LMFDB - Number field 21.5.34815740375535437995421466177.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 5*x^18 + 12*x^17 - 5*x^16 + 4*x^15 - 27*x^14 + 16*x^13 - 8*x^12 + 8*x^11 - 22*x^10 - 8*x^9 - 2*x^8 + 9*x^7 + 15*x^6 + 12*x^5 + 13*x^4 + 2*x^3 - x^2 - 2*x - 1)

gp: K = bnfinit(y^21 - 2*y^20 - 5*y^18 + 12*y^17 - 5*y^16 + 4*y^15 - 27*y^14 + 16*y^13 - 8*y^12 + 8*y^11 - 22*y^10 - 8*y^9 - 2*y^8 + 9*y^7 + 15*y^6 + 12*y^5 + 13*y^4 + 2*y^3 - y^2 - 2*y - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^20 - 5*x^18 + 12*x^17 - 5*x^16 + 4*x^15 - 27*x^14 + 16*x^13 - 8*x^12 + 8*x^11 - 22*x^10 - 8*x^9 - 2*x^8 + 9*x^7 + 15*x^6 + 12*x^5 + 13*x^4 + 2*x^3 - x^2 - 2*x - 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 2*x^20 - 5*x^18 + 12*x^17 - 5*x^16 + 4*x^15 - 27*x^14 + 16*x^13 - 8*x^12 + 8*x^11 - 22*x^10 - 8*x^9 - 2*x^8 + 9*x^7 + 15*x^6 + 12*x^5 + 13*x^4 + 2*x^3 - x^2 - 2*x - 1)

$ x^{21} - 2 x^{20} - 5 x^{18} + 12 x^{17} - 5 x^{16} + 4 x^{15} - 27 x^{14} + 16 x^{13} - 8 x^{12} + \cdots - 1 $ x^21 - 2*x^20 - 5*x^18 + 12*x^17 - 5*x^16 + 4*x^15 - 27*x^14 + 16*x^13 - 8*x^12 + 8*x^11 - 22*x^10 - 8*x^9 - 2*x^8 + 9*x^7 + 15*x^6 + 12*x^5 + 13*x^4 + 2*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:	$[5, 8]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$34815740375535437995421466177$ 34815740375535437995421466177 $\medspace = 3^{8}\cdot 37^{5}\cdot 2381^{5}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$22.86$	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:	$3^{1/2}37^{1/2}2381^{1/2}\approx 514.0924041454026$
Ramified primes:	$3$, $37$, $2381$ 3, 37, 2381	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{88097}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{17}-\frac{1}{3}a^{15}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{17}-\frac{1}{3}a^{16}-\frac{1}{3}a^{15}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{473307984537}a^{20}-\frac{12327440764}{157769328179}a^{19}-\frac{11630914336}{473307984537}a^{18}-\frac{84069340606}{473307984537}a^{17}-\frac{204542939242}{473307984537}a^{16}-\frac{15068508567}{157769328179}a^{15}-\frac{33293914006}{473307984537}a^{14}+\frac{162228257632}{473307984537}a^{13}-\frac{132410039999}{473307984537}a^{12}-\frac{77420821297}{473307984537}a^{11}+\frac{125959066994}{473307984537}a^{10}-\frac{235963177199}{473307984537}a^{9}+\frac{63132015346}{473307984537}a^{8}+\frac{82440484031}{473307984537}a^{7}-\frac{73393694513}{157769328179}a^{6}+\frac{4373278420}{473307984537}a^{5}-\frac{111945114004}{473307984537}a^{4}-\frac{90021523508}{473307984537}a^{3}-\frac{190660763158}{473307984537}a^{2}-\frac{183673740334}{473307984537}a-\frac{51957939668}{157769328179}$ 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, 1/3*a^18 - 1/3*a^17 - 1/3*a^15 - 1/3*a^12 - 1/3*a^11 - 1/3*a^10 - 1/3*a^8 - 1/3*a^7 - 1/3*a^6 + 1/3*a^3 + 1/3*a^2 - 1/3, 1/3*a^19 - 1/3*a^17 - 1/3*a^16 - 1/3*a^15 - 1/3*a^13 + 1/3*a^12 + 1/3*a^11 - 1/3*a^10 - 1/3*a^9 + 1/3*a^8 + 1/3*a^7 - 1/3*a^6 + 1/3*a^4 - 1/3*a^3 + 1/3*a^2 - 1/3*a - 1/3, 1/473307984537*a^20 - 12327440764/157769328179*a^19 - 11630914336/473307984537*a^18 - 84069340606/473307984537*a^17 - 204542939242/473307984537*a^16 - 15068508567/157769328179*a^15 - 33293914006/473307984537*a^14 + 162228257632/473307984537*a^13 - 132410039999/473307984537*a^12 - 77420821297/473307984537*a^11 + 125959066994/473307984537*a^10 - 235963177199/473307984537*a^9 + 63132015346/473307984537*a^8 + 82440484031/473307984537*a^7 - 73393694513/157769328179*a^6 + 4373278420/473307984537*a^5 - 111945114004/473307984537*a^4 - 90021523508/473307984537*a^3 - 190660763158/473307984537*a^2 - 183673740334/473307984537*a - 51957939668/157769328179

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:	$12$	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:	$a^{20}-2a^{19}-5a^{17}+12a^{16}-5a^{15}+4a^{14}-27a^{13}+16a^{12}-8a^{11}+8a^{10}-22a^{9}-8a^{8}-2a^{7}+9a^{6}+15a^{5}+12a^{4}+13a^{3}+2a^{2}-a-2$, $\frac{34217445404}{157769328179}a^{20}-\frac{488315448250}{473307984537}a^{19}+\frac{688206499904}{473307984537}a^{18}-\frac{799885111000}{473307984537}a^{17}+\frac{2729000289880}{473307984537}a^{16}-\frac{4530323171104}{473307984537}a^{15}+\frac{1179584476816}{157769328179}a^{14}-\frac{5018755805777}{473307984537}a^{13}+\frac{3332031702400}{157769328179}a^{12}-\frac{2983158922944}{157769328179}a^{11}+\frac{6302300973464}{473307984537}a^{10}-\frac{6097345552616}{473307984537}a^{9}+\frac{2269055692808}{157769328179}a^{8}-\frac{415140016492}{157769328179}a^{7}+\frac{1798353999248}{473307984537}a^{6}-\frac{260910661238}{157769328179}a^{5}-\frac{1669691233240}{473307984537}a^{4}-\frac{290165889644}{157769328179}a^{3}-\frac{2165208110960}{473307984537}a^{2}-\frac{3789990320}{473307984537}a-\frac{204922542064}{473307984537}$, $\frac{43779773836}{157769328179}a^{20}+\frac{24655088984}{473307984537}a^{19}-\frac{715906123534}{473307984537}a^{18}-\frac{449136551122}{473307984537}a^{17}+\frac{360378170134}{473307984537}a^{16}+\frac{3323491462355}{473307984537}a^{15}-\frac{718224225964}{157769328179}a^{14}-\frac{2998238272181}{473307984537}a^{13}-\frac{1589979019332}{157769328179}a^{12}+\frac{2077149117741}{157769328179}a^{11}-\frac{790637562229}{473307984537}a^{10}-\frac{3042596241026}{473307984537}a^{9}-\frac{1881793058754}{157769328179}a^{8}-\frac{475735967367}{157769328179}a^{7}+\frac{4631563365518}{473307984537}a^{6}+\frac{1073651551263}{157769328179}a^{5}+\frac{5032266586268}{473307984537}a^{4}+\frac{918187843347}{157769328179}a^{3}+\frac{1537436424694}{473307984537}a^{2}-\frac{1029376576805}{473307984537}a-\frac{1392310759006}{473307984537}$, $\frac{150633145937}{157769328179}a^{20}-\frac{684537044254}{473307984537}a^{19}-\frac{212338706539}{157769328179}a^{18}-\frac{1878741467849}{473307984537}a^{17}+\frac{4502513185711}{473307984537}a^{16}+\frac{986487772483}{473307984537}a^{15}-\frac{472389298710}{157769328179}a^{14}-\frac{11155870815266}{473307984537}a^{13}+\frac{2590660587188}{473307984537}a^{12}+\frac{3489459924725}{473307984537}a^{11}-\frac{224183705048}{473307984537}a^{10}-\frac{10113045526349}{473307984537}a^{9}-\frac{5945459663602}{473307984537}a^{8}-\frac{713330210422}{473307984537}a^{7}+\frac{5931583403095}{473307984537}a^{6}+\frac{2455986165804}{157769328179}a^{5}+\frac{6985439169842}{473307984537}a^{4}+\frac{5482994375359}{473307984537}a^{3}+\frac{1920823537379}{473307984537}a^{2}-\frac{1457496478397}{473307984537}a-\frac{1248168677273}{473307984537}$, $\frac{828000257488}{473307984537}a^{20}-\frac{1923957484594}{473307984537}a^{19}+\frac{153415103040}{157769328179}a^{18}-\frac{3958104434920}{473307984537}a^{17}+\frac{3754513308025}{157769328179}a^{16}-\frac{2381219310558}{157769328179}a^{15}+\frac{3737581702409}{473307984537}a^{14}-\frac{22921920714115}{473307984537}