Properties

Label 22.0.129...032.1
Degree $22$
Signature $[0, 11]$
Discriminant $-1.290\times 10^{41}$
Root discriminant \(73.90\)
Ramified primes $2,11$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $M_{22}:C_2$ (as 22T41)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763)
 
gp: K = bnfinit(y^22 - 11*y^20 + 99*y^18 - 33*y^16 - 726*y^14 + 23474*y^12 - 4096*y^11 + 25894*y^10 + 22528*y^9 + 141086*y^8 + 540672*y^7 + 2309285*y^6 + 315392*y^5 + 5775209*y^4 - 7434240*y^3 + 5021863*y^2 - 9168896*y + 5643763, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763)
 

\( x^{22} - 11 x^{20} + 99 x^{18} - 33 x^{16} - 726 x^{14} + 23474 x^{12} - 4096 x^{11} + 25894 x^{10} + 22528 x^{9} + 141086 x^{8} + 540672 x^{7} + 2309285 x^{6} + \cdots + 5643763 \) Copy content Toggle raw display

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:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-129045097915537907803232946851577207980032\) \(\medspace = -\,2^{57}\cdot 11^{23}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(73.90\)
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:   \(2\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-22}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{16}a^{6}+\frac{1}{16}a^{4}-\frac{1}{16}a^{2}-\frac{1}{16}$, $\frac{1}{16}a^{7}-\frac{1}{16}a^{5}-\frac{1}{8}a^{4}-\frac{1}{16}a^{3}+\frac{1}{16}a+\frac{1}{8}$, $\frac{1}{32}a^{8}-\frac{1}{16}a^{4}+\frac{1}{32}$, $\frac{1}{64}a^{9}-\frac{1}{64}a^{8}-\frac{1}{32}a^{7}-\frac{1}{32}a^{6}-\frac{1}{16}a^{5}+\frac{1}{32}a^{3}+\frac{1}{32}a^{2}+\frac{3}{64}a+\frac{1}{64}$, $\frac{1}{128}a^{10}+\frac{1}{128}a^{8}-\frac{1}{64}a^{6}+\frac{7}{64}a^{4}+\frac{1}{128}a^{2}-\frac{15}{128}$, $\frac{1}{1408}a^{11}-\frac{1}{128}a^{9}+\frac{1}{64}a^{7}-\frac{3}{64}a^{5}-\frac{5}{128}a^{3}-\frac{1}{4}a^{2}+\frac{39}{128}a+\frac{13}{44}$, $\frac{1}{2816}a^{12}-\frac{1}{256}a^{8}-\frac{7}{256}a^{4}+\frac{3}{11}a+\frac{101}{256}$, $\frac{1}{5632}a^{13}-\frac{1}{5632}a^{12}-\frac{1}{2816}a^{11}-\frac{1}{256}a^{10}+\frac{1}{512}a^{9}+\frac{7}{512}a^{8}-\frac{1}{128}a^{7}-\frac{3}{128}a^{6}+\frac{5}{512}a^{5}-\frac{53}{512}a^{4}-\frac{27}{256}a^{3}-\frac{243}{2816}a^{2}-\frac{1219}{5632}a-\frac{645}{5632}$, $\frac{1}{11264}a^{14}-\frac{1}{11264}a^{12}+\frac{3}{1024}a^{10}-\frac{11}{1024}a^{8}+\frac{17}{1024}a^{6}+\frac{127}{1024}a^{4}+\frac{3}{44}a^{3}+\frac{73}{1024}a^{2}+\frac{19}{44}a-\frac{209}{1024}$, $\frac{1}{22528}a^{15}-\frac{1}{22528}a^{14}-\frac{1}{22528}a^{13}-\frac{3}{22528}a^{12}+\frac{1}{22528}a^{11}+\frac{5}{2048}a^{10}-\frac{11}{2048}a^{9}-\frac{9}{2048}a^{8}+\frac{17}{2048}a^{7}+\frac{31}{2048}a^{6}-\frac{65}{2048}a^{5}+\frac{2319}{22528}a^{4}+\frac{1091}{22528}a^{3}-\frac{2187}{22528}a^{2}-\frac{8123}{22528}a+\frac{10327}{22528}$, $\frac{1}{22528}a^{16}-\frac{1}{256}a^{10}-