LMFDB - Number field 21.13.10423848562958827406082788615011739523268608.1

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 24*x^19 - 16*x^18 + 135*x^17 + 180*x^16 + 519*x^15 + 918*x^14 - 4653*x^13 - 13904*x^12 - 7236*x^11 + 16440*x^10 + 49612*x^9 + 101808*x^8 + 105621*x^7 - 51646*x^6 - 286524*x^5 - 369432*x^4 - 255248*x^3 - 102816*x^2 - 22848*x - 2176)

gp:K = bnfinit(y^21 - 24*y^19 - 16*y^18 + 135*y^17 + 180*y^16 + 519*y^15 + 918*y^14 - 4653*y^13 - 13904*y^12 - 7236*y^11 + 16440*y^10 + 49612*y^9 + 101808*y^8 + 105621*y^7 - 51646*y^6 - 286524*y^5 - 369432*y^4 - 255248*y^3 - 102816*y^2 - 22848*y - 2176, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 24*x^19 - 16*x^18 + 135*x^17 + 180*x^16 + 519*x^15 + 918*x^14 - 4653*x^13 - 13904*x^12 - 7236*x^11 + 16440*x^10 + 49612*x^9 + 101808*x^8 + 105621*x^7 - 51646*x^6 - 286524*x^5 - 369432*x^4 - 255248*x^3 - 102816*x^2 - 22848*x - 2176);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 24*x^19 - 16*x^18 + 135*x^17 + 180*x^16 + 519*x^15 + 918*x^14 - 4653*x^13 - 13904*x^12 - 7236*x^11 + 16440*x^10 + 49612*x^9 + 101808*x^8 + 105621*x^7 - 51646*x^6 - 286524*x^5 - 369432*x^4 - 255248*x^3 - 102816*x^2 - 22848*x - 2176)

$ x^{21} - 24 x^{19} - 16 x^{18} + 135 x^{17} + 180 x^{16} + 519 x^{15} + 918 x^{14} - 4653 x^{13} + \cdots - 2176 $ x^21 - 24*x^19 - 16*x^18 + 135*x^17 + 180*x^16 + 519*x^15 + 918*x^14 - 4653*x^13 - 13904*x^12 - 7236*x^11 + 16440*x^10 + 49612*x^9 + 101808*x^8 + 105621*x^7 - 51646*x^6 - 286524*x^5 - 369432*x^4 - 255248*x^3 - 102816*x^2 - 22848*x - 2176

