Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 33*x^20 + 363*x^18 + 1243*x^16 - 2310*x^14 - 14278*x^12 + 9878*x^10 + 25718*x^8 - 7051*x^6 - 363*x^4 + 143*x^2 - 1)
gp: K = bnfinit(y^22 + 33*y^20 + 363*y^18 + 1243*y^16 - 2310*y^14 - 14278*y^12 + 9878*y^10 + 25718*y^8 - 7051*y^6 - 363*y^4 + 143*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 33*x^20 + 363*x^18 + 1243*x^16 - 2310*x^14 - 14278*x^12 + 9878*x^10 + 25718*x^8 - 7051*x^6 - 363*x^4 + 143*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 33*x^20 + 363*x^18 + 1243*x^16 - 2310*x^14 - 14278*x^12 + 9878*x^10 + 25718*x^8 - 7051*x^6 - 363*x^4 + 143*x^2 - 1)
\( x^{22} + 33 x^{20} + 363 x^{18} + 1243 x^{16} - 2310 x^{14} - 14278 x^{12} + 9878 x^{10} + 25718 x^{8} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1543118794783990130660601200150793158656\)
\(\medspace = 2^{26}\cdot 7^{10}\cdot 11^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.44\) | 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: |
\(2\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{3}{8}a^{2}+\frac{3}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{8}-\frac{1}{8}a^{4}+\frac{1}{8}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{1}{16}a^{9}-\frac{1}{16}a^{8}+\frac{3}{16}a^{5}-\frac{3}{16}a^{4}-\frac{1}{2}a^{2}-\frac{5}{16}a-\frac{3}{16}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{12}+\frac{1}{16}a^{10}-\frac{1}{16}a^{8}+\frac{3}{16}a^{6}-\frac{3}{16}a^{4}-\frac{1}{2}a^{3}+\frac{3}{16}a^{2}-\frac{1}{2}a-\frac{3}{16}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{16}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{16}a^{4}-\frac{7}{16}a^{3}+\frac{3}{8}a^{2}-\frac{1}{8}a-\frac{1}{16}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{8}+\frac{1}{16}$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{16}+\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{2}+\frac{5}{32}a+\frac{3}{32}$, $\frac{1}{32}a^{18}-\frac{1}{32}a^{16}+\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{3}{32}a^{2}+\frac{1}{4}a+\frac{3}{32}$, $\frac{1}{32}a^{19}-\frac{1}{32}a^{16}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{9}{32}a^{3}+\frac{3}{8}a^{2}+\frac{1}{8}a-\frac{1}{32}$, $\frac{1}{28103676512}a^{20}-\frac{337131771}{28103676512}a^{18}+\frac{87854139}{3512959564}a^{16}+\frac{67220799}{14051838256}a^{14}+\frac{328995315}{7025919128}a^{12}-\frac{320991809}{3512959564}a^{10}-\frac{142906555}{14051838256}a^{8}-\frac{1}{4}a^{7}+\frac{3024972421}{14051838256}a^{6}-\frac{1}{4}a^{5}-\frac{4506759445}{28103676512}a^{4}-\frac{1}{4}a^{3}-\frac{3764689845}{28103676512}a^{2}-\frac{1}{4}a-\frac{5158704005}{14051838256}$, $\frac{1}{56207353024}a^{21}-\frac{1}{56207353024}a^{20}+\frac{67638515}{7025919128}a^{19}-\frac{67638515}{7025919128}a^{18}-\frac{175406779}{56207353024}a^{17}+\frac{175406779}{56207353024}a^{16}-\frac{202754773}{7025919128}a^{15}+\frac{202754773}{7025919128}a^{14}-\frac{220249261}{28103676512}a^{13}+\frac{220249261}{28103676512}a^{12}-\frac{320991809}{7025919128}a^{11}-\frac{278624041}{3512959564}a^{10}+\frac{1613573227}{28103676512}a^{9}+\frac{1899386337}{28103676512}a^{8}-\frac{1658916595}{7025919128}a^{7}+\frac{1658916595}{7025919128}a^{6}+\frac{4275639465}{56207353024}a^{5}-\frac{4275639465}{56207353024}a^{4}-\frac{1607542945}{3512959564}a^{3}+\frac{580366217}{7025919128}a^{2}-\frac{5926208555}{56207353024}a+\frac{27003965939}{56207353024}$