\frac{13}{1024}a^{8}+\frac{1}{128}a^{6}+\frac{3}{88}a^{5}-\frac{11}{256}a^{4}+\frac{63}{256}a^{2}-\frac{1}{8}a-\frac{305}{2048}$, $\frac{1}{495616}a^{17}-\frac{1}{495616}a^{16}-\frac{3}{247808}a^{15}-\frac{3}{247808}a^{14}-\frac{1}{247808}a^{13}+\frac{3}{247808}a^{12}-\frac{23}{247808}a^{11}+\frac{27}{22528}a^{10}+\frac{5}{704}a^{9}+\frac{119}{11264}a^{8}-\frac{443}{22528}a^{7}-\frac{7305}{247808}a^{6}+\frac{11265}{247808}a^{5}+\frac{5957}{247808}a^{4}-\frac{4317}{247808}a^{3}+\frac{16227}{247808}a^{2}-\frac{93633}{495616}a+\frac{83197}{495616}$, $\frac{1}{1982464}a^{18}+\frac{15}{1982464}a^{16}+\frac{1}{61952}a^{15}+\frac{5}{123904}a^{14}-\frac{1}{15488}a^{13}+\frac{17}{247808}a^{12}-\frac{9}{61952}a^{11}+\frac{29}{8192}a^{10}+\frac{9}{2816}a^{9}-\frac{1019}{90112}a^{8}+\frac{1311}{61952}a^{7}+\frac{277}{22528}a^{6}+\frac{105}{3872}a^{5}-\frac{6395}{123904}a^{4}-\frac{1303}{61952}a^{3}-\frac{463603}{1982464}a^{2}+\frac{6615}{30976}a-\frac{446805}{1982464}$, $\frac{1}{3964928}a^{19}-\frac{1}{3964928}a^{18}-\frac{1}{3964928}a^{17}+\frac{3}{360448}a^{16}-\frac{1}{123904}a^{15}+\frac{1}{61952}a^{14}-\frac{29}{495616}a^{13}-\frac{87}{495616}a^{12}+\frac{413}{1982464}a^{11}-\frac{199}{180224}a^{10}+\frac{973}{180224}a^{9}-\frac{20351}{1982464}a^{8}+\frac{9309}{495616}a^{7}-\frac{9457}{495616}a^{6}-\frac{173}{2816}a^{5}-\frac{12219}{123904}a^{4}+\frac{283133}{3964928}a^{3}+\frac{475235}{3964928}a^{2}-\frac{1418037}{3964928}a+\frac{975093}{3964928}$, $\frac{1}{55508992}a^{20}+\frac{3}{27754496}a^{19}-\frac{1}{6938624}a^{18}-\frac{1}{2523136}a^{17}-\frac{15}{7929856}a^{16}-\frac{9}{867328}a^{15}+\frac{151}{6938624}a^{14}-\frac{95}{3469312}a^{13}+\frac{347}{2523136}a^{12}-\frac{4905}{13877248}a^{11}-\frac{79}{57344}a^{10}-\frac{86587}{13877248}a^{9}-\frac{401253}{27754496}a^{8}-\frac{106017}{3469312}a^{7}-\frac{3545}{630784}a^{6}-\frac{2363}{108416}a^{5}+\frac{451867}{7929856}a^{4}-\frac{3284873}{27754496}a^{3}-\frac{9261}{247808}a^{2}+\frac{744971}{2523136}a-\frac{11498077}{55508992}$, $\frac{1}{26\!\cdots\!08}a^{21}+\frac{29332957}{26\!\cdots\!08}a^{20}+\frac{1379310087}{13\!\cdots\!04}a^{19}+\frac{108233505}{12\!\cdots\!64}a^{18}-\frac{9145271503}{26\!\cdots\!08}a^{17}-\frac{182926033075}{26\!\cdots\!08}a^{16}+\frac{72985534583}{33\!\cdots\!76}a^{15}-\frac{11763812079}{301030608879616}a^{14}+\frac{64293840679}{18\!\cdots\!72}a^{13}-\frac{1591522641235}{13\!\cdots\!04}a^{12}-\frac{32660940201}{946096199335936}a^{11}+\frac{12704392989469}{66\!\cdots\!52}a^{10}+\frac{68703102746933}{13\!\cdots\!04}a^{9}+\frac{31537558968001}{13\!\cdots\!04}a^{8}-\frac{82974342819}{3909488427008}a^{7}+\frac{24218622574535}{33\!\cdots\!76}a^{6}+\frac{822153941009917}{26\!\cdots\!08}a^{5}-\frac{24\!\cdots\!03}{26\!\cdots\!08}a^{4}+\frac{118366184568845}{12\!\cdots\!64}a^{3}+\frac{691414248872275}{13\!