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$21$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[13, 4]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$10423848562958827406082788615011739523268608$ 10423848562958827406082788615011739523268608 $\medspace = 2^{14}\cdot 3^{21}\cdot 17^{2}\cdot 29^{18}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$111.81$	sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:(1.0 * dK)^(1/degree(K))
Galois root discriminant:	not computed
Ramified primes:	$2$, $3$, $17$, $29$ 2, 3, 17, 29	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(\sqrt{3}) $
$\Aut(K/\Q)$:	$C_1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{8}a^{11}+\frac{3}{8}a^{9}+\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{64}a^{16}+\frac{1}{32}a^{15}+\frac{1}{16}a^{14}-\frac{1}{8}a^{13}-\frac{25}{64}a^{12}+\frac{1}{32}a^{11}-\frac{13}{64}a^{10}+\frac{7}{16}a^{9}+\frac{3}{64}a^{8}+\frac{3}{32}a^{7}-\frac{1}{2}a^{6}+\frac{3}{8}a^{5}-\frac{5}{16}a^{4}+\frac{1}{8}a^{3}+\frac{5}{64}a^{2}+\frac{3}{16}a+\frac{5}{16}$, $\frac{1}{512}a^{17}-\frac{5}{32}a^{14}+\frac{183}{512}a^{13}-\frac{51}{128}a^{12}-\frac{209}{512}a^{11}-\frac{37}{256}a^{10}-\frac{117}{512}a^{9}+\frac{3}{8}a^{8}+\frac{5}{128}a^{7}-\frac{13}{64}a^{6}-\frac{17}{128}a^{5}+\frac{15}{32}a^{4}-\frac{11}{512}a^{3}+\frac{33}{256}a^{2}+\frac{63}{128}a+\frac{27}{64}$, $\frac{1}{69632}a^{18}-\frac{1}{2048}a^{17}+\frac{3}{1088}a^{16}-\frac{77}{4352}a^{15}+\frac{1623}{69632}a^{14}+\frac{12659}{34816}a^{13}+\frac{15239}{69632}a^{12}-\frac{2849}{8704}a^{11}+\frac{14239}{69632}a^{10}-\frac{13915}{34816}a^{9}-\frac{5067}{17408}a^{8}-\frac{1769}{4352}a^{7}-\frac{6685}{17408}a^{6}+\frac{1087}{8704}a^{5}+\frac{8469}{69632}a^{4}+\frac{567}{8704}a^{3}+\frac{207}{512}a^{2}-\frac{45}{128}a+\frac{85}{256}$, $\frac{1}{557056}a^{19}-\frac{1}{139264}a^{18}+\frac{65}{139264}a^{17}-\frac{261}{34816}a^{16}+\frac{16887}{557056}a^{15}+\frac{11427}{69632}a^{14}-\frac{965}{557056}a^{13}+\frac{6117}{278528}a^{12}-\frac{218001}{557056}a^{11}+\frac{55771}{139264}a^{10}-\frac{231}{512}a^{9}+\frac{1273}{4096}a^{8}+\frac{69355}{139264}a^{7}-\frac{45}{17408}a^{6}-\frac{152795}{557056}a^{5}-\frac{105705}{278528}a^{4}-\frac{33191}{69632}a^{3}+\frac{815}{2048}a^{2}+\frac{569}{2048}a+\frac{363}{1024}$, $\frac{1}{4456448}a^{20}+\frac{1}{2228224}a^{19}-\frac{5}{1114112}a^{18}-\frac{327}{557056}a^{17}+\frac{12311}{4456448}a^{16}+\frac{2657}{131072}a^{15}-\frac{7221}{4456448}a^{14}-\frac{206517}{1114112}a^{13}+\frac{1002539}{4456448}a^{12}-\frac{545277}{2228224}a^{11}+\frac{65401}{557056}a^{10}-\frac{228295}{557056}a^{9}-\frac{549801}{1114112}a^{8}+\frac{29965}{557056}a^{7}-\frac{766619}{4456448}a^{6}-\frac{412093}{1114112}a^{5}+\frac{293047}{1114112}a^{4}+\frac{45701}{139264}a^{3}+\frac{2707}{16384}a^{2}-\frac{1109}{4096}a+\frac{321}{4096}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, 1/8*a^15 - 1/2*a^14 - 1/2*a^13 - 1/8*a^11 + 3/8*a^9 + 1/4*a^8 - 1/8*a^7 - 1/2*a^6 - 1/2*a^3 - 3/8*a - 1/4, 1/64*a^16 + 1/32*a^15 + 1/16*a^14 - 1/8*a^13 - 25/64*a^12 + 1/32*a^11 - 13/64*a^10 + 7/16*a^9 + 3/64*a^8 + 3/32*a^7 - 1/2*a^6 + 3/8*a^5 - 5/16*a^4 + 1/8*a^3 + 5/64*a^2 + 3/16*a + 5/16, 1/512*a^17 - 5/32*a^14 + 183/512*a^13 - 51/128*a^12 - 209/512*a^11 - 37/256*a^10 - 117/512*a^9 + 3/8*a^8 + 5/128*a^7 - 13/64*a^6 - 17/128*a^5 + 15/32*a^4 - 11/512*a^3 + 33/256*a^2 + 63/128*a + 27/64, 1/69632*a^18 - 1/2048*a^17 + 3/1088*a^16 - 77/4352*a^15 + 1623/69632*a^14 + 12659/34816*a^13 + 15239/69632*a^12 - 2849/8704*a^11 + 14239/69632*a^10 - 13915/34816*a^9 - 5067/17408*a^8 - 1769/4352*a^7 - 6685/17408*a^6 + 1087/8704*a^5 + 8469/69632*a^4 + 567/8704*a^3 + 207/512*a^2 - 45/128*a + 85/256, 1/557056*a^19 - 1/139264*a^18 + 65/139264*a^17 - 261/34816*a^16 + 16887/557056*a^15 + 11427/69632*a^14 - 965/557056*a^13 + 6117/278528*a^12 - 218001/557056*a^11 + 55771/139264*a^10 - 231/512*a^9 + 1273/4096*a^8 + 69355/139264*a^7 - 45/17408*a^6 - 152795/557056*a^5 - 105705/278528*a^4 - 33191/69632*a^3 + 815/2048*a^2 + 569/2048*a + 363/1024, 1/4456448*a^20 + 1/2228224*a^19 - 5/1114112*a^18 - 327/557056*a^17 + 12311/4456448*a^16 + 2657/131072*a^15 - 7221/4456448*a^14 - 206517/1114112*a^13 + 1002539/4456448*a^12 - 545277/2228224*a^11 + 65401/557056*a^10 - 228295/557056*a^9 - 549801/1114112*a^8 + 29965/557056*a^7 - 766619/4456448*a^6 - 412093/1114112*a^5 + 293047/1114112*a^4 + 45701/139264*a^3 + 2707/16384*a^2 - 1109/4096*a + 321/4096

