LMFDB - Number field 22.2.322170665670052381217287483857714996337890625.1

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129)

gp:K = bnfinit(y^22 - 22*y^20 + 209*y^18 - 1122*y^16 + 3740*y^14 - 8008*y^12 - 121*y^11 + 11011*y^10 + 1331*y^9 - 9438*y^8 - 5324*y^7 + 4719*y^6 + 9317*y^5 - 1210*y^4 - 6655*y^3 + 121*y^2 + 1331*y + 1129, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129)

$ x^{22} - 22 x^{20} + 209 x^{18} - 1122 x^{16} + 3740 x^{14} - 8008 x^{12} - 121 x^{11} + 11011 x^{10} + \cdots + 1129 $ x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129

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:	$322170665670052381217287483857714996337890625$ 322170665670052381217287483857714996337890625 $\medspace = 5^{16}\cdot 11^{32}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$105.46$	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:	$5^{3/4}11^{84/55}\approx 130.2321317975223$
Ramified primes:	$5$, $11$ 5, 11	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:	$\Q$
$\Aut(K/\Q)$:	$C_1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{45}a^{11}+\frac{4}{45}a^{9}-\frac{1}{45}a^{7}+\frac{13}{45}a^{5}+\frac{2}{9}a^{3}-\frac{1}{3}a^{2}-\frac{11}{45}a-\frac{8}{45}$, $\frac{1}{45}a^{12}+\frac{4}{45}a^{10}-\frac{1}{45}a^{8}-\frac{2}{45}a^{6}+\frac{1}{3}a^{5}+\frac{2}{9}a^{4}-\frac{11}{45}a^{2}+\frac{22}{45}a-\frac{1}{3}$, $\frac{1}{45}a^{13}-\frac{2}{45}a^{9}+\frac{2}{45}a^{7}+\frac{2}{5}a^{5}+\frac{1}{5}a^{3}+\frac{22}{45}a^{2}+\frac{14}{45}a+\frac{2}{45}$, $\frac{1}{45}a^{14}-\frac{2}{45}a^{10}+\frac{2}{45}a^{8}+\frac{1}{15}a^{6}+\frac{1}{3}a^{5}+\frac{1}{5}a^{4}-\frac{8}{45}a^{3}+\frac{14}{45}a^{2}-\frac{13}{45}a-\frac{1}{3}$, $\frac{1}{45}a^{15}-\frac{1}{9}a^{9}+\frac{1}{45}a^{7}+\frac{1}{9}a^{5}-\frac{8}{45}a^{4}+\frac{4}{45}a^{3}+\frac{17}{45}a^{2}-\frac{7}{45}a-\frac{16}{45}$, $\frac{1}{45}a^{16}-\frac{1}{9}a^{10}+\frac{1}{45}a^{8}+\frac{1}{9}a^{6}-\frac{8}{45}a^{5}+\frac{4}{45}a^{4}+\frac{17}{45}a^{3}-\frac{7}{45}a^{2}-\frac{16}{45}a$, $\frac{1}{675}a^{17}-\frac{4}{675}a^{16}-\frac{2}{675}a^{15}+\frac{4}{675}a^{14}-\frac{1}{675}a^{13}+\frac{4}{675}a^{12}-\frac{7}{675}a^{11}+\frac{58}{675}a^{10}-\frac{17}{135}a^{9}-\frac{2}{45}a^{8}+\frac{2}{75}a^{7}-\frac{28}{225}a^{6}-\frac{22}{75}a^{5}+\frac{37}{75}a^{4}-\frac{103}{225}a^{3}+\frac{43}{675}a^{2}+\frac{182}{675}a-\frac{14}{675}$, $\frac{1}{675}a^{18}-\frac{1}{225}a^{16}-\frac{4}{675}a^{15}-\frac{2}{225}a^{12}+\frac{14}{225}a^{10}-\frac{8}{135}a^{9}-\frac{34}{225}a^{8}+\frac{2}{75}a^{7}-\frac{8}{225}a^{6}+\frac{52}{225}a^{5}+\frac{41}{225}a^{4}-\frac{218}{675}a^{3}+\frac{68}{225}a^{2}-\frac{77}{225}a-\frac{41}{675}$, $\frac{1}{675}a^{19}-\frac{1}{675}a^{16}-\frac{2}{225}a^{15}-\frac{1}{225}a^{14}+\frac{2}{225