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: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$a$, $\frac{15239493745}{56207353024}a^{21}-\frac{7135019815}{56207353024}a^{20}+\frac{252828374143}{28103676512}a^{19}-\frac{118424913415}{28103676512}a^{18}+\frac{5623446565753}{56207353024}a^{17}-\frac{2636421426643}{56207353024}a^{16}+\frac{4990956483573}{14051838256}a^{15}-\frac{2347120291181}{14051838256}a^{14}-\frac{15767849621297}{28103676512}a^{13}+\frac{7296093683935}{28103676512}a^{12}-\frac{6969469018115}{1756479782}a^{11}+\frac{13057645383035}{7025919128}a^{10}+\frac{55138004912843}{28103676512}a^{9}-\frac{25136739963461}{28103676512}a^{8}+\frac{102506919571643}{14051838256}a^{7}-\frac{47978035570207}{14051838256}a^{6}-\frac{33937759304031}{56207353024}a^{5}+\frac{14168311796777}{56207353024}a^{4}-\frac{4942339400143}{28103676512}a^{3}+\frac{2416147902611}{28103676512}a^{2}+\frac{269434774785}{56207353024}a-\frac{87397493731}{56207353024}$, $\frac{13829016429}{28103676512}a^{20}+\frac{28680877007}{1756479782}a^{18}+\frac{5104125257255}{28103676512}a^{16}+\frac{566450764705}{878239891}a^{14}-\frac{14305750567701}{14051838256}a^{12}-\frac{12666464672687}{1756479782}a^{10}+\frac{49706003344057}{14051838256}a^{8}+\frac{11674586326927}{878239891}a^{6}-\frac{28969898630323}{28103676512}a^{4}-\frac{309032064060}{878239891}a^{2}+\frac{154928749623}{28103676512}$, $\frac{2786745157}{7025919128}a^{21}+\frac{184800634989}{14051838256}a^{19}+\frac{1026113032047}{7025919128}a^{17}+\frac{3625457580119}{7025919128}a^{15}-\frac{1465997751737}{1756479782}a^{13}-\frac{20348615026159}{3512959564}a^{11}+\frac{10552087497861}{3512959564}a^{9}+\frac{74811088603897}{7025919128}a^{7}-\frac{7771839665881}{7025919128}a^{5}-\frac{3758758433937}{14051838256}a^{3}+\frac{60131358647}{7025919128}a$, $\frac{28411121769}{56207353024}a^{21}+\frac{3989224057}{56207353024}a^{20}+\frac{58823051329}{3512959564}a^{19}+\frac{66265215339}{28103676512}a^{18}+\frac{10432746964951}{56207353024}a^{17}+\frac{1477509509933}{56207353024}a^{16}+\frac{2290252356403}{3512959564}a^{15}+\frac{1321577304315}{14051838256}a^{14}-\frac{30456696449883}{28103676512}a^{13}-\frac{4022640641733}{28103676512}a^{12}-\frac{12910387092507}{1756479782}a^{11}-\frac{3671740458879}{3512959564}a^{10}+\frac{113905823771071}{28103676512}a^{9}+\frac{13242805286975}{28103676512}a^{8}+\frac{11822392759182}{878239891}a^{7}+\frac{27210287236733}{14051838256}a^{6}-\frac{102758281526643}{56207353024}a^{5}-\frac{4524415113727}{56207353024}a^{4}-\frac{276486459317}{878239891}a^{3}-\frac{1829051317715}{28103676512}a^{2}+\frac{1530633336651}{56207353024}a-\frac{11418474747}{56207353024}$, $\frac{28411121769}{56207353024}a^{21}-\frac{3989224057}{56207353024}a^{20}+\frac{58823051329}{3512959564}a^{19}-\frac{66265215339}{28103676512}a^{18}+\frac{10432746964951}{56207353024}a^{17}-\frac{1477509509933}{56207353024}a^{16}+\frac{2290252356403}{3512959564}a^{15}-\frac{1321577304315}{14051838256}a^{14}-\frac{30456696449883}{28103676512}a^{13}+\frac{4022640641733}{28103676512}a^{12}-\frac{12910387092507}{1756479782}a^{11}+\frac{3671740458879}{3512959564}a^{10}+\frac{113905823771071}{28103676512}a^{9}-\frac{13242805286975}{28103676512}a^{8}+\frac{11822392759182}{878239891}a^{7}-\frac{27210287236733}{14051838256}a^{6}-\frac{102758281526643}{56207353024}a^{5}+\frac{4524415113727}{56207353024}a^{4}-\frac{276486459317}{878239891}a^{3}+\frac{1829051317715}{28103676512}a^{2}+\frac{1530633336651}{56207353024}a+\frac{11418474747}{56207353024}$, $\frac{6284522457}{14051838256}a^{21}+\frac{52019237457}{3512959564}a^{19}+\frac{2304086839955}{14051838256}a^{17}+\frac{2016249090761}{3512959564}a^{15}-\frac{851306652461}{878239891}a^{13}-\frac{5697980286669}{878239891}a^{11}+\frac{13010673138589}{3512959564}a^{9}+\frac{41677102872369}{3512959564}a^{7}-\frac{26040296591585}{14051838256}a^{5