LMFDB - Number field 21.1.1261412107834594448324876441.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 1)

gp: K = bnfinit(y^21 + 3*y^19 - y^18 + y^17 + 8*y^16 - 7*y^15 + 24*y^14 - 15*y^13 + 7*y^12 + 25*y^11 - 42*y^10 + 46*y^9 - 39*y^8 - 12*y^7 + 15*y^6 - 14*y^5 + y^4 + 10*y^3 - y^2 + 2*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 1)

$ x^{21} + 3 x^{19} - x^{18} + x^{17} + 8 x^{16} - 7 x^{15} + 24 x^{14} - 15 x^{13} + 7 x^{12} + 25 x^{11} + \cdots + 1 $ x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 1

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:	$[1, 10]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$1261412107834594448324876441$ 1261412107834594448324876441 $\medspace = 31^{7}\cdot 71^{9}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$19.52$	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:	$31^{1/2}71^{1/2}\approx 46.9148164229596$
Ramified primes:	$31$, $71$ 31, 71	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{2201}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{299}a^{19}+\frac{8}{23}a^{18}+\frac{122}{299}a^{17}-\frac{79}{299}a^{16}-\frac{41}{299}a^{15}+\frac{19}{299}a^{14}+\frac{119}{299}a^{13}-\frac{81}{299}a^{12}+\frac{132}{299}a^{11}-\frac{64}{299}a^{10}+\frac{120}{299}a^{9}+\frac{77}{299}a^{8}-\frac{4}{23}a^{7}+\frac{11}{299}a^{6}+\frac{40}{299}a^{5}+\frac{128}{299}a^{4}+\frac{131}{299}a^{3}+\frac{75}{299}a^{2}+\frac{142}{299}a+\frac{58}{299}$, $\frac{1}{31329330929}a^{20}+\frac{10588327}{31329330929}a^{19}-\frac{15216296701}{31329330929}a^{18}+\frac{10121876874}{31329330929}a^{17}-\frac{361413200}{1362144823}a^{16}+\frac{4705840582}{31329330929}a^{15}+\frac{5011990975}{31329330929}a^{14}-\frac{1631283704}{31329330929}a^{13}-\frac{9509682454}{31329330929}a^{12}+\frac{13199176568}{31329330929}a^{11}+\frac{982873072}{31329330929}a^{10}-\frac{1044585243}{2409948533}a^{9}+\frac{12174708862}{31329330929}a^{8}-\frac{13010489871}{31329330929}a^{7}-\frac{12732688830}{31329330929}a^{6}+\frac{5435101835}{31329330929}a^{5}-\frac{1553313246}{31329330929}a^{4}+\frac{6110604571}{31329330929}a^{3}-\frac{357606914}{31329330929}a^{2}-\frac{11723869589}{31329330929}a+\frac{13268663735}{31329330929}$ 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, a^15, a^16, a^17, a^18, 1/299*a^19 + 8/23*a^18 + 122/299*a^17 - 79/299*a^16 - 41/299*a^15 + 19/299*a^14 + 119/299*a^13 - 81/299*a^12 + 132/299*a^11 - 64/299*a^10 + 120/299*a^9 + 77/299*a^8 - 4/23*a^7 + 11/299*a^6 + 40/299*a^5 + 128/299*a^4 + 131/299*a^3 + 75/299*a^2 + 142/299*a + 58/299, 1/31329330929*a^20 + 10588327/31329330929*a^19 - 15216296701/31329330929*a^18 + 10121876874/31329330929*a^17 - 361413200/1362144823*a^16 + 4705840582/31329330929*a^15 + 5011990975/31329330929*a^14 - 1631283704/31329330929*a^13 - 9509682454/31329330929*a^12 + 13199176568/31329330929*a^11 + 982873072/31329330929*a^10 - 1044585243/2409948533*a^9 + 12174708862/31329330929*a^8 - 13010489871/31329330929*a^7 - 12732688830/31329330929*a^6 + 5435101835/31329330929*a^5 - 1553313246/31329330929*a^4 + 6110604571/31329330929*a^3 - 357606914/31329330929*a^2 - 11723869589/31329330929*a + 13268663735/31329330929