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:		$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:	$16$	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:	$\frac{5907613635}{4456448}a^{20}-\frac{2090116773}{2228224}a^{19}-\frac{34706659663}{1114112}a^{18}+\frac{27315171}{32768}a^{17}+\frac{794940281733}{4456448}a^{16}+\frac{250417803451}{2228224}a^{15}+\frac{2711440612449}{4456448}a^{14}+\frac{876147265989}{1114112}a^{13}-\frac{29968849674015}{4456448}a^{12}-\frac{30466926492111}{2228224}a^{11}+\frac{47375989083}{557056}a^{10}+\frac{12108030392347}{557056}a^{9}+\frac{56134630744677}{1114112}a^{8}+\frac{55316960726511}{557056}a^{7}+\frac{310777030162927}{4456448}a^{6}-\frac{131276897235963}{1114112}a^{5}-\frac{19428342453075}{65536}a^{4}-\frac{38982897618853}{139264}a^{3}-\frac{2297562530391}{16384}a^{2}-\frac{151665330771}{4096}a-\frac{16687624389}{4096}$, $\frac{1788013359}{2228224}a^{20}-\frac{666505935}{1114112}a^{19}-\frac{10479889623}{557056}a^{18}+\frac{19453041}{16384}a^{17}+\frac{239432206329}{2228224}a^{16}+\frac{71658937005}{1114112}a^{15}+\frac{821011878981}{2228224}a^{14}+\frac{16083303765}{34816}a^{13}-\frac{9087514065235}{2228224}a^{12}-\frac{9042858178965}{1114112}a^{11}+\frac{68839826337}{278528}a^{10}+\frac{3623764076015}{278528}a^{9}+\frac{16772172204321}{557056}a^{8}+\frac{16500690643449}{278528}a^{7}+\frac{90409860100747}{2228224}a^{6}-\frac{19974798705447}{278528}a^{5}-\frac{5782087696689}{32768}a^{4}-\frac{2869710950931}{17408}a^{3}-\frac{670574591703}{8192}a^{2}-\frac{21951238545}{1024}a-\frac{4792429667}{2048}$, $\frac{815896071}{2228224}a^{20}-\frac{316392885}{1114112}a^{19}-\frac{4774201803}{557056}a^{18}+\frac{12975363}{16384}a^{17}+\frac{108915111969}{2228224}a^{16}+\frac{31150438067}{1114112}a^{15}+\frac{374368510413}{2228224}a^{14}+\frac{114650088099}{557056}a^{13}-\frac{4154799602947}{2228224}a^{12}-\frac{4061888302959}{1114112}a^{11}+\frac{53592514443}{278528}a^{10}+\frac{1640459987311}{278528}a^{9}+\frac{7568216663409}{557056}a^{8}+\frac{7437880048071}{278528}a^{7}+\frac{39847526287555}{2228224}a^{6}-\frac{18346567307841}{557056}a^{5}-\frac{2602410309459}{32768}a^{4}-\frac{5111359098939}{69632}a^{3}-\frac{295653796227}{8192}a^{2}-\frac{19158686433}{2048}a-\frac{2068407213}{2048}$, $\frac{2967423741}{2228224}a^{20}-\frac{1051495479}{1114112}a^{19}-\frac{17431998513}{557056}a^{18}+\frac{14243769}{16384}a^{17}+\frac{399233903931}{2228224}a^{16}+\frac{125600078721}{1114112}a^{15}+\frac{1362043757439}{2228224}a^{14}+\frac{439704568497}{557056}a^{13}-\frac{15053973155825}{2228224}a^{12}-\frac{15295249416213}{1114112}a^{11}+\frac{26001018477}{278528}a^{10}+\frac{6079830166861}{278528}a^{9}+\frac{28187338420107}{557056}a^{8}+\frac{27774982805085}{278528}a^{7}+\frac{155944545712209}{2228224}a^{6}-\frac{65946113303499}{557056}a^{5}-\frac{9754434592521}{32768}a^{4}-\frac{19567650078737}{69632}a^{3}-\frac{1153067378049}{8192}a^{2}-\frac{76104013563}{2048}a-\frac{8372531527}{2048}$, $\frac{815896071}{2228224}a^{20}-\frac{316392885}{1114112}a^{19}-\frac{4774201803}{557056}a^{18}+\frac{12975363}{16384}a^{17}+\frac{108915111969}{2228224}a^{16}+\frac{31150438067}{1114112}a^{15}+\frac{374368510413}{2228224}a^{14}+\frac{114650088099}{557056}a^{13}-\frac{4154799602947}{2228224}a^{12}-\frac{4061888302959}{1114112}a^{11}+\frac{53592514443}{278528}a^{10}+\frac{1640459987311}{278528}a^{9}+\frac{7568216663409}{557056}a^{8}+\frac{7437880048071}{278528}a^{7}+\frac{39847526287555}{2228224}a^{6}-\frac{18346567307841}{557056}a^{5}-\frac{2602410309459}{32768}a^{4}-\frac{5111359098939}{69632}a^{3}-\frac{295653796227}{8192}a^{2}-\frac{19158686433}{2048}a-\frac{2068409261}{2048}$, $\frac{129140163}{262144}a^{20}-\frac{43046721}{131072}a^{19}-\frac{760492071}{65536}a^{18}-\frac{4782969}{32768}a^{17}+\frac{17459431173}{262144}a^{16}+\frac{5802804279}{131072}a^{15}+\frac{59286672225}{262144}a^{14}+\frac{19756555371}{65536}a^{13}-\frac{653573326095}{262144}a^{12}-\frac{679924637811}{131072}a^{11}-\frac{3486504465}{32768}a^{10}+\frac{267707371275}{32768}a^{9}+\frac{1244782279989}{65536}a^{8}+\frac{1228