}-\frac{895699493547}{3512959564}a^{3}+\frac{419581052945}{14051838256}a$, $\frac{2402919849}{14051838256}a^{20}+\frac{79806219705}{14051838256}a^{18}+\frac{889178953615}{14051838256}a^{16}+\frac{793756290487}{3512959564}a^{14}-\frac{305129843945}{878239891}a^{12}-\frac{17679870083457}{7025919128}a^{10}+\frac{4069950265295}{3512959564}a^{8}+\frac{16365822998339}{3512959564}a^{6}-\frac{3257703915753}{14051838256}a^{4}-\frac{1982078267087}{14051838256}a^{2}+\frac{42612360133}{14051838256}$, $\frac{3968996053}{28103676512}a^{21}+\frac{65346403637}{14051838256}a^{19}+\frac{1431309599731}{28103676512}a^{17}+\frac{603539585413}{3512959564}a^{15}-\frac{4771790300119}{14051838256}a^{13}-\frac{14021664716943}{7025919128}a^{11}+\frac{21707341654627}{14051838256}a^{9}+\frac{12519330488535}{3512959564}a^{7}-\frac{35877523865015}{28103676512}a^{5}-\frac{533839434999}{14051838256}a^{3}+\frac{786648357463}{28103676512}a$, $\frac{6740687229}{14051838256}a^{21}-\frac{678394277}{28103676512}a^{20}+\frac{444346369511}{28103676512}a^{19}-\frac{22756351019}{28103676512}a^{18}+\frac{2437912537533}{14051838256}a^{17}-\frac{258499317093}{28103676512}a^{16}+\frac{4139368597905}{7025919128}a^{15}-\frac{489532041797}{14051838256}a^{14}-\frac{15929429408669}{14051838256}a^{13}+\frac{544762971589}{14051838256}a^{12}-\frac{95704120769271}{14051838256}a^{11}+\frac{655398536535}{1756479782}a^{10}+\frac{70565218412335}{14051838256}a^{9}-\frac{614471145367}{14051838256}a^{8}+\frac{85813845573869}{7025919128}a^{7}-\frac{10109534535607}{14051838256}a^{6}-\frac{13731241820387}{3512959564}a^{5}-\frac{5514092478653}{28103676512}a^{4}-\frac{4551980288641}{28103676512}a^{3}+\frac{871171814323}{28103676512}a^{2}+\frac{149444194609}{1756479782}a+\frac{199375328715}{28103676512}$, $\frac{6740687229}{14051838256}a^{21}+\frac{678394277}{28103676512}a^{20}+\frac{444346369511}{28103676512}a^{19}+\frac{22756351019}{28103676512}a^{18}+\frac{2437912537533}{14051838256}a^{17}+\frac{258499317093}{28103676512}a^{16}+\frac{4139368597905}{7025919128}a^{15}+\frac{489532041797}{14051838256}a^{14}-\frac{15929429408669}{14051838256}a^{13}-\frac{544762971589}{14051838256}a^{12}-\frac{95704120769271}{14051838256}a^{11}-\frac{655398536535}{1756479782}a^{10}+\frac{70565218412335}{14051838256}a^{9}+\frac{614471145367}{14051838256}a^{8}+\frac{85813845573869}{7025919128}a^{7}+\frac{10109534535607}{14051838256}a^{6}-\frac{13731241820387}{3512959564}a^{5}+\frac{5514092478653}{28103676512}a^{4}-\frac{4551980288641}{28103676512}a^{3}-\frac{871171814323}{28103676512}a^{2}+\frac{149444194609}{1756479782}a-\frac{199375328715}{28103676512}$, $\frac{5920880463}{28103676512}a^{21}+\frac{252332927}{28103676512}a^{20}+\frac{197973111273}{28103676512}a^{19}+\frac{8110506717}{28103676512}a^{18}+\frac{558767088861}{7025919128}a^{17}+\frac{42217706669}{14051838256}a^{16}+\frac{1039349346057}{3512959564}a^{15}+\frac{117267322353}{14051838256}a^{14}-\frac{321054345467}{878239891}a^{13}-\frac{428342202839}{14051838256}a^{12}-\frac{44907191967255}{14051838256}a^{11}-\frac{775930343769}{7025919128}a^{10}+\frac{10307211078225}{14051838256}a^{9}+\frac{715989342549}{3512959564}a^{8}+\frac{5322747633779}{878239891}a^{7}+\frac{2355378566503}{14051838256}a^{6}+\frac{27955901133873}{28103676512}a^{5}-\frac{8002562048985}{28103676512}a^{4}-\frac{6909997616915}{28103676512}a^{3}+\frac{338696809175}{28103676512}a^{2}-\frac{539957429939}{14051838256}a+\frac{106396655363}{14051838256}$, $\frac{24892840779}{14051838256}a^{21}+\frac{206497100103}{3512959564}a^{19}+\frac{2296571248573}{3512959564}a^{17}+\frac{16306571325869}{7025919128}a^{15}-\frac{25778285040243}