}a^{13}-\frac{1}{225}a^{12}+\frac{2}{225}a^{11}+\frac{29}{675}a^{10}+\frac{1}{225}a^{9}-\frac{8}{75}a^{8}+\frac{1}{9}a^{7}-\frac{4}{75}a^{6}-\frac{97}{225}a^{5}-\frac{119}{675}a^{4}+\frac{104}{225}a^{3}-\frac{49}{225}a^{2}-\frac{34}{135}a+\frac{4}{25}$, $\frac{1}{675}a^{20}+\frac{1}{135}a^{16}-\frac{1}{135}a^{15}-\frac{1}{135}a^{14}-\frac{4}{675}a^{13}-\frac{1}{135}a^{12}+\frac{7}{675}a^{11}-\frac{44}{675}a^{10}+\frac{8}{675}a^{9}+\frac{1}{15}a^{8}-\frac{1}{225}a^{7}-\frac{2}{15}a^{6}+\frac{43}{675}a^{5}-\frac{17}{45}a^{4}+\frac{73}{225}a^{3}-\frac{277}{675}a^{2}+\frac{61}{135}a-\frac{119}{675}$, $\frac{1}{675}a^{21}+\frac{1}{135}a^{15}+\frac{2}{225}a^{14}+\frac{2}{675}a^{12}+\frac{2}{225}a^{11}+\frac{2}{75}a^{10}+\frac{16}{135}a^{9}-\frac{16}{225}a^{8}+\frac{2}{45}a^{7}-\frac{2}{675}a^{6}-\frac{4}{9}a^{5}+\frac{88}{225}a^{4}+\frac{248}{675}a^{3}-\frac{1}{3}a^{2}+\frac{112}{225}a-\frac{2}{27}$ 1, a, a^2, a^3, a^4, a^5, 1/3*a^6 - 1/3*a^5 - 1/3*a^3 + 1/3*a + 1/3, 1/3*a^7 - 1/3*a^5 - 1/3*a^4 - 1/3*a^3 + 1/3*a^2 - 1/3*a + 1/3, 1/3*a^8 + 1/3*a^5 - 1/3*a^4 - 1/3*a^2 - 1/3*a + 1/3, 1/3*a^9 - 1/3*a^2 - 1/3, 1/3*a^10 - 1/3*a^3 - 1/3*a, 1/45*a^11 + 4/45*a^9 - 1/45*a^7 + 13/45*a^5 + 2/9*a^3 - 1/3*a^2 - 11/45*a - 8/45, 1/45*a^12 + 4/45*a^10 - 1/45*a^8 - 2/45*a^6 + 1/3*a^5 + 2/9*a^4 - 11/45*a^2 + 22/45*a - 1/3, 1/45*a^13 - 2/45*a^9 + 2/45*a^7 + 2/5*a^5 + 1/5*a^3 + 22/45*a^2 + 14/45*a + 2/45, 1/45*a^14 - 2/45*a^10 + 2/45*a^8 + 1/15*a^6 + 1/3*a^5 + 1/5*a^4 - 8/45*a^3 + 14/45*a^2 - 13/45*a - 1/3, 1/45*a^15 - 1/9*a^9 + 1/45*a^7 + 1/9*a^5 - 8/45*a^4 + 4/45*a^3 + 17/45*a^2 - 7/45*a - 16/45, 1/45*a^16 - 1/9*a^10 + 1/45*a^8 + 1/9*a^6 - 8/45*a^5 + 4/45*a^4 + 17/45*a^3 - 7/45*a^2 - 16/45*a, 1/675*a^17 - 4/675*a^16 - 2/675*a^15 + 4/675*a^14 - 1/675*a^13 + 4/675*a^12 - 7/675*a^11 + 58/675*a^10 - 17/135*a^9 - 2/45*a^8 + 2/75*a^7 - 28/225*a^6 - 22/75*a^5 + 37/75*a^4 - 103/225*a^3 + 43/675*a^2 + 182/675*a - 14/675, 1/675*a^18 - 1/225*a^16 - 4/675*a^15 - 2/225*a^12 + 14/225*a^10 - 8/135*a^9 - 34/225*a^8 + 2/75*a^7 - 8/225*a^6 + 52/225*a^5 + 41/225*a^4 - 218/675*a^3 + 68/225*a^2 - 77/225*a - 41/675, 1/675*a^19 - 1/675*a^16 - 2/225*a^15 - 1/225*a^14 + 2/225*a^13 - 1/225*a^12 + 2/225*a^11 + 29/675*a^10 + 1/225*a^9 - 8/75*a^8 + 1/9*a^7 - 4/75*a^6 - 97/225*a^5 - 119/675*a^4 + 104/225*a^3 - 49/225*a^2 - 34/135*a + 4/25, 1/675*a^20 + 1/135*a^16 - 1/135*a^15 - 1/135*a^14 - 4/675*a^13 - 1/135*a^12 + 7/675*a^11 - 44/675*a^10 + 8/675*a^9 + 1/15*a^8 - 1/225*a^7 - 2/15*a^6 + 43/675*a^5 - 17/45*a^4 + 73/225*a^3 - 277/675*a^2 + 61/135*a - 119/675, 1/675*a^21 + 1/135*a^15 + 2/225*a^14 + 2/675*a^12 + 2/225*a^11 + 2/75*a^10 + 16/135*a^9 - 16/225*a^8 + 2/45*a^7 - 2/675*a^6 - 4/9*a^5 + 88/225*a^4 + 248/675*a^3 - 1/3*a^2 + 112/225*a - 2/27