a^{13}+\frac{20892263808824}{473307984537}a^{12}-\frac{3273323425729}{157769328179}a^{11}+\frac{7261198409603}{473307984537}a^{10}-\frac{20527533573040}{473307984537}a^{9}+\frac{680956995230}{473307984537}a^{8}+\frac{310153076721}{157769328179}a^{7}+\frac{2758440287281}{157769328179}a^{6}+\frac{9235766575054}{473307984537}a^{5}+\frac{5578256867854}{473307984537}a^{4}+\frac{2464762387240}{157769328179}a^{3}-\frac{2525680955194}{473307984537}a^{2}-\frac{455221699576}{157769328179}a-\frac{472231263225}{157769328179}$, $\frac{99957119579}{473307984537}a^{20}+\frac{151359217600}{473307984537}a^{19}-\frac{995385834203}{473307984537}a^{18}+\frac{378855786}{157769328179}a^{17}-\frac{82280560379}{157769328179}a^{16}+\frac{4862093783423}{473307984537}a^{15}-\frac{4450578176654}{473307984537}a^{14}-\frac{1353847398182}{473307984537}a^{13}-\frac{2281929064679}{157769328179}a^{12}+\frac{11600387140508}{473307984537}a^{11}-\frac{1581984117973}{157769328179}a^{10}-\frac{1441300750772}{473307984537}a^{9}-\frac{2447844166103}{157769328179}a^{8}+\frac{2982805711532}{473307984537}a^{7}+\frac{3554801462123}{473307984537}a^{6}+\frac{3836576112044}{473307984537}a^{5}+\frac{3514522507934}{473307984537}a^{4}+\frac{1873604780710}{473307984537}a^{3}+\frac{1138065749696}{473307984537}a^{2}-\frac{796501414868}{157769328179}a-\frac{1034266345876}{473307984537}$, $\frac{31272730154}{473307984537}a^{20}-\frac{127640749285}{473307984537}a^{19}+\frac{105889292660}{473307984537}a^{18}-\frac{103988991461}{473307984537}a^{17}+\frac{697789548311}{473307984537}a^{16}-\frac{276530159911}{157769328179}a^{15}+\frac{159273747952}{473307984537}a^{14}-\frac{331046351082}{157769328179}a^{13}+\frac{746437600077}{157769328179}a^{12}-\frac{721542563812}{473307984537}a^{11}+\frac{450163582408}{473307984537}a^{10}-\frac{427421915282}{157769328179}a^{9}+\frac{405765451200}{157769328179}a^{8}+\frac{851558603216}{473307984537}a^{7}+\frac{283006008816}{157769328179}a^{6}-\frac{613776862792}{473307984537}a^{5}-\frac{299829301077}{157769328179}a^{4}-\frac{325736134235}{473307984537}a^{3}-\frac{801619848512}{473307984537}a^{2}-\frac{624735022948}{473307984537}a-\frac{140998990872}{157769328179}$, $\frac{87834127040}{473307984537}a^{20}-\frac{245297979203}{473307984537}a^{19}+\frac{40644863359}{157769328179}a^{18}-\frac{420443530499}{473307984537}a^{17}+\frac{491794488907}{157769328179}a^{16}-\frac{434400467300}{157769328179}a^{15}+\frac{671949787003}{473307984537}a^{14}-\frac{3026948331683}{473307984537}a^{13}+\frac{3932344776235}{473307984537}a^{12}-\frac{718864705296}{157769328179}a^{11}+\frac{2119063031770}{473307984537}a^{10}-\frac{3947991382985}{473307984537}a^{9}+\frac{2363886606832}{473307984537}a^{8}-\frac{338523832182}{157769328179}a^{7}+\frac{826850598436}{157769328179}a^{6}-\frac{685297232788}{473307984537}a^{5}+\frac{214252305179}{473307984537}a^{4}-\frac{270382757380}{157769328179}a^{3}-\frac{792371097977}{473307984537}a^{2}-\frac{333317909437}{157769328179}a-\frac{45500002760}{157769328179}$, $\frac{200761779964}{157769328179}a^{20}-\frac{2307769299757}{473307984537}a^{19}+\frac{2457775764226}{473307984537}a^{18}-\frac{1082447434709}{157769328179}a^{17}+\frac{12040378716367}{473307984537}a^{16}-\frac{5741296410785}{157769328179}a^{15}+\frac{3265146525994}{157769328179}a^{14}-\frac{18047598438491}{473307984537}a^{13}+\frac{37352779215286}{473307984537}a^{12}-\frac