3 - 96/1081a, 427/1081a^21 - 2560/1081a^19 - 5513/1081a^17 + 35617/1081a^15 + 6416/1081a^13 - 2856/23a^11 + 3753/47a^9 + 78939/1081a^7 - 72614/1081a^5 + 6846/1081a^3 + 3136/1081a, 220/1081a^20 - 1398/1081a^18 - 2321/1081a^16 + 19198/1081a^14 - 4320/1081a^12 - 1459/23a^10 + 75416/1081a^8 + 20653/1081a^6 - 59729/1081a^4 + 18155/1081a^2 + 695/1081, 468/1081a^20 - 2997/1081a^18 - 4975/1081a^16 + 41905/1081a^14 - 7532/1081a^12 - 3304/23a^10 + 149000/1081a^8 + 66680/1081a^6 - 115315/1081a^4 + 29853/1081a^2 - 64/1081, 213/1081a^21 - 1524/1081a^19 - 1333/1081a^17 + 21317/1081a^15 - 16488/1081a^13 - 1607/23a^11 + 118597/1081a^9 + 6672/1081a^7 - 90128/1081a^5 + 37218/1081a^3 + 518/1081a, a + 1, 241/1081a^21 - 1/1081a^20 - 1772/1081a^19 + 76/1081a^18 - 1102/1081a^17 - 255/1081a^16 + 24074/1081a^15 - 75/47a^14 - 24498/1081a^13 + 3049/1081a^12 - 1644/23a^11 + 229/23a^10 + 155352/1081a^9 - 9583/1081a^8 - 34412/1081a^7 - 18125/1081a^6 - 103892/1081a^5 + 8260/1081a^4 + 69160/1081a^3 + 7571/1081a^2 - 4649/1081a - 220/1081, 220/1081a^20 - 1398/1081a^18 - 2321/1081a^16 + 19198/1081a^14 - 4320/1081a^12 - 1459/23a^10 + 75416/1081a^8 + 20653/1081a^6 - 59729/1081a^4 + 18155/1081a^2 + a + 695/1081, 573/1081a^21 - 459/1081a^20 - 3551/1081a^19 + 2642/1081a^18 - 6635/1081a^17 + 6424/1081a^16 + 48942/1081a^15 - 36109/1081a^14 - 2249/1081a^13 - 13000/1081a^12 - 3776/23a^11 + 2830/23a^10 + 157170/1081a^9 - 73046/1081a^8 + 67527/1081a^7 - 73836/1081a^6 - 5104/47a^5 + 66214/1081a^4 + 38833/1081a^3 - 11982/1081a^2 - 5728/1081a - 541/1081, 695/1081a^21 + 319/1081a^20 - 4645/1081a^19 - 63/47a^18 - 6263/1081a^17 - 6733/1081a^16 + 65094/1081a^15 + 19739/1081a^14 - 27367/1081a^13 + 40830/1081a^12 - 5090/23a^11 - 1732/23a^10 + 287267/1081a^9 - 72095/1081a^8 + 3702/47a^7 + 108270/1081a^6 - 230937/1081a^5 + 40253/1081a^4 + 55641/1081a^3 - 40615/1081a^2 + 11900/1081a - 1213/1081, 733/1081a^21 + 41/1081a^20 - 4619/1081a^19 - 531/1081a^18 - 8276/1081a^17 + 867/1081a^16 + 64891/1081a^15 + 8638/1081a^14 - 5143/1081a^13 - 17473/1081a^12 - 5224/23a^11 - 786/23a^10 + 209104/1081a^9 + 72081/1081a^8 + 134367/1081a^7 + 19043/1081a^6 - 169757/1081a^5 - 49610/1081a^4 + 19910/1081a^3 + 2750/1081a^2 + 2729/1081a - 239/1081, 1119/1081a^21 - 964/1081a^20 - 6648/1081a^19 + 5349/1081a^18 - 14656/1081a^17 + 14607/1081a^16 + 91870/1081a^15 - 72796/1081a^14 + 18746/1081a^13 - 42773/1081a^12 - 7294/23a^11 + 250a^10 + 223342/1081a^9 - 93598/1081a^8 + 194263/1081a^7 - 173539/1081a^6 - 192761/1081a^5 + 93759/1081a^4 + 24006/1081a^3 - 4510/1081a^2 + 5023/1081a - 204/1081 (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:	$ 1121614550.16 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 1121614550.16 \cdot 1}{2\cdot\sqrt{7198079267989980836471065337135104}}\cr\approx \mathstrut & 0.168789232782 \end{aligned}\] (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^22 - 7*x^20 - 7*x^18 + 97*x^16 - 67*x^14 - 338*x^12 + 512*x^10 + 14*x^8 - 362*x^6 + 166*x^4 - 9*x^2 - 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^22 - 7*x^20 - 7*x^18 + 97*x^16 - 67*x^14 - 338*x^12 + 512*x^10 + 14*x^8 - 362*x^6 + 166*x^4 - 9*x^2 - 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^22 - 7*x^20 - 7*x^18 + 97*x^16 - 67*x^14 - 338*x^12 + 512*x^10 + 14*x^8 - 362*x^6 + 166*x^4 - 9*x^2 - 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^22 - 7*x^20 - 7*x^18 + 97*x^16 - 67*x^14 - 338*x^12 + 512*x^10 + 14*x^8 - 362*x^6 + 166*x^4 - 9*x^2 - 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

$C_2^{10}:C_{11}$ (as 22T23):

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 solvable group of order 11264

The 104 conjugacy class representatives for $C_2^{10}:C_{11}$

Character table for $C_2^{10}:C_{11}$

Intermediate fields

$\Q(\zeta_{23})^+$

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 22 siblings:	data not computed
Degree 44 siblings:	data not computed
Minimal sibling:	22.10.7198079267989980836471065337135104.11

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.11.0.1}{11} }^{2}$	${\href{/padicField/5.11.0.1}{11} }^{2}$	${\href{/padicField/7.11.0.1}{11} }^{2}$	${\href{/padicField/11.11.0.1}{11} }^{2}$	${\href{/padicField/13.11.0.1}{11} }^{2}$	${\href{/padicField/17.11.0.1}{11} }^{2}$	${\href{/padicField/19.11.0.1}{11} }^{2}$	R	${\href{/padicField/29.11.0.1}{11} }^{2}$	${\href{/padicField/31.11.0.1}{11} }^{2}$	${\href{/padicField/37.11.0.1}{11} }^{2}$	${\href{/padicField/41.11.0.1}{11} }^{2}$	${\href{/padicField/43.11.0.1}{11} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{10}$	${\href{/padicField/53.11.0.1}{11} }^{2}$	${\href{/padicField/59.11.0.1}{11} }^{2}$

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
$2$ 2	2.11.2.22a49.2	$x^{22} + 2 x^{19} + 2 x^{18} + 2 x^{16} + 2 x^{14} + 2 x^{13} + 4 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 4 x^{7} + 4 x^{5} + x^{4} + 6 x^{3} + 4 x^{2} + 5$	$2$	$11$	$22$	not computed	not computed
$23$ 23	23.1.11.10a1.1	$x^{11} + 23$	$11$	$1$	$10$	$C_{11}$	$$[\ ]_{11}$$
$23$ 23	23.1.11.10a1.1	$x^{11} + 23$	$11$	$1$	$10$	$C_{11}$	$$[\ ]_{11}$$

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