510287675}{32768}a^{7}+\frac{7087858288623}{262144}a^{6}-\frac{2848702929345}{65536}a^{5}-\frac{7351303729623}{65536}a^{4}-\frac{878282252659}{8192}a^{3}-\frac{889130022255}{16384}a^{2}-\frac{59275334817}{4096}a-\frac{6586144217}{4096}$, $a+1$, $\frac{46916728039}{4456448}a^{20}-\frac{16281730653}{2228224}a^{19}-\frac{275848684347}{1114112}a^{18}+\frac{1894597931}{557056}a^{17}+\frac{6323116620609}{4456448}a^{16}+\frac{2028213344091}{2228224}a^{15}+\frac{21535001276525}{4456448}a^{14}+\frac{7030680587351}{1114112}a^{13}-\frac{237820132893571}{4456448}a^{12}-\frac{243632606528199}{2228224}a^{11}-\frac{165286196445}{557056}a^{10}+\frac{96524333463215}{557056}a^{9}+\frac{447925817405425}{1114112}a^{8}+\frac{441625053986383}{557056}a^{7}+\frac{25\cdots 31}{4456448}a^{6}-\frac{10\cdots 65}{1114112}a^{5}-\frac{26\cdots 31}{1114112}a^{4}-\frac{312720536456191}{139264}a^{3}-\frac{18494901183635}{16384}a^{2}-\frac{1224925329637}{4096}a-\frac{135221449797}{4096}$, $\frac{153354915413}{4456448}a^{20}-\frac{55285747707}{2228224}a^{19}-\frac{900204189049}{1114112}a^{18}+\frac{17825441165}{557056}a^{17}+\frac{20600604912547}{4456448}a^{16}+\frac{6375060748885}{2228224}a^{15}+\frac{70395357412647}{4456448}a^{14}+\frac{22505906661135}{1114112}a^{13}-\frac{778479439824441}{4456448}a^{12}-\frac{785477954433201}{2228224}a^{11}+\frac{2891157494005}{557056}a^{10}+\frac{313078433184957}{557056}a^{9}+\frac{14\cdots 87}{1114112}a^{8}+\frac{14\cdots 93}{557056}a^{7}+\frac{79\cdots 25}{4456448}a^{6}-\frac{34\cdots 37}{1114112}a^{5}-\frac{85\cdots 25}{1114112}a^{4}-\frac{10\cdots 43}{139264}a^{3}-\frac{58884243859377}{16384}a^{2}-\frac{3875778759745}{4096}a-\frac{425260682763}{4096}$, $\frac{13626143809}{2228224}a^{20}-\frac{4966113527}{1114112}a^{19}-\frac{79954354685}{557056}a^{18}+\frac{1892326121}{278528}a^{17}+\frac{1829168282839}{2228224}a^{16}+\frac{559490867369}{1114112}a^{15}+\frac{6252597718443}{2228224}a^{14}+\frac{1987740969255}{557056}a^{13}-\frac{69211436782037}{2228224}a^{12}-\frac{69509241619461}{1114112}a^{11}+\frac{361165816037}{278528}a^{10}+\frac{27764912683185}{278528}a^{9}+\frac{128496369455383}{557056}a^{8}+\frac{126524177798949}{278528}a^{7}+\frac{41207578360789}{131072}a^{6}-\frac{304011839753289}{557056}a^{5}-\frac{44387756885833}{32768}a^{4}-\frac{88464153520047}{69632}a^{3}-\frac{5182968764781}{8192}a^{2}-\frac{340127523841}{2048}a-\frac{37198258095}{2048}$, $\frac{4014028195}{557056}a^{20}-\frac{1292687511}{278528}a^{19}-\frac{23671551759}{139264}a^{18}-\frac{402205059}{69632}a^{17}+\frac{544297656805}{557056}a^{16}+\frac{185853649277}{278528}a^{15}+\frac{1842059989041}{557056}a^{14}+\frac{312308331445}{69632}a^{13}-\frac{20293282158423}{557056}a^{12}-\frac{21372193169501}{278528}a^{11}-\frac{179622169241}{69632}a^{10}+\frac{8376654600655}{69632}a^{9}+\frac{38978027812525}{139264}a^{8}+\frac{38505909809545}{69632}a^{7}+\frac{225133154060047}{557056}a^{6}-\frac{11034903265877}{17408}a^{5}-\frac{230721104340629}{139264}a^{4}-\frac{6929993164385}{4352}a^{3}-\frac{1658012939699}{2048}a^{2}-\frac{13870480287}{64}a-\frac{12372919927}{512}$, $\frac{1618592681}{4456448}a^{20}-\frac{589859023}{2228224}a^{19}-\frac{558567597}{65536}a^{18}+\frac{222580041}{557056}a^{17}+\frac{217127776047}{4456448}a^{16}+\frac{66580349393}{2228224}a^{15}+\frac{743448008707}{4456448}a^{14}+\frac{235992531439}{1114112}a^{13}-\frac{8217551544189}{4456448}a^{12}-\frac{8257334779661}{2228224}a^{11}+\frac{38103546185}{557056}a^{10}+\frac{3295505074177}{557056}a^{9}+\frac{15276035864335}{1114112}a^{8}+\frac{15037118365037}{557056}a^{7}+\frac{83386002343597}{4456448}a^{6}-\frac{2120143694945}{65536}a^{5}-\frac{89660621925785}{1114112}a^{4}-\frac{10528994928719}{139264}a^{3}-\frac{618126902085}{16384}a^{2}-\frac{40667429753}{4096}a-\frac{4461812319}{4096}$, $\frac{81800334245}{4456448}a^{20}-\frac{28300439771}{2228224}a^{19}-\frac{481011162185}{1114112}a^{18}+\frac{2814763021}{557056}a^{17}+\frac{11027487529939}{4456448}a^{16}+\frac{3546849013333}{2228224}a^{15}+\frac{37545826994455}{4456448}a^{14}+\frac{12278307469495}{1114112}a^{13}-\frac{414600771696393}{4456448}a^{12}-\frac{425236844540561}{2228224}a^{11}-\frac{428193834875}{557056}a^{10}+\frac{168397062124845}{557056}a^{9}+\frac{781527481619923}{1114112}a^{8}+\frac{770604217231457}{557056}a^{7}+\frac{43\cdots 