{28358363122625}{473307984537}a^{11}+\frac{3673909824884}{157769328179}a^{10}-\frac{15924814304627}{473307984537}a^{9}+\frac{18285859880230}{473307984537}a^{8}+\frac{2742380299078}{473307984537}a^{7}+\frac{125523156648}{157769328179}a^{6}-\frac{237771026140}{157769328179}a^{5}-\frac{4741501733923}{473307984537}a^{4}-\frac{145086622789}{473307984537}a^{3}-\frac{2521939065609}{157769328179}a^{2}+\frac{763568004826}{473307984537}a+\frac{620762930924}{157769328179}$, $\frac{168888661985}{473307984537}a^{20}+\frac{3669455650}{473307984537}a^{19}-\frac{923639549557}{473307984537}a^{18}-\frac{381569212513}{473307984537}a^{17}+\frac{161817231889}{157769328179}a^{16}+\frac{4114631858083}{473307984537}a^{15}-\frac{3664310624183}{473307984537}a^{14}-\frac{2802213087458}{473307984537}a^{13}-\frac{5537597491948}{473307984537}a^{12}+\frac{8970542208493}{473307984537}a^{11}-\frac{4063177861000}{473307984537}a^{10}-\frac{2484659758229}{473307984537}a^{9}-\frac{6985575040654}{473307984537}a^{8}+\frac{161152762918}{473307984537}a^{7}+\frac{3609066376201}{473307984537}a^{6}+\frac{4466439229628}{473307984537}a^{5}+\frac{4780461130154}{473307984537}a^{4}+\frac{3112481591975}{473307984537}a^{3}+\frac{673228627757}{157769328179}a^{2}-\frac{485704782679}{157769328179}a-\frac{1145516341061}{473307984537}$, $\frac{29679531267}{157769328179}a^{20}-\frac{268537056262}{473307984537}a^{19}+\frac{119559717134}{473307984537}a^{18}-\frac{227965412758}{473307984537}a^{17}+\frac{1315748466676}{473307984537}a^{16}-\frac{1248635263027}{473307984537}a^{15}-\frac{85448358951}{157769328179}a^{14}-\frac{1302556432499}{473307984537}a^{13}+\frac{1076485441760}{157769328179}a^{12}-\frac{283779424507}{157769328179}a^{11}-\frac{1566116953018}{473307984537}a^{10}-\frac{528625933001}{473307984537}a^{9}+\frac{398739024781}{157769328179}a^{8}+\frac{425260997872}{157769328179}a^{7}-\frac{128302325854}{473307984537}a^{6}+\frac{18729283173}{157769328179}a^{5}-\frac{112522953532}{473307984537}a^{4}+\frac{107217297689}{157769328179}a^{3}-\frac{930105982784}{473307984537}a^{2}-\frac{125574273401}{473307984537}a-\frac{79349690599}{473307984537}$, $\frac{340460846002}{473307984537}a^{20}-\frac{1135030470832}{473307984537}a^{19}+\frac{1225604301838}{473307984537}a^{18}-\frac{2354767998800}{473307984537}a^{17}+\frac{2102474245586}{157769328179}a^{16}-\frac{8556725183728}{473307984537}a^{15}+\frac{7559806997939}{473307984537}a^{14}-\frac{12538288718980}{473307984537}a^{13}+\frac{18052660946194}{473307984537}a^{12}-\frac{18287719940704}{473307984537}a^{11}+\frac{12299405861554}{473307984537}a^{10}-\frac{12298942089151}{473307984537}a^{9}+\frac{8470446117526}{473307984537}a^{8}-\frac{4618420446073}{473307984537}a^{7}+\frac{2686224766736}{473307984537}a^{6}+\frac{1495115775928}{473307984537}a^{5}+\frac{870740380216}{473307984537}a^{4}+\frac{2793470191657}{473307984537}a^{3}-\frac{801179066436}{157769328179}a^{2}+\frac{468963631863}{157769328179}a-\frac{201341254837}{473307984537}$ a^20 - 2a^19 - 5a^17 + 12a^16 - 5a^15 + 4a^14 - 27a^13 + 16a^12 - 8a^11 + 8a^10 - 22a^9 - 8a^8 - 2a^7 + 9a^6 + 15a^5 + 12a^4 + 13a^3 + 2a^2 - a - 2, 34217445404/157769328179a^20 - 488315448250/473307984537a^19 + 688206499904/473307984537a^18 - 799885111000/473307984537a^17 + 2729000289880/473307984537a^16 - 4530323171104/473307984537a^15 + 1179584476816/157769328179a^14 - 5018755805777/473307984537a^13 + 