LMFDB - Number field 22.2.716463968651672990739595264.1

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^22 + 4*x^20 + x^18 - 20*x^16 - 7*x^14 + 93*x^12 + 46*x^10 - 121*x^8 - 80*x^6 - 34*x^4 - 9*x^2 - 1)

gp:K = bnfinit(y^22 + 4*y^20 + y^18 - 20*y^16 - 7*y^14 + 93*y^12 + 46*y^10 - 121*y^8 - 80*y^6 - 34*y^4 - 9*y^2 - 1, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 4*x^20 + x^18 - 20*x^16 - 7*x^14 + 93*x^12 + 46*x^10 - 121*x^8 - 80*x^6 - 34*x^4 - 9*x^2 - 1);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 + 4*x^20 + x^18 - 20*x^16 - 7*x^14 + 93*x^12 + 46*x^10 - 121*x^8 - 80*x^6 - 34*x^4 - 9*x^2 - 1)

$ x^{22} + 4x^{20} + x^{18} - 20x^{16} - 7x^{14} + 93x^{12} + 46x^{10} - 121x^{8} - 80x^{6} - 34x^{4} - 9x^{2} - 1 $ x^22 + 4*x^20 + x^18 - 20*x^16 - 7*x^14 + 93*x^12 + 46*x^10 - 121*x^8 - 80*x^6 - 34*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:	$(2, 10)$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$716463968651672990739595264$ 716463968651672990739595264 $\medspace = 2^{22}\cdot 11^{4}\cdot 19^{4}\cdot 547^{4}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$16.62$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	not computed
Ramified primes:	$2$, $11$, $19$, $547$ 2, 11, 19, 547	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:automorphism_group(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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{431991212}a^{20}+\frac{19757011}{215995606}a^{18}-\frac{1}{4}a^{17}-\frac{12181834}{107997803}a^{16}-\frac{1}{4}a^{15}+\frac{19879178}{107997803}a^{14}+\frac{51748585}{215995606}a^{12}-\frac{1}{4}a^{11}-\frac{81565729}{215995606}a^{10}-\frac{1}{4}a^{9}-\frac{17188587}{215995606}a^{8}+\frac{559025}{107997803}a^{6}+\frac{1}{4}a^{5}-\frac{134894503}{431991212}a^{4}-\frac{1}{4}a^{3}-\frac{7104976}{107997803}a^{2}+\frac{1}{4}a-\frac{115094833}{431991212}$, $\frac{1}{431991212}a^{21}+\frac{19757011}{215995606}a^{19}-\frac{1}{4}a^{18}-\frac{12181834}{107997803}a^{17}-\frac{1}{4}a^{16}+\frac{19879178}{107997803}a^{15}+\frac{51748585}{215995606}a^{13}-\frac{1}{4}a^{12}-\frac{81565729}{215995606}a^{11}-\frac{1}{4}a^{10}-\frac{17188587}{215995606}a^{9}+\frac{559025}{107997803}a^{7}+\frac{1}{4}a^{6}-\frac{134894503}{431991212}a^{5}-\frac{1}{4}a^{4}-\frac{7104976}{107997803}a^{3}+\frac{1}{4}a^{2}-\frac{115094833}{431991212}a$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, 1/2*a^13 - 1/2*a^12 - 1/2*a^10 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2, 1/2*a^14 - 1/2*a^12 - 1/2*a^11 - 1/2*a^10 - 1/2*a^6 - 1/2*a^3 - 1/2*a - 1/2, 1/2*a^15 - 1/2*a^11 - 1/2*a^10 - 1/2*a^7 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^16 - 1/2*a^12 - 1/2*a^11 - 1/2*a^8 - 1/2*a^6 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^17 - 1/2*a^10 - 1/2*a^9 - 1/2*a^7 - 1/2*a^2 - 1/2, 1/2*a^18 - 1/2*a^11 - 1/2*a^10 - 1/2*a^8 - 1/2*a^3 - 1/2*a, 1/4*a^19 - 1/4*a^18 - 1/4*a^17 - 1/4*a^14 - 1/4*a^13 + 1/4*a^12 - 1/4*a^11 - 1/2*a^10 - 1/2*a^9 + 1/4*a^8 + 1/4*a^7 - 1/4*a^6 - 1/4*a^5 - 1/2*a^4 - 1/4*a^3 - 1/2*a^2 - 1/2*a + 1/4, 1/431991212*a^20 + 19757011/215995606*a^18 - 1/4*a^17 - 12181834/107997803*a^16 - 1/4*a^15 + 19879178/107997803*a^14 + 51748585/215995606*a^12 - 1/4*a^11 - 81565729/215995606*a^10 - 1/4*a^9 - 17188587/215995606*a^8 + 559025/107997803*a^6 + 1/4*a^5 - 134894503/431991212*a^4 - 