77}{4456448}a^{6}-\frac{18\cdots 09}{1114112}a^{5}-\frac{270886731944973}{65536}a^{4}-\frac{546057027582631}{139264}a^{3}-\frac{32310118280417}{16384}a^{2}-\frac{2140842461321}{4096}a-\frac{236428732715}{4096}$, $\frac{93137757103}{4456448}a^{20}-\frac{33984886633}{2228224}a^{19}-\frac{546425626443}{1114112}a^{18}+\frac{13109219247}{557056}a^{17}+\frac{12497031816121}{4456448}a^{16}+\frac{3822342368791}{2228224}a^{15}+\frac{2515285767989}{262144}a^{14}+\frac{13573911307553}{1114112}a^{13}-\frac{27823158273163}{262144}a^{12}-\frac{474902469863003}{2228224}a^{11}+\frac{2399318074679}{557056}a^{10}+\frac{189643791901927}{557056}a^{9}+\frac{878392464344025}{1114112}a^{8}+\frac{864760670136667}{557056}a^{7}+\frac{47\cdots 95}{4456448}a^{6}-\frac{20\cdots 07}{1114112}a^{5}-\frac{51\cdots 59}{1114112}a^{4}-\frac{604883238329761}{139264}a^{3}-\frac{35470333425651}{16384}a^{2}-\frac{2330490341959}{4096}a-\frac{255271871785}{4096}$, $\frac{27910026897}{557056}a^{20}-\frac{19519562363}{557056}a^{19}-\frac{20506144213}{17408}a^{18}+\frac{3094099045}{139264}a^{17}+\frac{3759367505943}{557056}a^{16}+\frac{2394495314567}{557056}a^{15}+\frac{753512430291}{32768}a^{14}+\frac{16662537672957}{557056}a^{13}-\frac{8324828227671}{32768}a^{12}-\frac{289086535341805}{557056}a^{11}+\frac{65519806995}{139264}a^{10}+\frac{57338944563305}{69632}a^{9}+\frac{265957228830845}{139264}a^{8}+\frac{524336383652059}{139264}a^{7}+\frac{14\cdots 01}{557056}a^{6}-\frac{24\cdots 77}{557056}a^{5}-\frac{31\cdots 33}{278528}a^{4}-\frac{741125928043257}{69632}a^{3}-\frac{1367716082329}{256}a^{2}-\frac{2894488123097}{2048}a-\frac{159526361965}{1024}$, $\frac{179359337319}{2228224}a^{20}-\frac{64244209311}{1114112}a^{19}-\frac{1053142761367}{557056}a^{18}+\frac{18502648713}{278528}a^{17}+\frac{24107301112321}{2228224}a^{16}+\frac{7507442184877}{1114112}a^{15}+\frac{82332227555597}{2228224}a^{14}+\frac{6604465670447}{139264}a^{13}-\frac{910256263823531}{2228224}a^{12}-\frac{920862163654245}{1114112}a^{11}+\frac{2685101666013}{278528}a^{10}+\frac{366651883540079}{278528}a^{9}+\frac{16\cdots 61}{557056}a^{8}+\frac{16\cdots 93}{278528}a^{7}+\frac{93\cdots 35}{2228224}a^{6}-\frac{19\cdots 09}{278528}a^{5}-\frac{99\cdots 37}{557056}a^{4}-\frac{147025206481349}{8704}a^{3}-\frac{69188487905999}{8192}a^{2}-\frac{2279571779087}{1024}a-\frac{500800038267}{2048}$ 5907613635/4456448a^20 - 2090116773/2228224a^19 - 34706659663/1114112a^18 + 27315171/32768a^17 + 794940281733/4456448a^16 + 250417803451/2228224a^15 + 2711440612449/4456448a^14 + 876147265989/1114112a^13 - 29968849674015/4456448a^12 - 30466926492111/2228224a^11 + 47375989083/557056a^10 + 12108030392347/557056a^9 + 56134630744677/1114112a^8 + 55316960726511/557056a^7 + 310777030162927/4456448a^6 - 131276897235963/1114112a^5 - 19428342453075/65536a^4 - 38982897618853/139264a^3 - 2297562530391/16384a^2 - 151665330771/4096a - 16687624389/4096, 1788013359/2228224a^20 - 666505935/1114112a^19 - 10479889623/557056a^18 + 19453041/16384a^17 + 239432206329/2228224a^16 + 71658937005/1114112a^15 + 821011878981/2228224a^14 + 16083303765/34816a^13 - 9087514065235/2228224a^12 - 9042858178965/1114112a^11 + 68839826337/278528a^10 + 3623764076015/278528a^9 + 16772172204321/557056a^8 + 16500690643449/278528a^7 + 90409860100747/2228224a^6 - 19974798705447/278528a^5 - 5782087696689/32768a^4 - 2869710950931/17408a^3 - 670574591703/8192a^2 - 21951238545/1024a - 4792429667/2048, 815896071/2228224a^20 - 316392885/1114112a^19 - 4774201803/557056a^18 + 12975363/16384a^17 + 108915111969/2228224a^16 + 31150438067/1114112a^15 + 374368510413/2228224a^14 + 114650088099/557056a^13 - 4154799602947/2228224a^12 - 4061888302959/1114112a^11 + 53592514443/278528a^10 + 1640459987311/278528a^9 + 7568216663409/557056a^8 + 7437880048071/278528a^7 + 39847526287555/2228224a^6 - 18346567307841/557056a^5 - 2602410309459/32768a^4 - 