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

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:	$10$	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:	$\frac{4563752185}{31329330929}a^{20}+\frac{9376610774}{31329330929}a^{19}+\frac{15390398243}{31329330929}a^{18}+\frac{20865861833}{31329330929}a^{17}-\frac{1365321368}{31329330929}a^{16}+\frac{36502357523}{31329330929}a^{15}+\frac{44278878928}{31329330929}a^{14}+\frac{57456225379}{31329330929}a^{13}+\frac{125253087898}{31329330929}a^{12}-\frac{64896560650}{31329330929}a^{11}+\frac{8095431879}{2409948533}a^{10}+\frac{64248911235}{31329330929}a^{9}-\frac{5755513301}{1362144823}a^{8}+\frac{125100013041}{31329330929}a^{7}-\frac{286721270684}{31329330929}a^{6}-\frac{157355245410}{31329330929}a^{5}+\frac{96194583689}{31329330929}a^{4}-\frac{27878235168}{31329330929}a^{3}+\frac{14550121054}{31329330929}a^{2}+\frac{89467270444}{31329330929}a+\frac{3693131261}{31329330929}$, $\frac{3701249904}{31329330929}a^{20}+\frac{90913567}{2409948533}a^{19}+\frac{12416450363}{31329330929}a^{18}-\frac{2799243279}{31329330929}a^{17}+\frac{4631520970}{31329330929}a^{16}+\frac{21454994463}{31329330929}a^{15}-\frac{18822068250}{31329330929}a^{14}+\frac{88577435723}{31329330929}a^{13}-\frac{62297932643}{31329330929}a^{12}+\frac{39369548843}{31329330929}a^{11}+\frac{27521414820}{31329330929}a^{10}-\frac{130794939692}{31329330929}a^{9}+\frac{11961218129}{2409948533}a^{8}-\frac{242713873573}{31329330929}a^{7}+\frac{15879364333}{31329330929}a^{6}-\frac{68494282144}{31329330929}a^{5}-\frac{62993751859}{31329330929}a^{4}+\frac{112809092198}{31329330929}a^{3}-\frac{12472315181}{31329330929}a^{2}+\frac{22526000438}{31329330929}a+\frac{3925723557}{2409948533}$, $\frac{1550285457}{31329330929}a^{20}-\frac{2537097573}{31329330929}a^{19}+\frac{10439822865}{31329330929}a^{18}-\frac{9419717090}{31329330929}a^{17}+\frac{20752842121}{31329330929}a^{16}+\frac{2210799582}{31329330929}a^{15}-\frac{28836441552}{31329330929}a^{14}+\frac{97708478817}{31329330929}a^{13}-\frac{131023662182}{31329330929}a^{12}+\frac{184055311894}{31329330929}a^{11}-\frac{77269160545}{31329330929}a^{10}-\frac{102634013609}{31329330929}a^{9}+\frac{305559106759}{31329330929}a^{8}-\frac{445746092284}{31329330929}a^{7}+\frac{26419696949}{2409948533}a^{6}-\frac{193413070548}{31329330929}a^{5}-\frac{9749646935}{2409948533}a^{4}+\frac{119267090499}{31329330929}a^{3}-\frac{46464807144}{31329330929}a^{2}+\frac{10852005353}{31329330929}a+\frac{44953602127}{31329330929}$, $\frac{3082306557}{31329330929}a^{20}+\frac{6406824785}{31329330929}a^{19}+\frac{10500154183}{31329330929}a^{18}+\frac{13927383138}{31329330929}a^{17}-\frac{764963206}{31329330929}a^{16}+\frac{23729651852}{31329330929}a^{15}+\frac{1372781560}{1362144823}a^{14}+\frac{39910840286}{31329330929}a^{13}+\frac{84669917338}{31329330929}a^{12}-\frac{41868882199}{31329330929}a^{11}+\frac{63678360433}{31329330929}a^{10}+\frac{49670355487}{31329330929}a^{9}-\frac{89000205447}{31329330929}a^{8}+\frac{83444145333}{31329330929}a^{7}-\frac{194281209787}{31329330929}a^{6}-\frac{8258356119}{2409948533}a^{5}+\frac{64620711114}{31329330929}a^{4}-\frac{19160260436}{31329330929}a^{3}+\frac{9771313920}{31329330929}a^{2}+\frac{15371689821}{31329330929}a+\frac{17203434425}{31329330929}$, $\frac{6780886188}{31329330929}a^{20}-\frac{6449821931}{31329330929}a^{19}+\frac{24683793956}{31329330929}a^{18}-\frac{22271499439}{31329330929}a^{17}+\frac{27797595642}{31329330929}a^{16}+\frac{53031899702}{31329330929}a^{15}-\frac{95893666031}{31329330929}a^{14}+\frac{239947216722}{31329330929}a^{13}-\frac{256146977329}{31329330929}a^{12}+\frac{235628744270}{31329330929}a^{11}+\frac{9710469