1/4*a^3 - 7104976/107997803*a^2 + 1/4*a - 115094833/431991212, 1/431991212*a^21 + 19757011/215995606*a^19 - 1/4*a^18 - 12181834/107997803*a^17 - 1/4*a^16 + 19879178/107997803*a^15 + 51748585/215995606*a^13 - 1/4*a^12 - 81565729/215995606*a^11 - 1/4*a^10 - 17188587/215995606*a^9 + 559025/107997803*a^7 + 1/4*a^6 - 134894503/431991212*a^5 - 1/4*a^4 - 7104976/107997803*a^3 + 1/4*a^2 - 115094833/431991212*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:	$11$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity sage:UK.torsion_generator() gp:K.tu[2] magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar:torsion_units_generator(OK)
Fundamental units:	$\frac{59479491}{215995606}a^{20}+\frac{119777647}{107997803}a^{18}+\frac{32631081}{107997803}a^{16}-\frac{594831634}{107997803}a^{14}-\frac{224868541}{107997803}a^{12}+\frac{2762127436}{107997803}a^{10}+\frac{1436154523}{107997803}a^{8}-\frac{3570183638}{107997803}a^{6}-\frac{4974261389}{215995606}a^{4}-\frac{1157994389}{107997803}a^{2}-\frac{499348735}{215995606}$, $\frac{17813216}{107997803}a^{21}-\frac{1950579}{431991212}a^{20}+\frac{139564123}{215995606}a^{19}+\frac{634871}{107997803}a^{18}+\frac{48758455}{431991212}a^{17}+\frac{13929455}{215995606}a^{16}-\frac{1422835025}{431991212}a^{15}+\frac{4074737}{215995606}a^{14}-\frac{90677920}{107997803}a^{13}-\frac{48297890}{107997803}a^{12}+\frac{6650233363}{431991212}a^{11}-\frac{21312055}{215995606}a^{10}+\frac{2630593413}{431991212}a^{9}+\frac{426887341}{215995606}a^{8}-\frac{2209352132}{107997803}a^{7}-\frac{76606387}{107997803}a^{6}-\frac{4529755251}{431991212}a^{5}-\frac{1302505903}{431991212}a^{4}-\frac{1977143779}{431991212}a^{3}+\frac{106908932}{107997803}a^{2}-\frac{1172497097}{431991212}a+\frac{267289633}{431991212}$, $\frac{132360061}{215995606}a^{20}+\frac{250891423}{107997803}a^{18}+\frac{15685202}{107997803}a^{16}-\frac{1321141584}{107997803}a^{14}-\frac{190411176}{107997803}a^{12}+\frac{6161915138}{107997803}a^{10}+\frac{1778123662}{107997803}a^{8}-\frac{8219168561}{107997803}a^{6}-\frac{7222648443}{215995606}a^{4}-\frac{1611233373}{107997803}a^{2}-\frac{701178693}{215995606}$, $a$, $\frac{634026491}{431991212}a^{21}-\frac{36441761}{431991212}a^{20}+\frac{2401960403}{431991212}a^{19}-\frac{33749157}{107997803}a^{18}+\frac{55358735}{215995606}a^{17}+\frac{11957017}{431991212}a^{16}-\frac{12759575615}{431991212}a^{15}+\frac{753254899}{431991212}a^{14}-\frac{1733420379}{431991212}a^{13}+\frac{5152740}{107997803}a^{12}+\frac{14912243825}{107997803}a^{11}-\frac{3560854187}{431991212}a^{10}+\frac{16580984871}{431991212}a^{9}-\frac{576223331}{431991212}a^{8}-\frac{81676584421}{431991212}a^{7}+\frac{2672210017}{215995606}a^{6}-\frac{33837561839}{431991212}a^{5}+\frac{594264513}{215995606}a^{4}-\frac{6127745411}{215995606}a^{3}-\frac{258114897}{431991212}a^{2}-\frac{459228158}{107997803}a+\frac{25086092}{107997803}$, $\frac{801690673}{431991212}a^{21}-\frac{28503477}{215995606}a^{20}+\frac{747069777}{107997803}a^{19}-\frac{246632021}{431991212}a^{18}-\frac{910444}{107997803}a^{17}-\frac{125360657}{431991212}a^{16}-\frac{7997806645}{215995606}a^{15}+\frac{572900283}{215995606}a^{14}-\frac{619293787}{215995606}a^{13}+\frac{777737887}{431991212}a^{12}+\frac{18678227455}{107997803}a^{11}-\frac{5299350857}{431991212}a^{10}+\frac{4107340897}{107997803}a^{9}-\frac{