5111359098939/69632a^3 - 295653796227/8192a^2 - 19158686433/2048a - 2068407213/2048, 2967423741/2228224a^20 - 1051495479/1114112a^19 - 17431998513/557056a^18 + 14243769/16384a^17 + 399233903931/2228224a^16 + 125600078721/1114112a^15 + 1362043757439/2228224a^14 + 439704568497/557056a^13 - 15053973155825/2228224a^12 - 15295249416213/1114112a^11 + 26001018477/278528a^10 + 6079830166861/278528a^9 + 28187338420107/557056a^8 + 27774982805085/278528a^7 + 155944545712209/2228224a^6 - 65946113303499/557056a^5 - 9754434592521/32768a^4 - 19567650078737/69632a^3 - 1153067378049/8192a^2 - 76104013563/2048a - 8372531527/2048, 815896071/2228224a^20 - 316392885/1114112a^19 - 4774201803/557056a^18 + 12975363/16384a^17 + 108915111969/2228224a^16 + 31150438067/1114112a^15 + 374368510413/2228224a^14 + 114650088099/557056a^13 - 4154799602947/2228224a^12 - 4061888302959/1114112a^11 + 53592514443/278528a^10 + 1640459987311/278528a^9 + 7568216663409/557056a^8 + 7437880048071/278528a^7 + 39847526287555/2228224a^6 - 18346567307841/557056a^5 - 2602410309459/32768a^4 - 5111359098939/69632a^3 - 295653796227/8192a^2 - 19158686433/2048a - 2068409261/2048, 129140163/262144a^20 - 43046721/131072a^19 - 760492071/65536a^18 - 4782969/32768a^17 + 17459431173/262144a^16 + 5802804279/131072a^15 + 59286672225/262144a^14 + 19756555371/65536a^13 - 653573326095/262144a^12 - 679924637811/131072a^11 - 3486504465/32768a^10 + 267707371275/32768a^9 + 1244782279989/65536a^8 + 1228510287675/32768a^7 + 7087858288623/262144a^6 - 2848702929345/65536a^5 - 7351303729623/65536a^4 - 878282252659/8192a^3 - 889130022255/16384a^2 - 59275334817/4096a - 6586144217/4096, a + 1, 46916728039/4456448a^20 - 16281730653/2228224a^19 - 275848684347/1114112a^18 + 1894597931/557056a^17 + 6323116620609/4456448a^16 + 2028213344091/2228224a^15 + 21535001276525/4456448a^14 + 7030680587351/1114112a^13 - 237820132893571/4456448a^12 - 243632606528199/2228224a^11 - 165286196445/557056a^10 + 96524333463215/557056a^9 + 447925817405425/1114112a^8 + 441625053986383/557056a^7 + 2503397903895331/4456448a^6 - 1040075936434165/1114112a^5 - 2638791961423931/1114112a^4 - 312720536456191/139264a^3 - 18494901183635/16384a^2 - 1224925329637/4096a - 135221449797/4096, 153354915413/4456448a^20 - 55285747707/2228224a^19 - 900204189049/1114112a^18 + 17825441165/557056a^17 + 20600604912547/4456448a^16 + 6375060748885/2228224a^15 + 70395357412647/4456448a^14 + 22505906661135/1114112a^13 - 778479439824441/4456448a^12 - 785477954433201/2228224a^11 + 2891157494005/557056a^10 + 313078433184957/557056a^9 + 1450564269596387/1114112a^8 + 1428616949296993/557056a^7 + 7956390708127625/4456448a^6 - 3414524392976137/1114112a^5 - 8523114216000925/1114112a^4 - 1002210250163343/139264a^3 - 58884243859377/16384a^2 - 3875778759745/4096a - 425260682763/4096, 13626143809/2228224a^20 - 4966113527/1114112a^19 - 79954354685/557056a^18 + 1892326121/278528a^17 + 1829168282839/2228224a^16 + 559490867369/1114112a^15 + 6252597718443/2228224a^14 + 1987740969255/557056a^13 - 69211436782037/2228224a^12 - 69509241619461/1114112a^11 + 361165816037/278528a^10 + 27764912683185/278528a^9 + 128496369455383/557056a^8 + 126524177798949/278528a^7 + 41207578360789/131072a^6 - 304011839753289/557056a^5 - 44387756885833/32768a^4 - 88464153520047/69632a^3 - 5182968764781/8192a^2 - 340127523841/2048a - 37198258095/2048, 4014028195/557056a^20 - 1292687511/278528a^19 - 23671551759/139264a^18 - 402205059/69632a^17 + 544297656805/557056a^16 + 185853649277/278528a^15 + 1842059989041/557056a^14 + 312308331445/69632a^13 - 20293282158423/557056a^12 - 21372193169501/278528a^11 - 179622169241/69632a^10 + 8376654600655/69632a^9 + 38978027812525/139264a^8 + 38505909809545/69632a^7 + 225133154060047/557056a^6 - 11034903265877/17408a^5 - 230721104340629/139264a^4 - 6929993164385/4352a^3 - 1658012939699/2048a^2 - 13870480287/64a - 12372919927/512, 1618592681/4456448a^20 - 589859023/2228224a^19 - 558567597/65536a^18 + 222580041/557056a^17 + 217127776047/4456448a^16 + 66580349393/2228224a^15 + 743448008707/4456448a^14 + 235992531439/1114112a^13 - 8217551544189/4456448a^12 - 