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:	$\frac{1}{45}a^{11}-\frac{11}{45}a^{9}+\frac{44}{45}a^{7}-\frac{77}{45}a^{5}+\frac{11}{9}a^{3}-\frac{11}{45}a-\frac{38}{45}$, $\frac{4}{675}a^{21}-\frac{13}{675}a^{20}-\frac{106}{675}a^{19}+\frac{49}{135}a^{18}+\frac{1181}{675}a^{17}-\frac{626}{225}a^{16}-\frac{7256}{675}a^{15}+\frac{7456}{675}a^{14}+\frac{1079}{27}a^{13}-\frac{70}{3}a^{12}-\frac{463}{5}a^{11}+\frac{13553}{675}a^{10}+\frac{90368}{675}a^{9}+\frac{953}{45}a^{8}-\frac{27431}{225}a^{7}-\frac{61616}{675}a^{6}+\frac{45689}{675}a^{5}+\frac{92033}{675}a^{4}+\frac{7234}{675}a^{3}-\frac{6106}{75}a^{2}-\frac{36794}{675}a-\frac{2317}{135}$, $\frac{8}{225}a^{21}+\frac{22}{675}a^{20}-\frac{32}{45}a^{19}-\frac{391}{675}a^{18}+\frac{154}{25}a^{17}+\frac{2951}{675}a^{16}-\frac{20407}{675}a^{15}-\frac{12299}{675}a^{14}+\frac{62189}{675}a^{13}+\frac{30676}{675}a^{12}-\frac{120758}{675}a^{11}-\frac{48299}{675}a^{10}+\frac{47902}{225}a^{9}+\frac{1309}{15}a^{8}-\frac{5689}{45}a^{7}-\frac{8974}{75}a^{6}+\frac{571}{675}a^{5}+\frac{3656}{25}a^{4}+\frac{8336}{675}a^{3}-\frac{62719}{675}a^{2}+\frac{13931}{675}a+\frac{244}{75}$, $\frac{2}{675}a^{21}-\frac{1}{675}a^{20}-\frac{16}{225}a^{19}+\frac{8}{225}a^{18}+\frac{493}{675}a^{17}-\frac{77}{225}a^{16}-\frac{2819}{675}a^{15}+\frac{361}{225}a^{14}+\frac{364}{25}a^{13}-\frac{2099}{675}a^{12}-\frac{21869}{675}a^{11}-\frac{44}{15}a^{10}+\frac{33118}{675}a^{9}+\frac{5984}{225}a^{8}-\frac{4339}{75}a^{7}-\frac{36664}{675}a^{6}+\frac{7531}{135}a^{5}+\frac{12697}{225}a^{4}-\frac{18143}{675}a^{3}-\frac{21052}{675}a^{2}-\frac{779}{75}a+\frac{1829}{675}$, $\frac{4}{675}a^{21}+\frac{19}{675}a^{20}-\frac{1}{9}a^{19}-\frac{23}{45}a^{18}+\frac{577}{675}a^{17}+\frac{2602}{675}a^{16}-\frac{2399}{675}a^{15}-\frac{10633}{675}a^{14}+\frac{6067}{675}a^{13}+\frac{25906}{675}a^{12}-\frac{3334}{225}a^{11}-\frac{39238}{675}a^{10}+\frac{2969}{225}a^{9}+\frac{12481}{225}a^{8}+\frac{736}{75}a^{7}-\frac{21971}{675}a^{6}-\frac{28589}{675}a^{5}+\frac{2369}{225}a^{4}+\frac{5029}{135}a^{3}-\frac{208}{75}a^{2}-\frac{5492}{675}a-\frac{1523}{225}$, $\frac{13}{135}a^{21}-\frac{62}{675}a^{20}-\frac{1334}{675}a^{19}+\frac{157}{75}a^{18}+\frac{11657}{675}a^{17}-\frac{13498}{675}a^{16}-\frac{56312}{675}a^{15}+\frac{4676}{45}a^{14}+\frac{54329}{225}a^{13}-\frac{215048}{675}a^{12}-\frac{285772}{675}a^{11}+\frac{386534}{675}a^{10}+\frac{300787}{675}a^{9}-\frac{114781}{225}a^{8}-\frac{84863}{225}a^{7}+\frac{1484}{675}a^{6}+\frac{75992}{135}a^{5}+\frac{192871}{675}a^{4}-\frac{384443}{675}a^{3}-\frac{901}{135}a^{2}+\frac{35296}{675}a+\frac{13762}{135}$, $\frac{1}{675}a^{21}-\frac{2}{675}a^{20}-\frac{26}{675}a^{19}+\frac{11}{135}a^{18}+\frac{278}{675}a^{17}-\frac{631}{675}a^{16}-\frac{319}{135}a^{15}+\frac{1337}{225}a^{14}+\frac{1753}{225}a^{13}-\frac{15688}{675}a^{12}-\frac{1871}{135}a^{11}+\frac{39311}{675}a^{10}+\frac{4681}{675}a^{9}-\frac{6949}{75}a^{8}+\frac{907}{45}a^{7}+\frac{58252}{675}a^{6}-\frac{24254}{675}a^{5}-\frac{25808}{675}a^{4}+\frac{2452}{225}a^{3}+\frac{1352}{135}a^{2}+\frac{6187}{675}a-\frac{6172