1103020000}{107997803}a^{8}-\frac{50319976517}{215995606}a^{7}+\frac{6600837179}{431991212}a^{6}-\frac{36132604921}{431991212}a^{5}+\frac{7151416495}{431991212}a^{4}-\frac{4616616930}{107997803}a^{3}+\frac{2578318901}{431991212}a^{2}-\frac{3120969985}{431991212}a+\frac{318554237}{215995606}$, $\frac{351040197}{431991212}a^{21}-\frac{36441761}{431991212}a^{20}+\frac{1313162497}{431991212}a^{19}-\frac{33749157}{107997803}a^{18}-\frac{252398}{107997803}a^{17}+\frac{11957017}{431991212}a^{16}-\frac{7072807835}{431991212}a^{15}+\frac{753254899}{431991212}a^{14}-\frac{642910429}{431991212}a^{13}+\frac{5152740}{107997803}a^{12}+\frac{16536812159}{215995606}a^{11}-\frac{3560854187}{431991212}a^{10}+\frac{7711858587}{431991212}a^{9}-\frac{576223331}{431991212}a^{8}-\frac{45778090687}{431991212}a^{7}+\frac{2672210017}{215995606}a^{6}-\frac{16985367579}{431991212}a^{5}+\frac{594264513}{215995606}a^{4}-\frac{1320840250}{107997803}a^{3}-\frac{258114897}{431991212}a^{2}-\frac{553347583}{215995606}a+\frac{25086092}{107997803}$, $\frac{93195361}{107997803}a^{21}-\frac{152291021}{215995606}a^{20}+\frac{1349780799}{431991212}a^{19}-\frac{1137771935}{431991212}a^{18}-\frac{155874005}{431991212}a^{17}-\frac{937443}{107997803}a^{16}-\frac{1865935893}{107997803}a^{15}+\frac{6091143021}{431991212}a^{14}+\frac{204812537}{431991212}a^{13}+\frac{519859115}{431991212}a^{12}+\frac{34917577773}{431991212}a^{11}-\frac{7112540512}{107997803}a^{10}+\frac{996105478}{107997803}a^{9}-\frac{6471854929}{431991212}a^{8}-\frac{48194637315}{431991212}a^{7}+\frac{38483969429}{431991212}a^{6}-\frac{12103586863}{431991212}a^{5}+\frac{3523951453}{107997803}a^{4}-\frac{6033547311}{431991212}a^{3}+\frac{1729650170}{107997803}a^{2}-\frac{275896905}{215995606}a+\frac{995525047}{431991212}$, $\frac{18202483}{107997803}a^{21}+\frac{234854093}{431991212}a^{20}+\frac{190978927}{215995606}a^{19}+\frac{211320155}{107997803}a^{18}+\frac{401680409}{431991212}a^{17}-\frac{59085827}{215995606}a^{16}-\frac{1496385929}{431991212}a^{15}-\frac{2351355807}{215995606}a^{14}-\frac{1168427557}{215995606}a^{13}+\frac{118228813}{215995606}a^{12}+\frac{6821923311}{431991212}a^{11}+\frac{11004044703}{215995606}a^{10}+\frac{11919926347}{431991212}a^{9}+\frac{989291329}{215995606}a^{8}-\frac{3923422209}{215995606}a^{7}-\frac{7615686486}{107997803}a^{6}-\frac{17876324265}{431991212}a^{5}-\frac{6749760449}{431991212}a^{4}-\frac{5488366909}{431991212}a^{3}-\frac{1842507285}{215995606}a^{2}-\frac{1827593389}{431991212}a-\frac{486676391}{431991212}$, $\frac{93195361}{107997803}a^{21}+\frac{152291021}{215995606}a^{20}+\frac{1349780799}{431991212}a^{19}+\frac{1137771935}{431991212}a^{18}-\frac{155874005}{431991212}a^{17}+\frac{937443}{107997803}a^{16}-\frac{1865935893}{107997803}a^{15}-\frac{6091143021}{431991212}a^{14}+\frac{204812537}{431991212}a^{13}-\frac{519859115}{431991212}a^{12}+\frac{34917577773}{431991212}a^{11}+\frac{7112540512}{107997803}a^{10}+\frac{996105478}{107997803}a^{9}+\frac{6471854929}{431991212}a^{8}-\frac{48194637315}{431991212}a^{7}-\frac{38483969429}{431991212}a^{6}-\frac{12103586863}{431991212}a^{5}-\frac{3523951453}{107997803}a^{4}-\frac{6033547311}{431991212}a^{3}-\frac{1729650170}{107997803}a^{2}-\frac{275896905}{215995606}a-\frac{995525047}{431991212}$, $\frac{354180669}{215995606}a^{21}+\frac{312134259}{215995606}