\cdots\!04}a^{2}-\frac{12\!\cdots\!51}{26\!\cdots\!08}a-\frac{4863114693803}{19521513324544}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{2}$, which has order $2$ (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:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{2399637741}{26\!\cdots\!08}a^{21}+\frac{4681511547}{26\!\cdots\!08}a^{20}-\frac{3925090341}{18\!\cdots\!72}a^{19}-\frac{4728503903}{18\!\cdots\!72}a^{18}+\frac{54495938379}{24\!\cdots\!28}a^{17}+\frac{571381036183}{26\!\cdots\!08}a^{16}-\frac{454122290157}{33\!\cdots\!76}a^{15}-\frac{178518637835}{33\!\cdots\!76}a^{14}+\frac{3514763349261}{13\!\cdots\!04}a^{13}+\frac{891303253051}{13\!\cdots\!04}a^{12}+\frac{12225555872567}{66\!\cdots\!52}a^{11}+\frac{10811014748333}{66\!\cdots\!52}a^{10}-\frac{280577052124851}{13\!\cdots\!04}a^{9}+\frac{83894180874723}{13\!\cdots\!04}a^{8}+\frac{95825830806163}{33\!\cdots\!76}a^{7}+\frac{6558823165511}{301030608879616}a^{6}+\frac{12\!\cdots\!21}{26\!\cdots\!08}a^{5}-\frac{70\!\cdots\!49}{26\!\cdots\!08}a^{4}-\frac{17\!\cdots\!03}{13\!\cdots\!04}a^{3}+\frac{66\!\cdots\!79}{13\!\cdots\!04}a^{2}-\frac{86\!\cdots\!55}{26\!\cdots\!08}a+\frac{63935659495535}{19521513324544}$, $\frac{8042064633}{37\!\cdots\!44}a^{21}+\frac{26925498267}{26\!\cdots\!08}a^{20}-\frac{306609091507}{13\!\cdots\!04}a^{19}-\frac{154330537107}{13\!\cdots\!04}a^{18}+\frac{5507605841399}{26\!\cdots\!08}a^{17}+\frac{400111425853}{37\!\cdots\!44}a^{16}-\frac{166787444319}{33\!\cdots\!76}a^{15}-\frac{278729637979}{33\!\cdots\!76}a^{14}-\frac{20645890862721}{13\!\cdots\!04}a^{13}-\frac{10039089933461}{13\!\cdots\!04}a^{12}+\frac{328463386031059}{66\!\cdots\!52}a^{11}+\frac{101357629141787}{66\!\cdots\!52}a^{10}+\frac{726325723651091}{13\!\cdots\!04}a^{9}+\frac{775306047756215}{13\!\cdots\!04}a^{8}+\frac{10\!\cdots\!13}{33\!\cdots\!76}a^{7}+\frac{41\!\cdots\!81}{33\!\cdots\!76}a^{6}+\frac{14\!\cdots\!11}{26\!\cdots\!08}a^{5}+\frac{10\!\cdots\!25}{37\!\cdots\!44}a^{4}+\frac{16\!\cdots\!49}{13\!\cdots\!04}a^{3}-\frac{22\!\cdots\!41}{18\!\cdots\!72}a^{2}+\frac{41\!\cdots\!15}{26\!\cdots\!08}a-\frac{413859198031837}{19521513324544}$, $\frac{18824011225}{13\!\cdots\!04}a^{21}+\frac{2907492667}{602061217759232}a^{20}-\frac{109220039263}{66\!\cdots\!52}a^{19}-\frac{198410780039}{33\!\cdots\!76}a^{18}+\frac{292620273971}{18\!\cdots\!72}a^{17}+\frac{3693010704917}{66\!\cdots\!52}a^{16}-\frac{326093980437}{16\!\cdots\!88}a^{15}-\frac{722009272403}{827834174418944}a^{14}-\frac{2693318069271}{66\!\cdots\!52}a^{13}-\frac{864358436397}{301030608879616}a^{12}+\frac{107484494211803}{33\!\cdots\!76}a^{11}+\frac{185900943315063}{16\!\cdots\!88}a^{10}-\frac{3139508030677}{602061217759232}a^{9}+\frac{9907362069313}{33\!\cdots\!76}a^{8}+\frac{680074733856451}{16\!\cdots\!88}a^{7}+\frac{11\!