316}{2409948533}a^{10}-\frac{440138297722}{31329330929}a^{9}+\frac{671264132819}{31329330929}a^{8}-\frac{635759664250}{31329330929}a^{7}+\frac{273215047455}{31329330929}a^{6}+\frac{81482309360}{31329330929}a^{5}-\frac{307542033904}{31329330929}a^{4}+\frac{73082365407}{31329330929}a^{3}+\frac{65859848220}{31329330929}a^{2}-\frac{24975856534}{31329330929}a+\frac{54459322941}{31329330929}$, $\frac{27620476372}{31329330929}a^{20}-\frac{250651304}{2409948533}a^{19}+\frac{72084602357}{31329330929}a^{18}-\frac{38383355174}{31329330929}a^{17}-\frac{383339869}{31329330929}a^{16}+\frac{225424603959}{31329330929}a^{15}-\frac{226083758560}{31329330929}a^{14}+\frac{596073322561}{31329330929}a^{13}-\frac{424742770124}{31329330929}a^{12}-\frac{4927230651}{31329330929}a^{11}+\frac{796039751258}{31329330929}a^{10}-\frac{1280809679053}{31329330929}a^{9}+\frac{84990951084}{2409948533}a^{8}-\frac{796764948333}{31329330929}a^{7}-\frac{653776910792}{31329330929}a^{6}+\frac{781150377602}{31329330929}a^{5}-\frac{235393791625}{31329330929}a^{4}-\frac{136836044655}{31329330929}a^{3}+\frac{16762040393}{1362144823}a^{2}-\frac{54291286014}{31329330929}a-\frac{3101378617}{2409948533}$, $\frac{5231421359}{31329330929}a^{20}+\frac{8931044714}{31329330929}a^{19}+\frac{14283890986}{31329330929}a^{18}+\frac{23448614086}{31329330929}a^{17}-\frac{6247694619}{31329330929}a^{16}+\frac{60412008067}{31329330929}a^{15}+\frac{36333962632}{31329330929}a^{14}+\frac{59751666575}{31329330929}a^{13}+\frac{157566563175}{31329330929}a^{12}-\frac{125278792344}{31329330929}a^{11}+\frac{263606878202}{31329330929}a^{10}+\frac{29289621}{2409948533}a^{9}-\frac{124237162511}{31329330929}a^{8}+\frac{283455474988}{31329330929}a^{7}-\frac{486116948154}{31329330929}a^{6}+\frac{69115428802}{31329330929}a^{5}+\frac{46844977329}{31329330929}a^{4}-\frac{138180296115}{31329330929}a^{3}+\frac{41927557487}{31329330929}a^{2}+\frac{47475498215}{31329330929}a-\frac{28971137075}{31329330929}$, $\frac{16212035159}{31329330929}a^{20}+\frac{1390243219}{31329330929}a^{19}+\frac{37262945565}{31329330929}a^{18}-\frac{20641922707}{31329330929}a^{17}-\frac{17903462425}{31329330929}a^{16}+\frac{123803585250}{31329330929}a^{15}-\frac{98290354920}{31329330929}a^{14}+\frac{296996797387}{31329330929}a^{13}-\frac{15460760581}{2409948533}a^{12}-\frac{112195938351}{31329330929}a^{11}+\frac{422745174630}{31329330929}a^{10}-\frac{589389478488}{31329330929}a^{9}+\frac{454860902404}{31329330929}a^{8}-\frac{351475991571}{31329330929}a^{7}-\frac{403513067549}{31329330929}a^{6}+\frac{378186714072}{31329330929}a^{5}+\frac{126338759852}{31329330929}a^{4}+\frac{39952584259}{31329330929}a^{3}+\frac{35185694767}{31329330929}a^{2}-\frac{38973855302}{31329330929}a-\frac{31249368417}{31329330929}$, $\frac{25515364744}{31329330929}a^{20}+\frac{24219627909}{31329330929}a^{19}+\frac{76313362445}{31329330929}a^{18}+\frac{39293613717}{31329330929}a^{17}-\frac{675778357}{31329330929}a^{16}+\frac{205314777202}{31329330929}a^{15}+\frac{1618509012}{2409948533}a^{14}+\frac{435391303485}{31329330929}a^{13}+\frac{142805086798}{31329330929}a^{12}-\frac{147739136840}{31329330929}a^{11}+\frac{636904011555}{31329330929}a^{10}-\frac{366427801309}{31329330929}a^{9}+\frac{108686212912}{31329330929}a^{8}-\frac{20654230231}{31329330929}a^{7}-\frac{994975429571}{31329330929}a^{6}-\frac{7524628015}{1362144823}a^{5}+\frac{291795133598}{31329330929}a^{4}-\frac{244745135844}{31329330929}a^{3}+\frac{21213911814}{2409948533}a^{2}+\frac{203447539020}{31329330929}a+\frac{35992145828}{31329330929}$, $\frac{14878734018}{31329330929}a^{20}-\frac{13642434279}{31329330929}a^{19}+\frac{34427353518}{31329330929}a^{18}-\frac{51787326158}{31329330