a^{20}+\frac{692638797}{107997803}a^{19}+\frac{1157187139}{215995606}a^{18}+\frac{109808727}{107997803}a^{17}-\frac{29767927}{215995606}a^{16}-\frac{7145182581}{215995606}a^{15}-\frac{6242455379}{215995606}a^{14}-\frac{919167608}{107997803}a^{13}-\frac{172184946}{107997803}a^{12}+\frac{16669764833}{107997803}a^{11}+\frac{29197135717}{215995606}a^{10}+\frac{6659558090}{107997803}a^{9}+\frac{2874222336}{107997803}a^{8}-\frac{45169806941}{215995606}a^{7}-\frac{19887983336}{107997803}a^{6}-\frac{12224398998}{107997803}a^{5}-\frac{6559539339}{107997803}a^{4}-\frac{4121853850}{107997803}a^{3}-\frac{3115514036}{107997803}a^{2}-\frac{2164077231}{215995606}a-\frac{1076489687}{215995606}$ 59479491/215995606a^20 + 119777647/107997803a^18 + 32631081/107997803a^16 - 594831634/107997803a^14 - 224868541/107997803a^12 + 2762127436/107997803a^10 + 1436154523/107997803a^8 - 3570183638/107997803a^6 - 4974261389/215995606a^4 - 1157994389/107997803a^2 - 499348735/215995606, 17813216/107997803a^21 - 1950579/431991212a^20 + 139564123/215995606a^19 + 634871/107997803a^18 + 48758455/431991212a^17 + 13929455/215995606a^16 - 1422835025/431991212a^15 + 4074737/215995606a^14 - 90677920/107997803a^13 - 48297890/107997803a^12 + 6650233363/431991212a^11 - 21312055/215995606a^10 + 2630593413/431991212a^9 + 426887341/215995606a^8 - 2209352132/107997803a^7 - 76606387/107997803a^6 - 4529755251/431991212a^5 - 1302505903/431991212a^4 - 1977143779/431991212a^3 + 106908932/107997803a^2 - 1172497097/431991212a + 267289633/431991212, 132360061/215995606a^20 + 250891423/107997803a^18 + 15685202/107997803a^16 - 1321141584/107997803a^14 - 190411176/107997803a^12 + 6161915138/107997803a^10 + 1778123662/107997803a^8 - 8219168561/107997803a^6 - 7222648443/215995606a^4 - 1611233373/107997803a^2 - 701178693/215995606, a, 634026491/431991212a^21 - 36441761/431991212a^20 + 2401960403/431991212a^19 - 33749157/107997803a^18 + 55358735/215995606a^17 + 11957017/431991212a^16 - 12759575615/431991212a^15 + 753254899/431991212a^14 - 1733420379/431991212a^13 + 5152740/107997803a^12 + 14912243825/107997803a^11 - 3560854187/431991212a^10 + 16580984871/431991212a^9 - 576223331/431991212a^8 - 81676584421/431991212a^7 + 2672210017/215995606a^6 - 33837561839/431991212a^5 + 594264513/215995606a^4 - 6127745411/215995606a^3 - 258114897/431991212a^2 - 459228158/107997803a + 25086092/107997803, 801690673/431991212a^21 - 28503477/215995606a^20 + 747069777/107997803a^19 - 246632021/431991212a^18 - 910444/107997803a^17 - 125360657/431991212a^16 - 7997806645/215995606a^15 + 572900283/215995606a^14 - 619293787/215995606a^13 + 777737887/431991212a^12 + 18678227455/107997803a^11 - 5299350857/431991212a^10 + 4107340897/107997803a^9 - 1103020000/107997803a^8 - 50319976517/215995606a^7 + 6600837179/431991212a^6 - 36132604921/431991212a^5 + 7151416495/431991212a^4 - 4616616930/107997803a^3 + 2578318901/431991212a^2 - 3120969985/431991212a + 318554237/215995606, 351040197/431991212a^21 - 36441761/431991212a^20 + 1313162497/431991212a^19 - 33749157/107997803a^18 - 252398/107997803a^17 + 11957017/431991212a^16 - 7072807835/431991212a^15 + 753254899/431991212a^14 - 642910429/431991212a^13 + 5152740/107997803a^12 + 16536812159/215995606a^11 - 3560854187/431991212a^10 + 7711858587/431991212a^9 - 576223331/431991212a^8 - 45778090687/431991212a^7 + 