8257334779661/2228224a^11 + 38103546185/557056a^10 + 3295505074177/557056a^9 + 15276035864335/1114112a^8 + 15037118365037/557056a^7 + 83386002343597/4456448a^6 - 2120143694945/65536a^5 - 89660621925785/1114112a^4 - 10528994928719/139264a^3 - 618126902085/16384a^2 - 40667429753/4096a - 4461812319/4096, 81800334245/4456448a^20 - 28300439771/2228224a^19 - 481011162185/1114112a^18 + 2814763021/557056a^17 + 11027487529939/4456448a^16 + 3546849013333/2228224a^15 + 37545826994455/4456448a^14 + 12278307469495/1114112a^13 - 414600771696393/4456448a^12 - 425236844540561/2228224a^11 - 428193834875/557056a^10 + 168397062124845/557056a^9 + 781527481619923/1114112a^8 + 770604217231457/557056a^7 + 4374109085373977/4456448a^6 - 1812835758001809/1114112a^5 - 270886731944973/65536a^4 - 546057027582631/139264a^3 - 32310118280417/16384a^2 - 2140842461321/4096a - 236428732715/4096, 93137757103/4456448a^20 - 33984886633/2228224a^19 - 546425626443/1114112a^18 + 13109219247/557056a^17 + 12497031816121/4456448a^16 + 3822342368791/2228224a^15 + 2515285767989/262144a^14 + 13573911307553/1114112a^13 - 27823158273163/262144a^12 - 474902469863003/2228224a^11 + 2399318074679/557056a^10 + 189643791901927/557056a^9 + 878392464344025/1114112a^8 + 864760670136667/557056a^7 + 4788694395011595/4456448a^6 - 2076182610881007/1114112a^5 - 5156366622942559/1114112a^4 - 604883238329761/139264a^3 - 35470333425651/16384a^2 - 2330490341959/4096a - 255271871785/4096, 27910026897/557056a^20 - 19519562363/557056a^19 - 20506144213/17408a^18 + 3094099045/139264a^17 + 3759367505943/557056a^16 + 2394495314567/557056a^15 + 753512430291/32768a^14 + 16662537672957/557056a^13 - 8324828227671/32768a^12 - 289086535341805/557056a^11 + 65519806995/139264a^10 + 57338944563305/69632a^9 + 265957228830845/139264a^8 + 524336383652059/139264a^7 + 1480832194543701/557056a^6 - 2477525906069477/557056a^5 - 3132157831698733/278528a^4 - 741125928043257/69632a^3 - 1367716082329/256a^2 - 2894488123097/2048a - 159526361965/1024, 179359337319/2228224a^20 - 64244209311/1114112a^19 - 1053142761367/557056a^18 + 18502648713/278528a^17 + 24107301112321/2228224a^16 + 7507442184877/1114112a^15 + 82332227555597/2228224a^14 + 6604465670447/139264a^13 - 910256263823531/2228224a^12 - 920862163654245/1114112a^11 + 2685101666013/278528a^10 + 366651883540079/278528a^9 + 1699281145134761/557056a^8 + 1673880738731193/278528a^7 + 9351341288652835/2228224a^6 - 1995223195275709/278528a^5 - 9988983484752137/557056a^4 - 147025206481349/8704a^3 - 69188487905999/8192a^2 - 2279571779087/1024a - 500800038267/2048 (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:	$ 166388749255000 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{4}\cdot 166388749255000 \cdot 1}{2\cdot\sqrt{10423848562958827406082788615011739523268608}}\cr\approx \mathstrut & 0.328995025914177 \end{aligned}\] (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^21 - 24*x^19 - 16*x^18 + 135*x^17 + 180*x^16 + 519*x^15 + 918*x^14 - 4653*x^13 - 13904*x^12 - 7236*x^11 + 16440*x^10 + 49612*x^9 + 101808*x^8 + 105621*x^7 - 51646*x^6 - 286524*x^5 - 369432*x^4 - 255248*x^3 - 102816*x^2 - 22848*x - 2176) 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^21 - 24*x^19 - 16*x^18 + 135*x^17 + 180*x^16 + 519*x^15 + 918*x^14 - 4653*x^13 - 13904*x^12 - 7236*x^11 + 16440*x^10 + 49612*x^9 + 101808*x^8 + 105621*x^7 - 51646*x^6 - 286524*x^5 - 369432*x^4 - 255248*x^3 - 102816*x^2 - 22848*x - 2176, 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^21 - 24*x^19 - 16*x^18 + 135*x^17 + 180*x^16 + 519*x^15 + 918*x^14 - 4653*x^13 - 13904*x^12 - 7236*x^11 + 16440*x^10 + 49612*x^9 + 101808*x^8 + 105621*x^7 - 51646*x^6 - 286524*x^5 - 369432*x^4 - 255248*x^3 - 102816*x^2 - 22848*x - 2176); 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^21 - 24*x^19 - 16*x^18 + 135*x^17 + 180*x^16 + 519*x^15 + 918*x^14 - 4653*x^13 - 13904*x^12 - 7236*x^11 + 16440*x^10 + 49612*x^9 + 101808*x^8 + 105621*x^7 - 51646*x^6 - 286524*x^5 - 369432*x^4 - 255248*x^3 - 102816*x^2 - 22848*x - 2176); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_3^7:C_2\wr C_7$ (as 21T123):