3332031702400/157769328179a^12 - 2983158922944/157769328179a^11 + 6302300973464/473307984537a^10 - 6097345552616/473307984537a^9 + 2269055692808/157769328179a^8 - 415140016492/157769328179a^7 + 1798353999248/473307984537a^6 - 260910661238/157769328179a^5 - 1669691233240/473307984537a^4 - 290165889644/157769328179a^3 - 2165208110960/473307984537a^2 - 3789990320/473307984537a - 204922542064/473307984537, 43779773836/157769328179a^20 + 24655088984/473307984537a^19 - 715906123534/473307984537a^18 - 449136551122/473307984537a^17 + 360378170134/473307984537a^16 + 3323491462355/473307984537a^15 - 718224225964/157769328179a^14 - 2998238272181/473307984537a^13 - 1589979019332/157769328179a^12 + 2077149117741/157769328179a^11 - 790637562229/473307984537a^10 - 3042596241026/473307984537a^9 - 1881793058754/157769328179a^8 - 475735967367/157769328179a^7 + 4631563365518/473307984537a^6 + 1073651551263/157769328179a^5 + 5032266586268/473307984537a^4 + 918187843347/157769328179a^3 + 1537436424694/473307984537a^2 - 1029376576805/473307984537a - 1392310759006/473307984537, 150633145937/157769328179a^20 - 684537044254/473307984537a^19 - 212338706539/157769328179a^18 - 1878741467849/473307984537a^17 + 4502513185711/473307984537a^16 + 986487772483/473307984537a^15 - 472389298710/157769328179a^14 - 11155870815266/473307984537a^13 + 2590660587188/473307984537a^12 + 3489459924725/473307984537a^11 - 224183705048/473307984537a^10 - 10113045526349/473307984537a^9 - 5945459663602/473307984537a^8 - 713330210422/473307984537a^7 + 5931583403095/473307984537a^6 + 2455986165804/157769328179a^5 + 6985439169842/473307984537a^4 + 5482994375359/473307984537a^3 + 1920823537379/473307984537a^2 - 1457496478397/473307984537a - 1248168677273/473307984537, 828000257488/473307984537a^20 - 1923957484594/473307984537a^19 + 153415103040/157769328179a^18 - 3958104434920/473307984537a^17 + 3754513308025/157769328179a^16 - 2381219310558/157769328179a^15 + 3737581702409/473307984537a^14 - 22921920714115/473307984537a^13 + 20892263808824/473307984537a^12 - 3273323425729/157769328179a^11 + 7261198409603/473307984537a^10 - 20527533573040/473307984537a^9 + 680956995230/473307984537a^8 + 310153076721/157769328179a^7 + 2758440287281/157769328179a^6 + 9235766575054/473307984537a^5 + 5578256867854/473307984537a^4 + 2464762387240/157769328179a^3 - 2525680955194/473307984537a^2 - 455221699576/157769328179a - 472231263225/157769328179, 99957119579/473307984537a^20 + 151359217600/473307984537a^19 - 995385834203/473307984537a^18 + 378855786/157769328179a^17 - 82280560379/157769328179a^16 + 4862093783423/473307984537a^15 - 4450578176654/473307984537a^14 - 1353847398182/473307984537a^13 - 2281929064679/157769328179a^12 + 11600387140508/473307984537a^11 - 1581984117973/157769328179a^10 - 1441300750772/473307984537a^9 - 2447844166103/157769328179a^8 + 2982805711532/473307984537a^7 + 3554801462123/473307984537a^6 + 3836576112044/473307984537a^5 + 3514522507934/473307984537a^4 + 1873604780710/473307984537a^3 + 1138065749696/473307984537a^2 - 796501414868/157769328179a - 1034266345876/473307984537, 31272730154/473307984537a^20 - 127640749285/473307984537a^19 + 105889292660/473307984537a^18 - 103988991461/473307984537a^17 + 697789548311/473307984537a^16 - 276530159911/157769328179a^15 + 159273747952/473307984537a^14 - 331046351082/157769328179a^13 + 746437600077/157769328179a^12 - 