2672210017/215995606a^6 - 16985367579/431991212a^5 + 594264513/215995606a^4 - 1320840250/107997803a^3 - 258114897/431991212a^2 - 553347583/215995606a + 25086092/107997803, 93195361/107997803a^21 - 152291021/215995606a^20 + 1349780799/431991212a^19 - 1137771935/431991212a^18 - 155874005/431991212a^17 - 937443/107997803a^16 - 1865935893/107997803a^15 + 6091143021/431991212a^14 + 204812537/431991212a^13 + 519859115/431991212a^12 + 34917577773/431991212a^11 - 7112540512/107997803a^10 + 996105478/107997803a^9 - 6471854929/431991212a^8 - 48194637315/431991212a^7 + 38483969429/431991212a^6 - 12103586863/431991212a^5 + 3523951453/107997803a^4 - 6033547311/431991212a^3 + 1729650170/107997803a^2 - 275896905/215995606a + 995525047/431991212, 18202483/107997803a^21 + 234854093/431991212a^20 + 190978927/215995606a^19 + 211320155/107997803a^18 + 401680409/431991212a^17 - 59085827/215995606a^16 - 1496385929/431991212a^15 - 2351355807/215995606a^14 - 1168427557/215995606a^13 + 118228813/215995606a^12 + 6821923311/431991212a^11 + 11004044703/215995606a^10 + 11919926347/431991212a^9 + 989291329/215995606a^8 - 3923422209/215995606a^7 - 7615686486/107997803a^6 - 17876324265/431991212a^5 - 6749760449/431991212a^4 - 5488366909/431991212a^3 - 1842507285/215995606a^2 - 1827593389/431991212a - 486676391/431991212, 93195361/107997803a^21 + 152291021/215995606a^20 + 1349780799/431991212a^19 + 1137771935/431991212a^18 - 155874005/431991212a^17 + 937443/107997803a^16 - 1865935893/107997803a^15 - 6091143021/431991212a^14 + 204812537/431991212a^13 - 519859115/431991212a^12 + 34917577773/431991212a^11 + 7112540512/107997803a^10 + 996105478/107997803a^9 + 6471854929/431991212a^8 - 48194637315/431991212a^7 - 38483969429/431991212a^6 - 12103586863/431991212a^5 - 3523951453/107997803a^4 - 6033547311/431991212a^3 - 1729650170/107997803a^2 - 275896905/215995606a - 995525047/431991212, 354180669/215995606a^21 + 312134259/215995606a^20 + 692638797/107997803a^19 + 1157187139/215995606a^18 + 109808727/107997803a^17 - 29767927/215995606a^16 - 7145182581/215995606a^15 - 6242455379/215995606a^14 - 919167608/107997803a^13 - 172184946/107997803a^12 + 16669764833/107997803a^11 + 29197135717/215995606a^10 + 6659558090/107997803a^9 + 2874222336/107997803a^8 - 45169806941/215995606a^7 - 19887983336/107997803a^6 - 12224398998/107997803a^5 - 6559539339/107997803a^4 - 4121853850/107997803a^3 - 3115514036/107997803a^2 - 2164077231/215995606a - 1076489687/215995606 (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:	$ 38833.928533 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 38833.928533 \cdot 1}{2\cdot\sqrt{716463968651672990739595264}}\cr\approx \mathstrut & 0.27825490525 \end{aligned}\] (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 + 4*x^20 + x^18 - 20*x^16 - 7*x^14 + 93*x^12 + 46*x^10 - 121*x^8 - 80*x^6 - 34*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 + 4*x^20 + x^18 - 20*x^16 - 7*x^14 + 93*x^12 + 46*x^10 - 121*x^8 - 80*x^6 - 34*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 + 4*x^20 + x^18 - 20*x^16 - 7*x^14 + 93*x^12 + 46*x^10 - 121*x^8 - 80*x^6 - 34*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 = polynomial_ring(QQ); K, a = number_field(x^22 + 4*x^20 + x^18 - 20*x^16 - 7*x^14 + 93*x^12 + 46*x^10 - 121*x^8 - 80*x^6 - 34*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