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 1959552

The 333 conjugacy class representatives for $C_3^7:C_2\wr C_7$

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

Intermediate fields

7.7.594823321.1

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(b)]

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

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

Sibling fields

Degree 42 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$	${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }$	${\href{/padicField/11.7.0.1}{7} }^{3}$	$21$	R	${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$	$21$	R	${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.7.0.1}{7} }$	$21$	${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{12}$	${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$	${\href{/padicField/47.7.0.1}{7} }^{3}$	${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }$	${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{9}$

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	2.7.1.0a1.1	$x^{7} + x + 1$	$1$	$7$	$0$	$C_7$	$$[\ ]^{7}$$
$2$ 2	2.7.2.14a11.1	$x^{14} + 2 x^{12} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + x^{2} + 2 x + 3$	$2$	$7$	$14$	14T9	$$[2, 2, 2, 2]^{7}$$
$3$ 3	3.7.3.21a17.1	$x^{21} + 6 x^{16} + 3 x^{14} + 12 x^{11} + 3 x^{10} + 18 x^{9} + 3 x^{7} + 8 x^{6} + 6 x^{5} + 24 x^{4} + 3 x^{3} + 12 x^{2} + 4$	$3$	$7$	$21$	not computed	not computed
$17$ 17	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	17.2.1.0a1.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	17.2.1.0a1.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	17.1.3.2a1.1	$x^{3} + 17$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
	17.3.1.0a1.1	$x^{3} + x + 14$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
	17.3.1.0a1.1	$x^{3} + x + 14$	$1$	$3$	$0$	$C_3$	$$[\ ]^{3}$$
$29$ 29	29.1.7.6a1.1	$x^{7} + 29$	$7$	$1$	$6$	$C_7$	$$[\ ]_{7}$$
$29$ 29	29.2.7.12a1.2	$x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970880 x^{7} + 2815755824 x^{6} + 684383616 x^{5} + 93865632 x^{4} + 7773696 x^{3} + 387520 x^{2} + 10752 x + 157$	$7$	$2$	$12$	$C_{14}$	$$[\ ]_{7}^{2}$$

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