\cdots\!45}{827834174418944}a^{6}+\frac{10\!\cdots\!31}{18\!\cdots\!72}a^{5}+\frac{68\!\cdots\!49}{66\!\cdots\!52}a^{4}+\frac{47\!\cdots\!15}{946096199335936}a^{3}+\frac{13\!\cdots\!93}{33\!\cdots\!76}a^{2}-\frac{32\!\cdots\!57}{12\!\cdots\!64}a-\frac{8121958765301}{697196904448}$, $\frac{15382545917}{37\!\cdots\!44}a^{21}-\frac{19567397097}{26\!\cdots\!08}a^{20}-\frac{671622264175}{13\!\cdots\!04}a^{19}+\frac{154907825357}{13\!\cdots\!04}a^{18}+\frac{12579740977779}{26\!\cdots\!08}a^{17}-\frac{447366643015}{37\!\cdots\!44}a^{16}-\frac{2674238565755}{33\!\cdots\!76}a^{15}+\frac{1463757075433}{33\!\cdots\!76}a^{14}-\frac{26240203140517}{13\!\cdots\!04}a^{13}-\frac{1995023502905}{13\!\cdots\!04}a^{12}+\frac{650470023547247}{66\!\cdots\!52}a^{11}-\frac{243384111983261}{66\!\cdots\!52}a^{10}-\frac{411311100596225}{13\!\cdots\!04}a^{9}+\frac{25\!\cdots\!39}{13\!\cdots\!04}a^{8}+\frac{17\!\cdots\!73}{33\!\cdots\!76}a^{7}+\frac{61\!\cdots\!77}{33\!\cdots\!76}a^{6}+\frac{21\!\cdots\!47}{26\!\cdots\!08}a^{5}-\frac{11\!\cdots\!39}{37\!\cdots\!44}a^{4}+\frac{16\!\cdots\!57}{13\!\cdots\!04}a^{3}-\frac{47\!\cdots\!89}{18\!\cdots\!72}a^{2}+\frac{48\!\cdots\!55}{26\!\cdots\!08}a+\frac{150034654393895}{19521513324544}$, $\frac{84525090535}{26\!\cdots\!08}a^{21}-\frac{27484265525}{26\!\cdots\!08}a^{20}-\frac{229542191843}{13\!\cdots\!04}a^{19}+\frac{282711686937}{13\!\cdots\!04}a^{18}+\frac{2716105175167}{26\!\cdots\!08}a^{17}-\frac{5531915655197}{26\!\cdots\!08}a^{16}+\frac{895055535295}{473048099667968}a^{15}+\frac{443181892291}{473048099667968}a^{14}-\frac{67786991579625}{13\!\cdots\!04}a^{13}+\frac{16190085829691}{13\!\cdots\!04}a^{12}+\frac{441327357348435}{66\!\cdots\!52}a^{11}-\frac{324935475038793}{66\!\cdots\!52}a^{10}+\frac{68\!\cdots\!83}{13\!\cdots\!04}a^{9}+\frac{388198146454169}{18\!\cdots\!72}a^{8}+\frac{13\!\cdots\!53}{33\!\cdots\!76}a^{7}+\frac{11\!\cdots\!39}{473048099667968}a^{6}+\frac{26\!\cdots\!19}{26\!\cdots\!08}a^{5}+\frac{21\!\cdots\!87}{26\!\cdots\!08}a^{4}+\frac{83\!\cdots\!93}{13\!\cdots\!04}a^{3}-\frac{13\!\cdots\!79}{13\!\cdots\!04}a^{2}+\frac{25\!\cdots\!23}{26\!\cdots\!08}a-\frac{17\!\cdots\!89}{19521513324544}$, $\frac{82951707}{24\!\cdots\!28}a^{21}+\frac{885338725}{24\!\cdots\!28}a^{20}-\frac{4769124051}{13\!\cdots\!04}a^{19}-\frac{53244607261}{13\!\cdots\!04}a^{18}+\frac{119290236861}{26\!\cdots\!08}a^{17}+\frac{832862639267}{26\!\cdots\!08}a^{16}-\frac{84476155161}{33\!\cdots\!76}a^{15}+\frac{47391285625}{33\!\cdots\!76}a^{14}+\frac{1943848936041}{13\!\cdots\!04}a^{13}-\frac{7742718476393}{13\!\cdots\!04}a^{12}-\frac{1396187971793}{66\!\cdots\!52}a^{11}+\frac{546742620645}{86008745394176}a^{10}-\frac{5177562414813}{12\!\cdots\!64}a^{9}+\frac{100773227065311}{13\!\cdots\!04}a^{8}+\frac{132541652764911}{33\!\cdots\!76}a^{7}+\frac{8465106202321}{33\!\cdots\!76}a^{6}+\frac{787100704831685}{26\!\cdots\!08}a^{5}+\frac{90\!\cdots\!95}{26\!\cdots\!08}a^{4}-\frac{74\!\cdots\!55}{13\!\cdots\!04}a^{3}+\frac{75\!\cdots\!91}{13\!\cdots\!04}a^{2}-\frac{24\!\cdots\!91}{26\!\cdots\!08}a+\frac{11441500970843}{19521513324544}$, $\frac{15703124403}{13\!\cdots\!04}a^{21}-\frac{246600173}{473048099667968}a^{20}-\frac{103035244803}{66\!\cdots\!52}a^{19}+\frac{30617112599}{33\!\cdots\!76}a^{18}+\frac{1994450602115}{13\!\cdots\!04}a^{17}-\frac{21425571799}{206958543604736}a^{16}-\frac{593532869095}{16\!\cdots\!88}a^{15}+\frac{121519186827}{206958543604736}a^{14}-\frac{1802514521901}{66\!\cdots\!52}a^{13}-\frac{267755340689}{150515304439808}a^{12}+\frac{100635675937651}{33\!\cdots\!76}a^{11}-\frac{25346643865661}{16\!\cdots\!88}a^{10}-\frac{36287658932919}{946096199335936}a^{9}+\frac{27615013861839}{206958543604736}a^{8}+\frac{266404093812421}{16\!\cdots\!88}a^{7}+\frac{93212895976945}{413917087209472}a^{6}+\frac{33\!\cdots\!23}{13\!\cdots\!04}a^{5}-\frac{41\!\cdots\!23}{33\!\cdots\!76}a^{4}+\frac{14\!\cdots\!21}{66\!\cdots\!52}a^{3}-\frac{89\!\cdots\!09}{33\!\cdots\!76}a^{2}+\frac{12238508130203}{172017490788352}a+\frac{384353346173}{305023645696}$, $\frac{24852128453}{827834174418944}a^{21}+\frac{34187880631}{66\!\cdots\!52}a^{20}-\frac{2429147167347}{66\!\cdots\!52}a^{19}-\frac{363131421721}{66\!\cdots\!52}a^{18}+\frac{3207785978813}{946096199335936}a^{17}+\frac{1757341850833}{33\!\cdots\!76}a^{16}-\frac{994893197181}{206958543604736}a^{15}-\frac{554072497235}{827834174418944}a^{14}-\frac{15316794992927}{827834174418944}a^{13}+\frac{11560453025603}{33\!\cdots\!76}a^{12}+\frac{218060232462283}{301030608879616}a^{11}-\frac{116486173840097}{33\!\cdots\!76}a^{10}-\frac{212984654710501}{33\!\cdots\!76}a^{9}+\frac{17\!\cdots\!59}{16\!\cdots\!88}a^{8}+\frac{29\!\cdots\!05}{827834174418944}a^{7}+\frac{71\!\cdots\!11}{413917087209472}a^{6}+\frac{79\!\cdots\!83}{118262024916992}a^{5}-\frac{22\!\cdots\!45}{66\!\cdots\!52}a^{4}+\frac{96\!\cdots\!23}{946096199335936}a^{3}-\frac{11\!\cdots\!17}{66\!\cdots\!52}a^{2}-\frac{53\!\cdots\!69}{66\!\cdots\!52}a+\frac{3346716876301}{31690768384}$, $\frac{32623917783}{26\!\cdots\!08}a^{21}+\frac{1900883381513}{26\!\cdots\!08}a^{20}+\frac{424321734385}{13\!\cdots\!04}a^{19}-\frac{13569940968665}{13\!\cdots\!04}a^{18}-\frac{16426916533817}{26\!\cdots\!08}a^{17}+\frac{278171626953321}{26\!\cdots\!08}a^{16}+\frac{27545464230609}{33\!\cdots\!76}a^{15}-\frac{123127503713633}{33\!\cdots\!76}a^{14}-\frac{595310189571257}{13\!\cdots\!04}a^{13}+\frac{962909583912345}{13\!\cdots\!04}a^{12}+\frac{13\!\cdots\!15}{66\!\cdots\!52}a^{11}+\frac{13\!\cdots\!83}{946096199335936}a^{10}+\frac{432925410932899}{13\!\cdots\!04}a^{9}-\frac{42\!\cdots\!23}{13\!\cdots\!04}a^{8}+\frac{48\!\cdots\!57}{33\!\cdots\!76}a^{7}+\frac{81\!