}{675}$, $\frac{127}{675}a^{21}-\frac{16}{225}a^{20}-\frac{97}{25}a^{19}+\frac{964}{675}a^{18}+\frac{22906}{675}a^{17}-\frac{8047}{675}a^{16}-\frac{110419}{675}a^{15}+\frac{35728}{675}a^{14}+\frac{318827}{675}a^{13}-\frac{9911}{75}a^{12}-\frac{22402}{27}a^{11}+\frac{105583}{675}a^{10}+\frac{194978}{225}a^{9}+\frac{7636}{75}a^{8}-\frac{124682}{225}a^{7}-\frac{17783}{27}a^{6}+\frac{23594}{75}a^{5}+\frac{165853}{225}a^{4}-\frac{347}{5}a^{3}-\frac{71927}{675}a^{2}-\frac{12623}{135}a+\frac{1168}{75}$, $\frac{116}{675}a^{21}+\frac{61}{225}a^{20}-\frac{2222}{675}a^{19}-\frac{3533}{675}a^{18}+\frac{661}{25}a^{17}+\frac{28613}{675}a^{16}-\frac{579}{5}a^{15}-\frac{2807}{15}a^{14}+\frac{22616}{75}a^{13}+\frac{331264}{675}a^{12}-\frac{11952}{25}a^{11}-\frac{542557}{675}a^{10}+\frac{94696}{225}a^{9}+\frac{206359}{225}a^{8}+\frac{6871}{225}a^{7}-\frac{555331}{675}a^{6}-\frac{140341}{225}a^{5}+\frac{304504}{675}a^{4}+\frac{373952}{675}a^{3}-\frac{27011}{225}a^{2}-\frac{125402}{675}a-\frac{7982}{75}$, $\frac{146}{675}a^{21}-\frac{23}{135}a^{20}-\frac{1036}{225}a^{19}+\frac{821}{225}a^{18}+\frac{634}{15}a^{17}-\frac{22676}{675}a^{16}-\frac{49373}{225}a^{15}+\frac{4724}{27}a^{14}+\frac{480217}{675}a^{13}-\frac{129274}{225}a^{12}-\frac{1014607}{675}a^{11}+\frac{840212}{675}a^{10}+\frac{1423541}{675}a^{9}-\frac{43584}{25}a^{8}-\frac{456232}{225}a^{7}+\frac{953639}{675}a^{6}+\frac{979991}{675}a^{5}-\frac{8027}{15}a^{4}-\frac{484817}{675}a^{3}+\frac{3898}{675}a^{2}+\frac{133678}{675}a+\frac{60883}{675}$, $\frac{1298}{675}a^{21}-\frac{2227}{675}a^{20}-\frac{8039}{225}a^{19}+\frac{40619}{675}a^{18}+\frac{7579}{27}a^{17}-461a^{16}-\frac{821254}{675}a^{15}+\frac{48272}{25}a^{14}+\frac{715562}{225}a^{13}-\frac{3252722}{675}a^{12}-\frac{3467963}{675}a^{11}+\frac{1594744}{225}a^{10}+\frac{3666263}{675}a^{9}-\frac{1098026}{225}a^{8}-\frac{141561}{25}a^{7}-\frac{123119}{135}a^{6}+\frac{963442}{135}a^{5}+\frac{715229}{225}a^{4}-\frac{229016}{45}a^{3}-\frac{391201}{675}a^{2}+\frac{67823}{225}a+\frac{575068}{675}$ 1/45a^11 - 11/45a^9 + 44/45a^7 - 77/45a^5 + 11/9a^3 - 11/45a - 38/45, 4/675a^21 - 13/675a^20 - 106/675a^19 + 49/135a^18 + 1181/675a^17 - 626/225a^16 - 7256/675a^15 + 7456/675a^14 + 1079/27a^13 - 70/3a^12 - 463/5a^11 + 13553/675a^10 + 90368/675a^9 + 953/45a^8 - 27431/225a^7 - 61616/675a^6 + 45689/675a^5 + 92033/675a^4 + 7234/675a^3 - 6106/75a^2 - 36794/675a - 2317/135, 8/225a^21 + 22/675a^20 - 32/45a^19 - 391/675a^18 + 154/25a^17 + 2951/675a^16 - 20407/675a^15 - 12299/675a^14 + 62189/675a^13 + 30676/675a^12 - 120758/675a^11 - 48299/675a^10 + 47902/225a^9 + 1309/15a^8 - 5689/45a^7 - 8974/75a^6 + 571/675a^5 + 3656/25a^4 + 8336/675a^3 - 62719/675a^2 + 13931/675a + 244/75, 2/675a^21 - 1/675a^20 - 16/225a^19 + 8/225a^18 + 493/675a^17 - 77/225a^16 - 2819/675a^15 + 361/225a^14 + 364/25a^13 - 2099/675a^12 - 21869/675a^11 - 44/15a^10 + 33118/675a^9 + 5984/225a^8 - 4339/75a^7 - 36664/675a^6 + 7531/135a^5 + 12697/225a^4 - 18143/675a^3 - 21052/675a^2 - 779/75a + 1829/675, 4/675a^21 + 19/675a^20 - 1/9a^19 - 23/45a^18 + 577/675a^17 + 2602/675a^16 - 2399/675a^15 - 10633/675a^14 + 6067/675a^13 + 25906/675a^12 - 3334/225a^11 - 39238/675a^10 + 2969/225a^9 + 12481/225a^8 + 736/75a^7 - 21971/675a^6 - 28589/675a^5 + 2369/225a^4 + 5029/135a^3 - 208/75a^2 - 5492/675a - 1523/225, 13/135a^21 - 62/675a^20 - 1334/675a^19 + 157/75a^18 + 11657/675a^17 - 13498/675a^16 - 56312/675a^15 + 4676/45a^14 + 54329/225a^13 - 215048/675a^12 - 285772/675a^11 + 386534/675a^10 + 300787/675a^9 - 114781/225a^8 - 84863/225a^7 + 1484/675a^6 + 75992/135a^5 + 192871/675a^4 - 384443/675a^3 - 901/135a^2 + 35296/675a + 13762/135, 1/675a^21 - 2/675a^20 - 26/675a^19 + 11/135a^18 + 278/675a^17 - 631/675a^16 - 319/135a^15 + 1337/225a^14 + 1753/225a^13 - 15688/675a^12 - 1871/135a^11 + 39311/675a^10 + 4681/675a^9 - 6949/75a^8 + 907/45a^7 + 58252/675a^6 - 24254/675a^5 - 25808/675a^4 + 2452/225a^3 + 1352/135a^2 + 6187/675a - 6172/675, 127/675a^21 - 16/225a^20 - 97/25a^19 + 964/675a^18 + 22906/675a^17 - 8047/675a^16 - 110419/675a^15 + 35728/675a^14 + 318827/675a^13 - 9911/75a^12 - 22402/27a^11 + 105583/675a^10 + 194978/225a^9 + 7636/75a^8 - 124682/225a^7 - 17783/27a^6 + 23594/75a^5 + 165853/225a^4 - 347/5a^3 - 71927/675a^2 - 12623/135a + 1168/75, 116/675a^21 + 61/225a^20 - 2222/675a^19 - 3533/675a^18 + 661/25a^17 + 28613/675a^16 - 579/5a^15 - 2807/15a^14 + 22616/75a^13 + 331264/675a^12 - 11952/25a^11 - 542557/675a^10 + 94696/225a^9 + 206359/225a^8 + 6871/225a^7 - 555331/675a^6 - 140341/225a^5 + 304504/675a^4 + 373952/675a^3 - 27011/225a^2 - 125402/675a - 7982/75, 146/675a^21 - 23/135a^20 - 1036/225a^19 + 821/225a^18 + 634/15a^17 - 22676/675a^16 - 49373/225a^15 + 4724/27a^14 + 480217/675a^13 - 129274/225a^12 - 1014607/675a^11 + 840212/675a^10 + 1423541/675a^9 - 43584/25a^8 - 456232/225a^7 + 953639/675a^6 + 979991/675a^5 - 8027/15a^4 - 484817/675a^3 + 3898/675a^2 + 133678/675a + 60883/675, 1298/675a^21 - 2227/675a^20 - 8039/225a^19 + 40619/675a^18 + 7579/27a^17 - 461a^16 - 821254/675a^15 + 48272/25a^14 + 715562/225a^13 - 3252722/675a^12 - 3467963/675a^11 + 1594744/225a^10 + 3666263/675a^9 - 1098026/225a^8 - 141561/25a^7 - 123119/135a^6 + 963442/135a^5 + 715229/225a^4 - 229016/45a^3 - 391201/675a^2 + 67823/225a + 575068/675 (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:	$ 40057640608100 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 40057640608100 \cdot 1}{2\cdot\sqrt{322170665670052381217287483857714996337890625}}\cr\approx \mathstrut & 0.428026884288700 \end{aligned}\] (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 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129) 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 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129, 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 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129); 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 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 121*x^11 + 11011*x^10 + 1331*x^9 - 9438*x^8 - 5324*x^7 + 4719*x^6 + 9317*x^5 - 1210*x^4 - 6655*x^3 + 121*x^2 + 1331*x + 1129); 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