929}a^{17}+\frac{6276801527}{31329330929}a^{16}+\frac{129199846831}{31329330929}a^{15}-\frac{15755359932}{2409948533}a^{14}+\frac{370436903693}{31329330929}a^{13}-\frac{444636732694}{31329330929}a^{12}+\frac{104989956519}{31329330929}a^{11}+\frac{486575005419}{31329330929}a^{10}-\frac{928719539836}{31329330929}a^{9}+\frac{958789100691}{31329330929}a^{8}-\frac{675286833933}{31329330929}a^{7}-\frac{46237180284}{31329330929}a^{6}+\frac{698698397498}{31329330929}a^{5}-\frac{201603452852}{31329330929}a^{4}-\frac{153015015980}{31329330929}a^{3}+\frac{12733688094}{2409948533}a^{2}-\frac{78023261709}{31329330929}a-\frac{2599377288}{1362144823}$ 4563752185/31329330929a^20 + 9376610774/31329330929a^19 + 15390398243/31329330929a^18 + 20865861833/31329330929a^17 - 1365321368/31329330929a^16 + 36502357523/31329330929a^15 + 44278878928/31329330929a^14 + 57456225379/31329330929a^13 + 125253087898/31329330929a^12 - 64896560650/31329330929a^11 + 8095431879/2409948533a^10 + 64248911235/31329330929a^9 - 5755513301/1362144823a^8 + 125100013041/31329330929a^7 - 286721270684/31329330929a^6 - 157355245410/31329330929a^5 + 96194583689/31329330929a^4 - 27878235168/31329330929a^3 + 14550121054/31329330929a^2 + 89467270444/31329330929a + 3693131261/31329330929, 3701249904/31329330929a^20 + 90913567/2409948533a^19 + 12416450363/31329330929a^18 - 2799243279/31329330929a^17 + 4631520970/31329330929a^16 + 21454994463/31329330929a^15 - 18822068250/31329330929a^14 + 88577435723/31329330929a^13 - 62297932643/31329330929a^12 + 39369548843/31329330929a^11 + 27521414820/31329330929a^10 - 130794939692/31329330929a^9 + 11961218129/2409948533a^8 - 242713873573/31329330929a^7 + 15879364333/31329330929a^6 - 68494282144/31329330929a^5 - 62993751859/31329330929a^4 + 112809092198/31329330929a^3 - 12472315181/31329330929a^2 + 22526000438/31329330929a + 3925723557/2409948533, 1550285457/31329330929a^20 - 2537097573/31329330929a^19 + 10439822865/31329330929a^18 - 9419717090/31329330929a^17 + 20752842121/31329330929a^16 + 2210799582/31329330929a^15 - 28836441552/31329330929a^14 + 97708478817/31329330929a^13 - 131023662182/31329330929a^12 + 184055311894/31329330929a^11 - 77269160545/31329330929a^10 - 102634013609/31329330929a^9 + 305559106759/31329330929a^8 - 445746092284/31329330929a^7 + 26419696949/2409948533a^6 - 193413070548/31329330929a^5 - 9749646935/2409948533a^4 + 119267090499/31329330929a^3 - 46464807144/31329330929a^2 + 10852005353/31329330929a + 44953602127/31329330929, 3082306557/31329330929a^20 + 6406824785/31329330929a^19 + 10500154183/31329330929a^18 + 13927383138/31329330929a^17 - 764963206/31329330929a^16 + 23729651852/31329330929a^15 + 1372781560/1362144823a^14 + 39910840286/31329330929a^13 + 84669917338/31329330929a^12 - 41868882199/31329330929a^11 + 63678360433/31329330929a^10 + 49670355487/31329330929a^9 - 89000205447/31329330929a^8 + 83444145333/31329330929a^7 - 194281209787/31329330929a^6 - 8258356119/2409948533a^5 + 64620711114/31329330929a^4 - 19160260436/31329330929a^3 + 9771313920/31329330929a^2 + 15371689821/31329330929a + 17203434425/31329330929, 6780886188/31329330929a^20 - 6449821931/31329330929a^19 + 24683793956/31329330929a^18 - 22271499439/31329330929a^17 + 27797595642/31329330929a^16 + 53031899702/31329330929a^15 - 95893666031/31329330929a^14 + 239947216722/31329330929a^13 - 256146977329/31329330929a^12 + 235628744270/31329330929a^11 + 9710469316/2409948533a^10 - 440138297722/31329330929a^9 + 671264132819/31329330929a^8 - 635759664250/31329330929a^7 + 273215047455/31329330929a^6 + 81482309360/31329330929a^5 - 307542033904/31329330929a^4 + 73082365407/31329330929a^3 + 65859848220/31329330929a^2 - 24975856534/31329330929a + 