{7025919128}a^{13}-\frac{182362120966653}{7025919128}a^{11}+\frac{11220108332265}{878239891}a^{9}+\frac{336076599054387}{7025919128}a^{7}-\frac{53114529780437}{14051838256}a^{5}-\frac{8774390348969}{7025919128}a^{3}+\frac{75525046451}{3512959564}a$, $\frac{32426820351}{14051838256}a^{21}+\frac{14723927719}{14051838256}a^{20}+\frac{1076424029261}{14051838256}a^{19}+\frac{977403970333}{28103676512}a^{18}+\frac{5990672954487}{7025919128}a^{17}+\frac{10876277649101}{28103676512}a^{16}+\frac{10662064647539}{3512959564}a^{15}+\frac{19340874071045}{14051838256}a^{14}-\frac{33286539191353}{7025919128}a^{13}-\frac{30313230100159}{14051838256}a^{12}-\frac{119002072050603}{3512959564}a^{11}-\frac{26997954959637}{1756479782}a^{10}+\frac{113683970315015}{7025919128}a^{9}+\frac{52146724110079}{7025919128}a^{8}+\frac{219678547862003}{3512959564}a^{7}+\frac{398589766705835}{14051838256}a^{6}-\frac{57293919286677}{14051838256}a^{5}-\frac{1764850162159}{878239891}a^{4}-\frac{23944559477729}{14051838256}a^{3}-\frac{22428279036765}{28103676512}a^{2}-\frac{32190822123}{3512959564}a+\frac{196996299703}{28103676512}$, $\frac{80226859937}{56207353024}a^{21}+\frac{19053893027}{56207353024}a^{20}+\frac{658569719383}{14051838256}a^{19}+\frac{317546956983}{28103676512}a^{18}+\frac{28683550746495}{56207353024}a^{17}+\frac{7126639288819}{56207353024}a^{16}+\frac{11853660922359}{7025919128}a^{15}+\frac{6508148883239}{14051838256}a^{14}-\frac{101422626458293}{28103676512}a^{13}-\frac{17760607902509}{28103676512}a^{12}-\frac{139877641459355}{7025919128}a^{11}-\frac{35513287180081}{7025919128}a^{10}+\frac{493142760964537}{28103676512}a^{9}+\frac{48160193211799}{28103676512}a^{8}+\frac{246916289653633}{7025919128}a^{7}+\frac{132426210177277}{14051838256}a^{6}-\frac{922046298639447}{56207353024}a^{5}+\frac{31804755974631}{56207353024}a^{4}-\frac{2821090209001}{14051838256}a^{3}-\frac{8256446237147}{28103676512}a^{2}+\frac{20507639669711}{56207353024}a-\frac{1607039608401}{56207353024}$
| 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: | \( 720857418211 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 720857418211 \cdot 1}{2\cdot\sqrt{1543118794783990130660601200150793158656}}\cr\approx \mathstrut & 0.578094533584251 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 33*x^20 + 363*x^18 + 1243*x^16 - 2310*x^14 - 14278*x^12 + 9878*x^10 + 25718*x^8 - 7051*x^6 - 363*x^4 + 143*x^2 - 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^22 + 33*x^20 + 363*x^18 + 1243*x^16 - 2310*x^14 - 14278*x^12 + 9878*x^10 + 25718*x^8 - 7051*x^6 - 363*x^4 + 143*x^2 - 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^22 + 33*x^20 + 363*x^18 + 1243*x^16 - 2310*x^14 - 14278*x^12 + 9878*x^10 + 25718*x^8 - 7051*x^6 - 363*x^4 + 143*x^2 - 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^22 + 33*x^20 + 363*x^18 + 1243*x^16 - 2310*x^14 - 14278*x^12 + 9878*x^10 + 25718*x^8 - 7051*x^6 - 363*x^4 + 143*x^2 - 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
$C_2^{10}.F_{11}$ (as 22T34):
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 112640 |
| The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
| Character table for $C_2^{10}.F_{11}$ |
Intermediate fields
| 11.11.4910318845910094848.1 |
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 22 sibling: | data not computed |
| Degree 44 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 | $20{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $22$ | $1$ | $26$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.20.10.1 | $x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
|
\(11\)
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |