LMFDB - Number field 21.3.4297390112987454978404646912.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4)

gp: K = bnfinit(y^21 - y^20 + 9*y^19 - 7*y^18 + 35*y^17 - 27*y^16 + 83*y^15 - 61*y^14 + 123*y^13 - 111*y^12 + 163*y^11 - 225*y^10 + 213*y^9 - 253*y^8 + 177*y^7 - 167*y^6 + 192*y^5 - 180*y^4 + 136*y^3 - 76*y^2 - 12*y + 4, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4)

$ x^{21} - x^{20} + 9 x^{19} - 7 x^{18} + 35 x^{17} - 27 x^{16} + 83 x^{15} - 61 x^{14} + 123 x^{13} + \cdots + 4 $ x^21 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$21$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[3, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-4297390112987454978404646912$ -4297390112987454978404646912 $\medspace = -\,2^{38}\cdot 3^{18}\cdot 7^{9}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$20.70$	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:	$2^{13/6}3^{6/7}7^{1/2}\approx 30.46083139496543$
Ramified primes:	$2$, $3$, $7$ 2, 3, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-7}) $
$\card{ \Aut(K/\Q) }$:	$1$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{6}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{3}{8}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{16}-\frac{1}{8}a^{12}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{3}{8}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{17}-\frac{1}{8}a^{13}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{14}-\frac{1}{8}a^{10}+\frac{1}{8}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{3064}a^{19}+\frac{21}{766}a^{18}+\frac{151}{3064}a^{17}-\frac{121}{3064}a^{16}+\frac{45}{1532}a^{15}+\frac{291}{3064}a^{14}+\frac{69}{1532}a^{13}+\frac{43}{766}a^{12}-\frac{375}{3064}a^{11}-\frac{21}{766}a^{10}+\frac{617}{3064}a^{9}-\frac{707}{3064}a^{8}+\frac{21}{1532}a^{7}-\frac{1287}{3064}a^{6}-\frac{109}{1532}a^{5}+\frac{175}{383}a^{4}+\frac{77}{1532}a^{3}-\frac{43}{766}a^{2}-\frac{174}{383}a-\frac{116}{383}$, $\frac{1}{641043952}a^{20}-\frac{4635}{320521976}a^{19}+\frac{32363831}{641043952}a^{18}+\frac{8251891}{320521976}a^{17}+\frac{1206609}{641043952}a^{16}+\frac{12188533}{320521976}a^{15}-\frac{75715165}{641043952}a^{14}+\frac{30270037}{320521976}a^{13}-\frac{11344357}{641043952}a^{12}-\frac{28886933}{320521976}a^{11}-\frac{33953279}{641043952}a^{10}-\frac{13960699}{320521976}a^{9}+\frac{109565635}{641043952}a^{8}+\frac{67665735}{320521976}a^{7}+\frac{49630213}{641043952}a^{6}+\frac{12984823}{320521976}a^{5}+\frac{14117977}{160260988}a^{4}+\frac{39150567}{80130494}a^{3}+\frac{17535969}{40065247}a^{2}+\frac{17297159}{80130494}a+\frac{76042019}{160260988}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/2*a^7 - 1/2*a^3, 1/2*a^8 - 1/2*a^4, 1/2*a^9 - 1/2*a^5, 1/4*a^10 - 1/4*a^9 - 1/4*a^8 - 1/4*a^7 + 1/4*a^6 + 1/4*a^5 + 1/4*a^4 - 1/4*a^3 - 1/2*a - 1/2, 1/4*a^11 - 1/2*a^6 - 1/2*a^4 - 1/4*a^3 - 1/2*a^2 - 1/2, 1/4*a^12 - 1/2*a^5 - 1/4*a^4 - 1/2*a, 1/4*a^13 - 1/2*a^6 - 1/4*a^5 - 1/2*a^2, 1/4*a^14 - 1/4*a^6, 1/8*a^15 - 1/8*a^14 - 1/8*a^13 - 1/8*a^12 - 1/4*a^8 + 1/8*a^7 - 1/8*a^6 + 3/8*a^5 + 3/8*a^4 + 1/4*a^3 - 1/4*a^2 + 1/4*a - 1/2, 1/8*a^16 - 1/8*a^12 - 1/4*a^9 - 1/8*a^8 - 1/2*a^6 - 1/2*a^5 - 3/8*a^4 - 1/2*a^2 - 1/4*a - 1/2, 1/8*a^17 - 1/8*a^13 + 1/8*a^9 - 1/4*a^8 - 1/4*a^7 - 1/4*a^6 + 3/8*a^5 + 1/4*a^4 - 1/4*a^3 - 1/4*a^2 - 1/2, 1/8*a^18 - 1/8*a^14 - 1/8*a^10 + 1/8*a^6 - 1/2*a^4 - 1/2, 1/3064*a^19 + 21/766*a^18 + 151/3064*a^17 - 121/3064*a^16 + 45/1532*a^15 + 291/3064*a^14 + 69/1532*a^13 + 43/766*a^12 - 375/3064*a^11 - 21/766*a^10 + 617/3064*a^9 - 707/3064*a^8 + 21/1532*a^7 - 1287/3064*a^6 - 109/1532*a^5 + 175/383*a^4 + 77/1532*a^3 - 43/766*a^2 - 174/383*a - 116/383, 1/641043952*a^20 - 4635/320521976*a^19 + 32363831/641043952*a^18 + 8251891/320521976*a^17 + 1206609/641043952*a^16 + 12188533/320521976*a^15 - 75715165/641043952*a^14 + 30270037/320521976*a^13 - 11344357/641043952*a^12 - 28886933/320521976*a^11 - 33953279/641043952*a^10 - 13960699/320521976*a^9 + 109565635/641043952*a^8 + 67665735/320521976*a^7 + 49630213/641043952*a^6 + 12984823/320521976*a^5 + 14117977/160260988*a^4 + 39150567/80130494*a^3 + 17535969/40065247*a^2 + 17297159/80130494*a + 76042019/160260988

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:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{7298367}{160260988}a^{20}-\frac{5317739}{320521976}a^{19}+\frac{69521399}{160260988}a^{18}-\frac{41820393}{320521976}a^{17}+\frac{72029592}{40065247}a^{16}-\frac{215602813}{320521976}a^{15}+\frac{173862131}{40065247}a^{14}-\frac{629878269}{320521976}a^{13}+\frac{268895337}{40065247}a^{12}-\frac{1442421937}{320521976}a^{11}+\frac{1249220975}{160260988}a^{10}-\frac{3406593485}{320521976}a^{9}+\frac{1470658295}{160260988}a^{8}-\frac{4655273341}{320521976}a^{7}+\frac{1371825041}{160260988}a^{6}-\frac{2916362561}{320521976}a^{5}+\frac{757047227}{80130494}a^{4}-\frac{1406862175}{160260988}a^{3}+\frac{1333068003}{160260988}a^{2}-\frac{237182819}{40065247}a-\frac{11799}{80130494}$, $\frac{2358957}{320521976}a^{20}+\frac{663451}{80130494}a^{19}+\frac{20463161}{320521976}a^{18}+\frac{11023817}{320521976}a^{17}+\frac{40345281}{160260988}a^{16}-\frac{1342171}{80130494}a^{15}+\frac{149177731}{320521976}a^{14}-\frac{89326751}{320521976}a^{13}+\frac{19542791}{40065247}a^{12}-\frac{30633800}{40065247}a^{11}-\frac{74004143}{320521976}a^{10}-\frac{305576879}{320521976}a^{9}-\frac{43423253}{160260988}a^{8}-\frac{45037958}{40065247}a^{7}+\frac{508956409}{320521976}a^{6}+\frac{417684767}{320521976}a^{5}+\frac{217116259}{320521976}a^{4}+\frac{38570557}{40065247}a^{3}+\frac{31516433}{160260988}a^{2}-\frac{318897821}{160260988}a-\frac{45306455}{40065247}$, $\frac{25365579}{641043952}a^{20}-\frac{6307759}{320521976}a^{19}+\frac{222054505}{641043952}a^{18}-\frac{16228917}{160260988}a^{17}+\frac{863625735}{641043952}a^{16}-\frac{118533657}{320521976}a^{15}+\frac{2037796529}{641043952}a^{14}-\frac{107790995}{160260988}a^{13}+\frac{3021342357}{641043952}a^{12}-\frac{497720537}{320521976}a^{11}+\frac{3692932939}{641043952}a^{10}-\frac{792925657}{160260988}a^{9}+\frac{3504237285}{641043952}a^{8}-\frac{1962828611}{320521976}a^{7}+\frac{1882714207}{641043952}a^{6}-\frac{749998409}{160260988}a^{5}+\frac{144993791}{40065247}a^{4}-\frac{314080175}{80130494}a^{3}+\frac{257028159}{160260988}a^{2}-\frac{54129779}{80130494}a-\frac{115234759}{160260988}$, $\frac{31883957}{641043952}a^{20}-\frac{25275683}{320521976}a^{19}+\frac{317664455}{641043952}a^{18}-\frac{178234281}{320521976}a^{17}+\frac{1330003551}{641043952}a^{16}-\frac{630062123}{320521976}a^{15}+\frac{3437524331}{641043952}a^{14}-\frac{1372799777}{320521976}a^{13}+\frac{5484831941}{641043952}a^{12}-\frac{2249328343}{320521976}a^{11}+\frac{7693585557}{641043952}a^{10}-\frac{4563840691}{320521976}a^{9}+\frac{10364166625}{641043952}a^{8}-\frac{5671746155}{320521976}a^{7}+\frac{7647174665}{641043952}a^{6}-\frac{4107847589}{320521976}a^{5}+\frac{4053220517}{320521976}a^{4}-\frac{1030725227}{80130494}a^{3}+\frac{904364425}{80130494}a^{2}-\frac{819083001}{160260988}a-\frac{168416531}{160260988}$, $\frac{2492346}{40065247}a^{20}-\frac{7006945}{80130494}a^{19}+\frac{84257545}{160260988}a^{18}-\frac{195970389}{320521976}a^{17}+\frac{75977319}{40065247}a^{16}-\frac{731503125}{320521976}a^{15}+\frac{1381930849}{320521976}a^{14}-\frac{789903623}{160260988}a^{13}+\frac{1923309473}{320521976}a^{12}-\frac{1346310913}{160260988}a^{11}+\frac{366503889}{40065247}a^{10}-\frac{4784423675}{320521976}a^{9}+\frac{2188269479}{160260988}a^{8}-\frac{4422629777}{320521976}a^{7}+\frac{3955328921}{320521976}a^{6}-\frac{381210344}{40065247}a^{5}+\frac{4431486613}{320521976}a^{4}-\frac{442437007}{40065247}a^{3}+\frac{323859310}{40065247}a^{2}-\frac{682372791}{160260988}a+\frac{6757650}{40065247}$, $\frac{41702755}{641043952}a^{20}-\frac{44716237}{320521976}a^{19}+\frac{405930135}{641043952}a^{18}-\frac{328687455}{320521976}a^{17}+\frac{1628142219}{641043952}a^{16}-\frac{584390979}{160260988}a^{15}+\frac{4125210885}{641043952}a^{14}-\frac{621258929}{80130494}a^{13}+\frac{6406233787}{641043952}a^{12}-\frac{3734058587}{320521976}a^{11}+\frac{9595141645}{641043952}a^{10}-\frac{6571880651}{320521976}a^{9}+\frac{15044140025}{641043952}a^{8}-\frac{3567372123}{160260988}a^{7}+\frac{11634797299}{641043952}a^{6}-\frac{605443860}{40065247}a^{5}+\frac{5817206193}{320521976}a^{4}-\frac{1484477399}{80130494}a^{3}+\frac{2483861545}{160260988}a^{2}-\frac{682506381}{160260988}a-\frac{8269645}{160260988}$, $\frac{19246287}{160260988}a^{20}-\frac{4836997}{160260988}a^{19}+\frac{299180709}{320521976}a^{18}-\frac{19007995}{160260988}a^{17}+\frac{261654771}{80130494}a^{16}-\frac{127370837}{160260988}a^{15}+\frac{2185842241}{320521976}a^{14}-\frac{302050377}{160260988}a^{13}+\frac{710963555}{80130494}a^{12}-\frac{1023577021}{160260988}a^{11}+\frac{3317885695}{320521976}a^{10}-\frac{2442375573}{160260988}a^{9}+\frac{343624179}{40065247}a^{8}-\frac{2314172913}{160260988}a^{7}+\frac{2852116959}{320521976}a^{6}-\frac{1606486529}{160260988}a^{5}+\frac{1859704107}{160260988}a^{4}-\frac{299018706}{40065247}a^{3}+\frac{345651389}{80130494}a^{2}-\frac{350828881}{80130494}a-\frac{130964611}{80130494}$, $\frac{12597441}{641043952}a^{20}-\frac{10622059}{320521976}a^{19}+\frac{81299117}{641043952}a^{18}-\frac{29294311}{160260988}a^{17}+\frac{198178175}{641043952}a^{16}-\frac{21818420}{40065247}a^{15}+\frac{311717515}{641043952}a^{14}-\frac{234456943}{320521976}a^{13}+\frac{49055231}{641043952}a^{12}-\frac{306811813}{320521976}a^{11}+\frac{429921999}{641043952}a^{10}-\frac{58773802}{40065247}a^{9}-\frac{88218567}{641043952}a^{8}+\frac{318159471}{160260988}a^{7}-\frac{1605865127}{641043952}a^{6}+\frac{84631921}{320521976}a^{5}-\frac{118018497}{160260988}a^{4}+\frac{140565639}{160260988}a^{3}-\frac{265585217}{80130494}a^{2}+\frac{163511940}{40065247}a-\frac{126215127}{160260988}$, $\frac{894203}{8781424}a^{20}-\frac{185687}{4390712}a^{19}+\frac{7404829}{8781424}a^{18}-\frac{521471}{2195356}a^{17}+\frac{27350491}{8781424}a^{16}-\frac{5096231}{4390712}a^{15}+\frac{61072977}{8781424}a^{14}-\frac{1589202}{548839}a^{13}+\frac{86682749}{8781424}a^{12}-\frac{32259673}{4390712}a^{11}+\frac{108620987}{8781424}a^{10}-\frac{18437039}{1097678}a^{9}+\frac{110255221}{8781424}a^{8}-\frac{77129835}{4390712}a^{7}+\frac{103977947}{8781424}a^{6}-\frac{28932881}{2195356}a^{5}+\frac{33008393}{2195356}a^{4}-\frac{26120065}{2195356}a^{3}+\frac{13062777}{2195356}a^{2}-\frac{2989584}{548839}a-\frac{756577}{2195356}$, $\frac{556455}{4390712}a^{20}-\frac{86606}{548839}a^{19}+\frac{5029573}{4390712}a^{18}-\frac{2420239}{2195356}a^{17}+\frac{2429336}{548839}a^{16}-\frac{8907269}{2195356}a^{15}+\frac{46239427}{4390712}a^{14}-\frac{4816723}{548839}a^{13}+\frac{16947219}{1097678}a^{12}-\frac{16439621}{1097678}a^{11}+\frac{92620785}{4390712}a^{10}-\frac{65235865}{2195356}a^{9}+\frac{63575583}{2195356}a^{8}-\frac{36002857}{1097678}a^{7}+\frac{98981251}{4390712}a^{6}-\frac{48274851}{2195356}a^{5}+\frac{109664327}{4390712}a^{4}-\frac{53630441}{2195356}a^{3}+\frac{10700280}{548839}a^{2}-\frac{17709295}{2195356}a-\frac{1981345}{1097678}$, $\frac{789425}{160260988}a^{20}-\frac{7084685}{80130494}a^{19}+\frac{2671206}{40065247}a^{18}-\frac{202809021}{320521976}a^{17}+\frac{71065233}{320521976}a^{16}-\frac{161325283}{80130494}a^{15}+\frac{29905196}{40065247}a^{14}-\frac{1199888335}{320521976}a^{13}+\frac{346639485}{320521976}a^{12}-\frac{323138313}{80130494}a^{11}+\frac{603712027}{160260988}a^{10}-\frac{1658799033}{320521976}a^{9}+\frac{2878724607}{320521976}a^{8}-\frac{119062320}{40065247}a^{7}+\frac{442520481}{80130494}a^{6}-\frac{949783549}{320521976}a^{5}+\frac{1571506893}{320521976}a^{4}-\frac{465405613}{80130494}a^{3}+\frac{524278931}{160260988}a^{2}-\frac{32156339}{160260988}a+\frac{99253571}{80130494}$ 7298367/160260988a^20 - 5317739/320521976a^19 + 69521399/160260988a^18 - 41820393/320521976a^17 + 72029592/40065247a^16 - 215602813/320521976a^15 + 173862131/40065247a^14 - 629878269/320521976a^13 + 268895337/40065247a^12 - 1442421937/320521976a^11 + 1249220975/160260988a^10 - 3406593485/320521976a^9 + 1470658295/160260988a^8 - 4655273341/320521976a^7 + 1371825041/160260988a^6 - 2916362561/320521976a^5 + 757047227/80130494a^4 - 1406862175/160260988a^3 + 1333068003/160260988a^2 - 237182819/40065247a - 11799/80130494, 2358957/320521976a^20 + 663451/80130494a^19 + 20463161/320521976a^18 + 11023817/320521976a^17 + 40345281/160260988a^16 - 1342171/80130494a^15 + 149177731/320521976a^14 - 89326751/320521976a^13 + 19542791/40065247a^12 - 30633800/40065247a^11 - 74004143/320521976a^10 - 305576879/320521976a^9 - 43423253/160260988a^8 - 45037958/40065247a^7 + 508956409/320521976a^6 + 417684767/320521976a^5 + 217116259/320521976a^4 + 38570557/40065247a^3 + 31516433/160260988a^2 - 318897821/160260988a - 45306455/40065247, 25365579/641043952a^20 - 6307759/320521976a^19 + 222054505/641043952a^18 - 16228917/160260988a^17 + 863625735/641043952a^16 - 118533657/320521976a^15 + 2037796529/641043952a^14 - 107790995/160260988a^13 + 3021342357/641043952a^12 - 497720537/320521976a^11 + 3692932939/641043952a^10 - 792925657/160260988a^9 + 3504237285/641043952a^8 - 1962828611/320521976a^7 + 1882714207/641043952a^6 - 749998409/160260988a^5 + 144993791/40065247a^4 - 314080175/80130494a^3 + 257028159/160260988a^2 - 54129779/80130494a - 115234759/160260988, 31883957/641043952a^20 - 25275683/320521976a^19 + 317664455/641043952a^18 - 178234281/320521976a^17 + 1330003551/641043952a^16 - 630062123/320521976a^15 + 3437524331/641043952a^14 - 1372799777/320521976a^13 + 5484831941/641043952a^12 - 2249328343/320521976a^11 + 7693585557/641043952a^10 - 4563840691/320521976a^9 + 10364166625/641043952a^8 - 5671746155/320521976a^7 + 7647174665/641043952a^6 - 4107847589/320521976a^5 + 4053220517/320521976a^4 - 1030725227/80130494a^3 + 904364425/80130494a^2 - 819083001/160260988a - 168416531/160260988, 2492346/40065247a^20 - 7006945/80130494a^19 + 84257545/160260988a^18 - 195970389/320521976a^17 + 75977319/40065247a^16 - 731503125/320521976a^15 + 1381930849/320521976a^14 - 789903623/160260988a^13 + 1923309473/320521976a^12 - 1346310913/160260988a^11 + 366503889/40065247a^10 - 4784423675/320521976a^9 + 2188269479/160260988a^8 - 4422629777/320521976a^7 + 3955328921/320521976a^6 - 381210344/40065247a^5 + 4431486613/320521976a^4 - 442437007/40065247a^3 + 323859310/40065247a^2 - 682372791/160260988a + 6757650/40065247, 41702755/641043952a^20 - 44716237/320521976a^19 + 405930135/641043952a^18 - 328687455/320521976a^17 + 1628142219/641043952a^16 - 584390979/160260988a^15 + 4125210885/641043952a^14 - 621258929/80130494a^13 + 6406233787/641043952a^12 - 3734058587/320521976a^11 + 9595141645/641043952a^10 - 6571880651/320521976a^9 + 15044140025/641043952a^8 - 3567372123/160260988a^7 + 11634797299/641043952a^6 - 605443860/40065247a^5 + 5817206193/320521976a^4 - 1484477399/80130494a^3 + 2483861545/160260988a^2 - 682506381/160260988a - 8269645/160260988, 19246287/160260988a^20 - 4836997/160260988a^19 + 299180709/320521976a^18 - 19007995/160260988a^17 + 261654771/80130494a^16 - 127370837/160260988a^15 + 2185842241/320521976a^14 - 302050377/160260988a^13 + 710963555/80130494a^12 - 1023577021/160260988a^11 + 3317885695/320521976a^10 - 2442375573/160260988a^9 + 343624179/40065247a^8 - 2314172913/160260988a^7 + 2852116959/320521976a^6 - 1606486529/160260988a^5 + 1859704107/160260988a^4 - 299018706/40065247a^3 + 345651389/80130494a^2 - 350828881/80130494a - 130964611/80130494, 12597441/641043952a^20 - 10622059/320521976a^19 + 81299117/641043952a^18 - 29294311/160260988a^17 + 198178175/641043952a^16 - 21818420/40065247a^15 + 311717515/641043952a^14 - 234456943/320521976a^13 + 49055231/641043952a^12 - 306811813/320521976a^11 + 429921999/641043952a^10 - 58773802/40065247a^9 - 88218567/641043952a^8 + 318159471/160260988a^7 - 1605865127/641043952a^6 + 84631921/320521976a^5 - 118018497/160260988a^4 + 140565639/160260988a^3 - 265585217/80130494a^2 + 163511940/40065247a - 126215127/160260988, 894203/8781424a^20 - 185687/4390712a^19 + 7404829/8781424a^18 - 521471/2195356a^17 + 27350491/8781424a^16 - 5096231/4390712a^15 + 61072977/8781424a^14 - 1589202/548839a^13 + 86682749/8781424a^12 - 32259673/4390712a^11 + 108620987/8781424a^10 - 18437039/1097678a^9 + 110255221/8781424a^8 - 77129835/4390712a^7 + 103977947/8781424a^6 - 28932881/2195356a^5 + 33008393/2195356a^4 - 26120065/2195356a^3 + 13062777/2195356a^2 - 2989584/548839a - 756577/2195356, 556455/4390712a^20 - 86606/548839a^19 + 5029573/4390712a^18 - 2420239/2195356a^17 + 2429336/548839a^16 - 8907269/2195356a^15 + 46239427/4390712a^14 - 4816723/548839a^13 + 16947219/1097678a^12 - 16439621/1097678a^11 + 92620785/4390712a^10 - 65235865/2195356a^9 + 63575583/2195356a^8 - 36002857/1097678a^7 + 98981251/4390712a^6 - 48274851/2195356a^5 + 109664327/4390712a^4 - 53630441/2195356a^3 + 10700280/548839a^2 - 17709295/2195356a - 1981345/1097678, 789425/160260988a^20 - 7084685/80130494a^19 + 2671206/40065247a^18 - 202809021/320521976a^17 + 71065233/320521976a^16 - 161325283/80130494a^15 + 29905196/40065247a^14 - 1199888335/320521976a^13 + 346639485/320521976a^12 - 323138313/80130494a^11 + 603712027/160260988a^10 - 1658799033/320521976a^9 + 2878724607/320521976a^8 - 119062320/40065247a^7 + 442520481/80130494a^6 - 949783549/320521976a^5 + 1571506893/320521976a^4 - 465405613/80130494a^3 + 524278931/160260988a^2 - 32156339/160260988a + 99253571/80130494 (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:	$ 1790438.21987 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 1790438.21987 \cdot 1}{2\cdot\sqrt{4297390112987454978404646912}}\cr\approx \mathstrut & 1.66738448630 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4)

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 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4, 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 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4);

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 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4);

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

$\PGL(2,7)$ (as 21T20):

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 non-solvable group of order 336

The 9 conjugacy class representatives for $\PGL(2,7)$

Character table for $\PGL(2,7)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 8 sibling:	8.2.16387080192.5
Degree 14 sibling:	data not computed
Degree 16 sibling:	data not computed
Degree 24 sibling:	data not computed
Degree 28 siblings:	data not computed
Degree 42 siblings:	data not computed
Minimal sibling:	8.2.16387080192.5

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	${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$	R	${\href{/padicField/11.7.0.1}{7} }^{3}$	${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$	${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$	${\href{/padicField/23.3.0.1}{3} }^{7}$	${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$	${\href{/padicField/43.7.0.1}{7} }^{3}$	${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.3.0.1}{3} }^{7}$	${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.6.10.2	$x^{6} + 2 x^{5} + 4 x^{4} + 4 x + 2$	$6$	$1$	$10$	$S_4$	$[8/3, 8/3]_{3}^{2}$
	2.12.26.65	$x^{12} + 4 x^{3} + 2$	$12$	$1$	$26$	$S_4$	$[8/3, 8/3]_{3}^{2}$
$3$ 3		Deg $21$	$7$	$3$	$18$
$7$ 7	7.3.0.1	$x^{3} + 6 x^{2} + 4$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

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