\cdots\!99}{33\!\cdots\!76}a^{6}+\frac{12\!\cdots\!75}{26\!\cdots\!08}a^{5}+\frac{28\!\cdots\!97}{26\!\cdots\!08}a^{4}-\frac{61\!\cdots\!31}{13\!\cdots\!04}a^{3}+\frac{64\!\cdots\!67}{13\!\cdots\!04}a^{2}-\frac{38\!\cdots\!05}{26\!\cdots\!08}a+\frac{11\!\cdots\!65}{19521513324544}$, $\frac{52000861959}{66\!\cdots\!52}a^{21}+\frac{67651105897}{66\!\cdots\!52}a^{20}-\frac{368043493425}{33\!\cdots\!76}a^{19}-\frac{48041480275}{301030608879616}a^{18}+\frac{7471957134291}{66\!\cdots\!52}a^{17}+\frac{11551969132801}{66\!\cdots\!52}a^{16}-\frac{448026839039}{118262024916992}a^{15}-\frac{87425479115}{10751093174272}a^{14}+\frac{19761588626959}{33\!\cdots\!76}a^{13}+\frac{90803456973681}{33\!\cdots\!76}a^{12}+\frac{282152490815753}{16\!\cdots\!88}a^{11}+\frac{172647485205997}{16\!\cdots\!88}a^{10}-\frac{13\!\cdots\!57}{33\!\cdots\!76}a^{9}-\frac{70822792146949}{473048099667968}a^{8}+\frac{208885484476313}{75257652219904}a^{7}+\frac{819416151372135}{118262024916992}a^{6}+\frac{95\!\cdots\!83}{66\!\cdots\!52}a^{5}-\frac{13\!\cdots\!75}{66\!\cdots\!52}a^{4}-\frac{10\!\cdots\!23}{301030608879616}a^{3}-\frac{49\!\cdots\!85}{33\!\cdots\!76}a^{2}-\frac{34\!\cdots\!21}{66\!\cdots\!52}a+\frac{58611071190073}{4880378331136}$ Copy content Toggle raw display (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:  \( 1485021311930 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 1485021311930 \cdot 2}{2\cdot\sqrt{129045097915537907803232946851577207980032}}\cr\approx \mathstrut & 2.49080932282076 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763)
 
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 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763, 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 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763);
 
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 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763);
 
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

$M_{22}:C_2$ (as 22T41):

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 887040
The 21 conjugacy class representatives for $M_{22}:C_2$
Character table for $M_{22}:C_2$ is not computed

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 44 sibling: 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 ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.11.0.1}{11} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.9.2$x^{4} + 2 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.8.22.24$x^{8} - 4 x^{6} + 16 x^{5} + 68 x^{4} - 32 x^{3} + 8 x^{2} + 416 x + 724$$4$$2$$22$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
2.8.26.40$x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{4} + 8 x^{3} + 10$$8$$1$$26$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
\(11\) Copy content Toggle raw display Deg $22$$22$$1$$23$