$C_2^{10}.A_{11}$ (as 22T49):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A non-solvable group of order 20437401600

The 200 conjugacy class representatives for $C_2^{10}.A_{11}$

Character table for $C_2^{10}.A_{11}$

Intermediate fields

11.3.836463893056.1

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(L)]

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

oscar:subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.11.0.1}{11} }^{2}$	${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$	${\href{/padicField/7.11.0.1}{11} }^{2}$	R	${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$	${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$	R	${\href{/padicField/23.11.0.1}{11} }^{2}$	${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$	${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$	${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.11.0.1}{11} }^{2}$	${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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	2.2.2.4a1.1	$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$	$2$	$2$	$4$	$C_2^2$	$$[2]^{2}$$
$2$ 2	2.3.6.18a1.2	$x^{18} + 6 x^{16} + 6 x^{15} + 15 x^{14} + 30 x^{13} + 35 x^{12} + 60 x^{11} + 75 x^{10} + 80 x^{9} + 96 x^{8} + 90 x^{7} + 78 x^{6} + 66 x^{5} + 49 x^{4} + 32 x^{3} + 17 x^{2} + 12 x + 7$	$6$	$3$	$18$	18T33	$$[\frac{4}{3}, \frac{4}{3}]_{3}^{6}$$
$11$ 11	11.1.3.2a1.1	$x^{3} + 11$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
	11.3.1.0a1.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	11.3.1.0a1.1	$x^{3} + 2 x + 9$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	11.1.3.2a1.1	$x^{3} + 11$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
	11.4.1.0a1.1	$x^{4} + 8 x^{2} + 10 x + 2$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	11.6.1.0a1.1	$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
$19$ 19	19.2.2.2a1.2	$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	19.2.2.2a1.2	$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	19.7.1.0a1.1	$x^{7} + 6 x + 17$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
	19.7.1.0a1.1	$x^{7} + 6 x + 17$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
$547$ 547	$\Q_{547}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{547}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$
		Deg $10$	$1$	$10$	$0$	$C_{10}$	$$[\ ]^{10}$$

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