$C_{11}^2:C_{20}$ (as 22T18):

comment:Galois group

sage:K.galois_group(type='pari')

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 solvable group of order 2420

The 26 conjugacy class representatives for $C_{11}^2:C_{20}$

Character table for $C_{11}^2:C_{20}$

Intermediate fields

$\Q(\sqrt{5}) $

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]

Sibling fields

Degree 22 siblings:	data not computed
Degree 44 siblings:	data not computed
Minimal sibling:	22.2.1502949932221620345262426910400390625.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$20{,}\,{\href{/padicField/2.2.0.1}{2} }$	$20{,}\,{\href{/padicField/3.2.0.1}{2} }$	R	$20{,}\,{\href{/padicField/7.2.0.1}{2} }$	R	$20{,}\,{\href{/padicField/13.2.0.1}{2} }$	$20{,}\,{\href{/padicField/17.2.0.1}{2} }$	${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$	${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	$20{,}\,{\href{/padicField/37.2.0.1}{2} }$	${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$	${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.2.0.1}{2} }$	$20{,}\,{\href{/padicField/47.2.0.1}{2} }$	$20{,}\,{\href{/padicField/53.2.0.1}{2} }$	${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
$5$ 5	5.1.2.1a1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
$5$ 5	5.5.4.15a1.4	$x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 86$	$4$	$5$	$15$	20T1	not computed
$11$ 11	11.1.11.16a2.1	$x^{11} + 22 x^{6} + 11$	$11$	$1$	$16$	$C_{11}:C_5$	$$[\frac{8}{5}]_{5}$$
$11$ 11	11.1.11.16a2.1	$x^{11} + 22 x^{6} + 11$	$11$	$1$	$16$	$C_{11}:C_5$	$$[\frac{8}{5}]_{5}$$

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