721542563812/473307984537a^11 + 450163582408/473307984537a^10 - 427421915282/157769328179a^9 + 405765451200/157769328179a^8 + 851558603216/473307984537a^7 + 283006008816/157769328179a^6 - 613776862792/473307984537a^5 - 299829301077/157769328179a^4 - 325736134235/473307984537a^3 - 801619848512/473307984537a^2 - 624735022948/473307984537a - 140998990872/157769328179, 87834127040/473307984537a^20 - 245297979203/473307984537a^19 + 40644863359/157769328179a^18 - 420443530499/473307984537a^17 + 491794488907/157769328179a^16 - 434400467300/157769328179a^15 + 671949787003/473307984537a^14 - 3026948331683/473307984537a^13 + 3932344776235/473307984537a^12 - 718864705296/157769328179a^11 + 2119063031770/473307984537a^10 - 3947991382985/473307984537a^9 + 2363886606832/473307984537a^8 - 338523832182/157769328179a^7 + 826850598436/157769328179a^6 - 685297232788/473307984537a^5 + 214252305179/473307984537a^4 - 270382757380/157769328179a^3 - 792371097977/473307984537a^2 - 333317909437/157769328179a - 45500002760/157769328179, 200761779964/157769328179a^20 - 2307769299757/473307984537a^19 + 2457775764226/473307984537a^18 - 1082447434709/157769328179a^17 + 12040378716367/473307984537a^16 - 5741296410785/157769328179a^15 + 3265146525994/157769328179a^14 - 18047598438491/473307984537a^13 + 37352779215286/473307984537a^12 - 28358363122625/473307984537a^11 + 3673909824884/157769328179a^10 - 15924814304627/473307984537a^9 + 18285859880230/473307984537a^8 + 2742380299078/473307984537a^7 + 125523156648/157769328179a^6 - 237771026140/157769328179a^5 - 4741501733923/473307984537a^4 - 145086622789/473307984537a^3 - 2521939065609/157769328179a^2 + 763568004826/473307984537a + 620762930924/157769328179, 168888661985/473307984537a^20 + 3669455650/473307984537a^19 - 923639549557/473307984537a^18 - 381569212513/473307984537a^17 + 161817231889/157769328179a^16 + 4114631858083/473307984537a^15 - 3664310624183/473307984537a^14 - 2802213087458/473307984537a^13 - 5537597491948/473307984537a^12 + 8970542208493/473307984537a^11 - 4063177861000/473307984537a^10 - 2484659758229/473307984537a^9 - 6985575040654/473307984537a^8 + 161152762918/473307984537a^7 + 3609066376201/473307984537a^6 + 4466439229628/473307984537a^5 + 4780461130154/473307984537a^4 + 3112481591975/473307984537a^3 + 673228627757/157769328179a^2 - 485704782679/157769328179a - 1145516341061/473307984537, 29679531267/157769328179a^20 - 268537056262/473307984537a^19 + 119559717134/473307984537a^18 - 227965412758/473307984537a^17 + 1315748466676/473307984537a^16 - 1248635263027/473307984537a^15 - 85448358951/157769328179a^14 - 1302556432499/473307984537a^13 + 1076485441760/157769328179a^12 - 283779424507/157769328179a^11 - 1566116953018/473307984537a^10 - 528625933001/473307984537a^9 + 398739024781/157769328179a^8 + 425260997872/157769328179a^7 - 128302325854/473307984537a^6 + 18729283173/157769328179a^5 - 112522953532/473307984537a^4 + 107217297689/157769328179a^3 - 930105982784/473307984537a^2 - 125574273401/473307984537a - 79349690599/473307984537, 340460846002/473307984537a^20 - 1135030470832/473307984537a^19 + 1225604301838/473307984537a^18 - 2354767998800/473307984537a^17 + 2102474245586/157769328179a^16 - 8556725183728/473307984537a^15 + 7559806997939/473307984537a^14 - 12538288718980/473307984537a^13 + 18052660946194/473307984537a^12 - 18287719940704/473307984537a^11 + 12299405861554/473307984537a^10 - 12298942089151/473307984537a^9 + 8470446117526/473307984537a^8 - 4618420446073/473307984537a^7 + 2686224766736/473307984537a^6 + 1495115775928/473307984537a^5 + 870740380216/473307984537a^4 + 2793470191657/473307984537a^3 - 801179066436/157769328179a^2 + 468963631863/157769328179*a - 201341254837/473307984537 (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:	$ 1358971.01083 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 1358971.01083 \cdot 1}{2\cdot\sqrt{34815740375535437995421466177}}\cr\approx \mathstrut & 0.283061826407 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 5*x^18 + 12*x^17 - 5*x^16 + 4*x^15 - 27*x^14 + 16*x^13 - 8*x^12 + 8*x^11 - 22*x^10 - 8*x^9 - 2*x^8 + 9*x^7 + 15*x^6 + 12*x^5 + 13*x^4 + 2*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 - 2*x^20 - 5*x^18 + 12*x^17 - 5*x^16 + 4*x^15 - 27*x^14 + 16*x^13 - 8*x^12 + 8*x^11 - 22*x^10 - 8*x^9 - 2*x^8 + 9*x^7 + 15*x^6 + 12*x^5 + 13*x^4 + 2*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 - 2*x^20 - 5*x^18 + 12*x^17 - 5*x^16 + 4*x^15 - 27*x^14 + 16*x^13 - 8*x^12 + 8*x^11 - 22*x^10 - 8*x^9 - 2*x^8 + 9*x^7 + 15*x^6 + 12*x^5 + 13*x^4 + 2*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 - 2*x^20 - 5*x^18 + 12*x^17 - 5*x^16 + 4*x^15 - 27*x^14 + 16*x^13 - 8*x^12 + 8*x^11 - 22*x^10 - 8*x^9 - 2*x^8 + 9*x^7 + 15*x^6 + 12*x^5 + 13*x^4 + 2*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

$S_7$ (as 21T38):

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 5040

The 15 conjugacy class representatives for $S_7$

Character table for $S_7$

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 7 sibling:	7.3.792873.1
Degree 14 sibling:	deg 14
Degree 30 sibling:	data not computed
Degree 35 sibling:	data not computed
Degree 42 siblings:	data not computed
Minimal sibling:	7.3.792873.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.7.0.1}{7} }^{3}$	R	${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$	${\href{/padicField/7.7.0.1}{7} }^{3}$	${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$	${\href{/padicField/13.7.0.1}{7} }^{3}$	${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$	${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$	R	${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$	${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.3.0.1}{3} }$	${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$	${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
$3$ 3	3.2.0.1	$x^{2} + 2 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	3.3.0.1	$x^{3} + 2 x + 1$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	3.4.2.2	$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	3.12.6.1	$x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$	$2$	$6$	$6$	$C_{12}$	$[\ ]_{2}^{6}$
$37$ 37	$\Q_{37}$	$x + 35$	$1$	$1$	$0$	Trivial	$[\ ]$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	37.4.0.1	$x^{4} + 6 x^{2} + 24 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	37.4.0.1	$x^{4} + 6 x^{2} + 24 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	37.8.4.1	$x^{8} + 3700 x^{7} + 5133910 x^{6} + 3166256548 x^{5} + 732510094073 x^{4} + 136269235536 x^{3} + 4476368972260 x^{2} + 17928293629116 x + 2173698901413$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$2381$ 2381	$\Q_{2381}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $8$	$2$	$4$	$4$

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