54459322941/31329330929, 27620476372/31329330929a^20 - 250651304/2409948533a^19 + 72084602357/31329330929a^18 - 38383355174/31329330929a^17 - 383339869/31329330929a^16 + 225424603959/31329330929a^15 - 226083758560/31329330929a^14 + 596073322561/31329330929a^13 - 424742770124/31329330929a^12 - 4927230651/31329330929a^11 + 796039751258/31329330929a^10 - 1280809679053/31329330929a^9 + 84990951084/2409948533a^8 - 796764948333/31329330929a^7 - 653776910792/31329330929a^6 + 781150377602/31329330929a^5 - 235393791625/31329330929a^4 - 136836044655/31329330929a^3 + 16762040393/1362144823a^2 - 54291286014/31329330929a - 3101378617/2409948533, 5231421359/31329330929a^20 + 8931044714/31329330929a^19 + 14283890986/31329330929a^18 + 23448614086/31329330929a^17 - 6247694619/31329330929a^16 + 60412008067/31329330929a^15 + 36333962632/31329330929a^14 + 59751666575/31329330929a^13 + 157566563175/31329330929a^12 - 125278792344/31329330929a^11 + 263606878202/31329330929a^10 + 29289621/2409948533a^9 - 124237162511/31329330929a^8 + 283455474988/31329330929a^7 - 486116948154/31329330929a^6 + 69115428802/31329330929a^5 + 46844977329/31329330929a^4 - 138180296115/31329330929a^3 + 41927557487/31329330929a^2 + 47475498215/31329330929a - 28971137075/31329330929, 16212035159/31329330929a^20 + 1390243219/31329330929a^19 + 37262945565/31329330929a^18 - 20641922707/31329330929a^17 - 17903462425/31329330929a^16 + 123803585250/31329330929a^15 - 98290354920/31329330929a^14 + 296996797387/31329330929a^13 - 15460760581/2409948533a^12 - 112195938351/31329330929a^11 + 422745174630/31329330929a^10 - 589389478488/31329330929a^9 + 454860902404/31329330929a^8 - 351475991571/31329330929a^7 - 403513067549/31329330929a^6 + 378186714072/31329330929a^5 + 126338759852/31329330929a^4 + 39952584259/31329330929a^3 + 35185694767/31329330929a^2 - 38973855302/31329330929a - 31249368417/31329330929, 25515364744/31329330929a^20 + 24219627909/31329330929a^19 + 76313362445/31329330929a^18 + 39293613717/31329330929a^17 - 675778357/31329330929a^16 + 205314777202/31329330929a^15 + 1618509012/2409948533a^14 + 435391303485/31329330929a^13 + 142805086798/31329330929a^12 - 147739136840/31329330929a^11 + 636904011555/31329330929a^10 - 366427801309/31329330929a^9 + 108686212912/31329330929a^8 - 20654230231/31329330929a^7 - 994975429571/31329330929a^6 - 7524628015/1362144823a^5 + 291795133598/31329330929a^4 - 244745135844/31329330929a^3 + 21213911814/2409948533a^2 + 203447539020/31329330929a + 35992145828/31329330929, 14878734018/31329330929a^20 - 13642434279/31329330929a^19 + 34427353518/31329330929a^18 - 51787326158/31329330929a^17 + 6276801527/31329330929a^16 + 129199846831/31329330929a^15 - 15755359932/2409948533a^14 + 370436903693/31329330929a^13 - 444636732694/31329330929a^12 + 104989956519/31329330929a^11 + 486575005419/31329330929a^10 - 928719539836/31329330929a^9 + 958789100691/31329330929a^8 - 675286833933/31329330929a^7 - 46237180284/31329330929a^6 + 698698397498/31329330929a^5 - 201603452852/31329330929a^4 - 153015015980/31329330929a^3 + 12733688094/2409948533a^2 - 78023261709/31329330929a - 2599377288/1362144823 (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:	$ 80948.9311635 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 80948.9311635 \cdot 1}{2\cdot\sqrt{1261412107834594448324876441}}\cr\approx \mathstrut & 0.218565345970 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 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))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 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)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 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)));

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

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 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

$S_3\times D_7$ (as 21T8):

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 84

The 15 conjugacy class representatives for $S_3\times D_7$

Character table for $S_3\times D_7$

Intermediate fields

3.1.31.1, 7.1.357911.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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	$21$	${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.7.0.1}{7} }$	$21$	${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$	${\href{/padicField/11.2.0.1}{2} }^{10}{,}\,{\href{/padicField/11.1.0.1}{1} }$	${\href{/padicField/13.2.0.1}{2} }^{10}{,}\,{\href{/padicField/13.1.0.1}{1} }$	${\href{/padicField/17.2.0.1}{2} }^{10}{,}\,{\href{/padicField/17.1.0.1}{1} }$	$21$	${\href{/padicField/23.2.0.1}{2} }^{10}{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.7.0.1}{7} }$	R	${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$	${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$	${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$	${\href{/padicField/47.2.0.1}{2} }^{9}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$	${\href{/padicField/53.2.0.1}{2} }^{10}{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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
$31$ 31	$\Q_{31}$	$x + 28$	$1$	$1$	$0$	Trivial	$[\ ]$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.4.2.1	$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	31.4.2.1	$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	31.4.2.1	$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$71$ 71	71.3.0.1	$x^{3} + 4 x + 64$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	71.6.3.2	$x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	71.6.3.2	$x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	71.6.3.2	$x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

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